Algebraic quantum field theory
International Nuclear Information System (INIS)
The basic assumption that the complete information relevant for a relativistic, local quantum theory is contained in the net structure of the local observables of this theory results first of all in a concise formulation of the algebraic structure of the superselection theory and an intrinsic formulation of charge composition, charge conjugation and the statistics of an algebraic quantum field theory. In a next step, the locality of massive particles together with their spectral properties are wed for the formulation of a selection criterion which opens the access to the massive, non-abelian quantum gauge theories. The role of the electric charge as a superselection rule results in the introduction of charge classes which in term lead to a set of quantum states with optimum localization properties. Finally, the asymptotic observables of quantum electrodynamics are investigated within the framework of algebraic quantum field theory. (author)
Lectures on algebraic quantum field theory and operator algebras
Energy Technology Data Exchange (ETDEWEB)
Schroer, Bert [Berlin Univ. (Germany). Institut fuer Theoretische Physik. E-mail: schroer@cbpf.br
2001-04-01
In this series of lectures directed towards a mainly mathematically oriented audience I try to motivate the use of operator algebra methods in quantum field theory. Therefore a title as why mathematicians are/should be interested in algebraic quantum field theory would be equally fitting. besides a presentation of the framework and the main results of local quantum physics these notes may serve as a guide to frontier research problems in mathematical. (author)
Quantum groups and algebraic geometry in conformal field theory
International Nuclear Information System (INIS)
The classification of two-dimensional conformal field theories is described with algebraic geometry and group theory. This classification is necessary in a consistent formulation of a string theory. (author). 130 refs.; 4 figs.; schemes
Quantum field theories on algebraic curves. I. Additive bosons
International Nuclear Information System (INIS)
Using Serre's adelic interpretation of cohomology, we develop a 'differential and integral calculus' on an algebraic curve X over an algebraically closed field k of constants of characteristic zero, define algebraic analogues of additive multi-valued functions on X and prove the corresponding generalized residue theorem. Using the representation theory of the global Heisenberg algebra and lattice Lie algebra, we formulate quantum field theories of additive and charged bosons on an algebraic curve X. These theories are naturally connected with the algebraic de Rham theorem. We prove that an extension of global symmetries (Witten's additive Ward identities) from the k-vector space of rational functions on X to the vector space of additive multi-valued functions uniquely determines these quantum theories of additive and charged bosons.
Quantum field theories on algebraic curves. I. Additive bosons
Takhtajan, Leon A.
2013-04-01
Using Serre's adelic interpretation of cohomology, we develop a `differential and integral calculus' on an algebraic curve X over an algebraically closed field k of constants of characteristic zero, define algebraic analogues of additive multi-valued functions on X and prove the corresponding generalized residue theorem. Using the representation theory of the global Heisenberg algebra and lattice Lie algebra, we formulate quantum field theories of additive and charged bosons on an algebraic curve X. These theories are naturally connected with the algebraic de Rham theorem. We prove that an extension of global symmetries (Witten's additive Ward identities) from the k-vector space of rational functions on X to the vector space of additive multi-valued functions uniquely determines these quantum theories of additive and charged bosons.
Algebraic structures and eigenstates for integrable collective field theories
International Nuclear Information System (INIS)
Conditions for the construction of polynomial eigen-operators for the Hamiltonian of collective string field theories are explored. Such eigen-operators arise for only one monomial potential v(x)=μx2 in the collective field theory. They form a w∞-algebra isomorphic to the algebra of vertex operators in 2d gravity. Polynomial potentials of orders only strictly larger or smaller than 2 have no non-zero-energy polynomial eigen-operators. This analysis leads us to consider a particular potential ν(x)=μx2+g/x2. A Lie algebra of polynomial eigen-operators is then constructed for this potential. It is a symmetric 2-index Lie algebra, also represented as a subalgebra of U(sl(2)). (orig.)
Path operator algebras in conformal quantum field theories
International Nuclear Information System (INIS)
Two different kinds of path algebras and methods from noncommutative geometry are applied to conformal field theory: Fusion rings and modular invariants of extended chiral algebras are analyzed in terms of essential paths which are a path description of intertwiners. As an example, the ADE classification of modular invariants for minimal models is reproduced. The analysis of two-step extensions is included. Path algebras based on a path space interpretation of character identities can be applied to the analysis of fusion rings as well. In particular, factorization properties of character identities and therefore of the corresponding path spaces are - by means of K-theory - related to the factorization of the fusion ring of Virasoro- and W-algebras. Examples from nonsupersymmetric as well as N=2 supersymmetric minimal models are discussed. (orig.)
Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes
Hack, Thomas-Paul
2015-01-01
This monograph provides a largely self--contained and broadly accessible exposition of two cosmological applications of algebraic quantum field theory (QFT) in curved spacetime: a fundamental analysis of the cosmological evolution according to the Standard Model of Cosmology and a fundamental study of the perturbations in Inflation. The two central sections of the book dealing with these applications are preceded by sections containing a pedagogical introduction to the subject as well as introductory material on the construction of linear QFTs on general curved spacetimes with and without gauge symmetry in the algebraic approach, physically meaningful quantum states on general curved spacetimes, and the backreaction of quantum fields in curved spacetimes via the semiclassical Einstein equation. The target reader should have a basic understanding of General Relativity and QFT on Minkowski spacetime, but does not need to have a background in QFT on curved spacetimes or the algebraic approach to QFT. In particul...
Cosmological applications of algebraic quantum field theory in curved spacetimes
Hack, Thomas-Paul
2016-01-01
This book provides a largely self-contained and broadly accessible exposition on two cosmological applications of algebraic quantum field theory (QFT) in curved spacetime: a fundamental analysis of the cosmological evolution according to the Standard Model of Cosmology; and a fundamental study of the perturbations in inflation. The two central sections of the book dealing with these applications are preceded by sections providing a pedagogical introduction to the subject. Introductory material on the construction of linear QFTs on general curved spacetimes with and without gauge symmetry in the algebraic approach, physically meaningful quantum states on general curved spacetimes, and the backreaction of quantum fields in curved spacetimes via the semiclassical Einstein equation is also given. The reader should have a basic understanding of General Relativity and QFT on Minkowski spacetime, but no background in QFT on curved spacetimes or the algebraic approach to QFT is required.
Perturbative algebraic quantum field theory at finite temperature
Energy Technology Data Exchange (ETDEWEB)
Lindner, Falk
2013-08-15
We present the algebraic approach to perturbative quantum field theory for the real scalar field in Minkowski spacetime. In this work we put a special emphasis on the inherent state-independence of the framework and provide a detailed analysis of the state space. The dynamics of the interacting system is constructed in a novel way by virtue of the time-slice axiom in causal perturbation theory. This method sheds new light in the connection between quantum statistical dynamics and perturbative quantum field theory. In particular it allows the explicit construction of the KMS and vacuum state for the interacting, massive Klein-Gordon field which implies the absence of infrared divergences of the interacting theory at finite temperature, in particular for the interacting Wightman and time-ordered functions.
Samuel, Pierre
2008-01-01
Algebraic number theory introduces students not only to new algebraic notions but also to related concepts: groups, rings, fields, ideals, quotient rings and quotient fields, homomorphisms and isomorphisms, modules, and vector spaces. Author Pierre Samuel notes that students benefit from their studies of algebraic number theory by encountering many concepts fundamental to other branches of mathematics - algebraic geometry, in particular.This book assumes a knowledge of basic algebra but supplements its teachings with brief, clear explanations of integrality, algebraic extensions of fields, Gal
Hopf Algebra Structure of a Model Quantum Field Theory
Solomon, A I; Blasiak, P; Horzela, A; Penson, K A
2006-01-01
Recent elegant work on the structure of Perturbative Quantum Field Theory (PQFT) has revealed an astonishing interplay between analysis(Riemann Zeta functions), topology (Knot theory), combinatorial graph theory (Feynman Diagrams) and algebra (Hopf structure). The difficulty inherent in the complexities of a fully-fledged field theory such as PQFT means that the essential beauty of the relationships between these areas can be somewhat obscured. Our intention is to display some, although not all, of these structures in the context of a simple zero-dimensional field theory; i.e. a quantum theory of non-commuting operators which do not depend on spacetime. The combinatorial properties of these boson creation and annihilation operators, which is our chosen example, may be described by graphs, analogous to the Feynman diagrams of PQFT, which we show possess a Hopf algebra structure. Our approach is based on the partition function for a boson gas. In a subsequent note in these Proceedings we sketch the relationship...
Short-distance analysis for algebraic euclidean field theory
Schlingemann, D
1999-01-01
Recently D. Buchholz and R. Verch have proposed a method for implementing in algebraic quantum field theory ideas from renormalization group analysis of short-distance (high energy) behavior by passing to certain scaling limit theories. Buchholz and Verch distinguish between different types of theories where the limit is unique, degenerate, or classical, and the method allows in principle to extract the `ultraparticle' content of a given model, i.e. to identify particles (like quarks and gluons) that are not visible at finite distances due to `confinement'. It is therefore of great importance for the physical interpretation of the theory. The method has been illustrated in a simple model in with some rather surprising results. This paper will focus on the question how the short distance behavior of models defined by euclidean means is reflected in the corresponding behavior of their Minkowski counterparts. More specifically, we shall prove that if a euclidean theory has some short distance limit, then it is p...
Perturbative algebraic quantum field theory an introduction for mathematicians
Rejzner, Kasia
2016-01-01
Perturbative Algebraic Quantum Field Theory (pAQFT), the subject of this book, is a complete and mathematically rigorous treatment of perturbative quantum field theory (pQFT) that doesn’t require the use of divergent quantities. We discuss in detail the examples of scalar fields and gauge theories and generalize them to QFT on curved spacetimes. pQFT models describe a wide range of physical phenomena and have remarkable agreement with experimental results. Despite this success, the theory suffers from many conceptual problems. pAQFT is a good candidate to solve many, if not all of these conceptual problems. Chapters 1-3 provide some background in mathematics and physics. Chapter 4 concerns classical theory of the scalar field, which is subsequently quantized in chapters 5 and 6. Chapter 7 covers gauge theory and chapter 8 discusses QFT on curved spacetimes and effective quantum gravity. The book aims to be accessible researchers and graduate students interested in the mathematical foundations of pQFT are th...
Current algebra and conformal field theory on a figure eight
Balachandran, A P; Sen-Gupta, K; Marmo, G; Salomonson, P; Simoni, A; Stern, A
1993-01-01
We examine the dynamics of a free massless scalar field on a figure eight network. Upon requiring the scalar field to have a well defined value at the junction of the network, it is seen that the conserved currents of the theory satisfy Kirchhoff's law, that is that the current flowing into the junction equals the current flowing out. We obtain the corresponding current algebra and show that, unlike on a circle, the left- and right-moving currents on the figure eight do not in general commute in quantum theory. Since a free scalar field theory on a one dimensional spatial manifold exhibits conformal symmetry, it is natural to ask whether an analogous symmetry can be defined for the figure eight. We find that, unlike in the case of a manifold, the action plus boundary conditions for the network are not invariant under separate conformal transformations associated with left- and right-movers. Instead, the system is, at best, invariant under only a single set of transformations. Its conserved current is also fou...
An Algebraic Construction of Boundary Quantum Field Theory
Longo, Roberto; Witten, Edward
2011-04-01
We build up local, time translation covariant Boundary Quantum Field Theory nets of von Neumann algebras {mathcal A_V} on the Minkowski half-plane M + starting with a local conformal net {mathcal A} of von Neumann algebras on {mathbb R} and an element V of a unitary semigroup {mathcal E(mathcal A)} associated with {mathcal A}. The case V = 1 reduces to the net {mathcal A_+} considered by Rehren and one of the authors; if the vacuum character of {mathcal A} is summable, {mathcal A_V} is locally isomorphic to {mathcal A_+}. We discuss the structure of the semigroup {mathcal E(mathcal A)}. By using a one-particle version of Borchers theorem and standard subspace analysis, we provide an abstract analog of the Beurling-Lax theorem that allows us to describe, in particular, all unitaries on the one-particle Hilbert space whose second quantization promotion belongs to {mathcal E(mathcal A^{(0)})} with {mathcal A^{(0)}} the U(1)-current net. Each such unitary is attached to a scattering function or, more generally, to a symmetric inner function. We then obtain families of models via any Buchholz-Mack-Todorov extension of {mathcal A^{(0)}}. A further family of models comes from the Ising model.
Algebraic extensions of fields
McCarthy, Paul J
1991-01-01
""...clear, unsophisticated and direct..."" - MathThis textbook is intended to prepare graduate students for the further study of fields, especially algebraic number theory and class field theory. It presumes some familiarity with topology and a solid background in abstract algebra. Chapter 1 contains the basic results concerning algebraic extensions. In addition to separable and inseparable extensions and normal extensions, there are sections on finite fields, algebraically closed fields, primitive elements, and norms and traces. Chapter 2 is devoted to Galois theory. Besides the fundamenta
Quantum field theory on toroidal topology: Algebraic structure and applications
Energy Technology Data Exchange (ETDEWEB)
Khanna, F.C., E-mail: khannaf@uvic.ca [Department of Physics and Astronomy, University of Victoria, Victoria, BC V8P 5C2 (Canada); TRIUMF, Vancouver, BC, V6T 2A3 (Canada); Malbouisson, A.P.C., E-mail: adolfo@cbpf.br [Centro Brasileiro de Pesquisas Físicas/MCT, 22290-180, Rio de Janeiro, RJ (Brazil); Malbouisson, J.M.C., E-mail: jmalboui@ufba.br [Instituto de Física, Universidade Federal da Bahia, 40210-340, Salvador, BA (Brazil); Santana, A.E., E-mail: asantana@unb.br [International Center for Condensed Matter Physics, Instituto de Física, Universidade de Brasília, 70910-900, Brasília, DF (Brazil)
2014-06-01
The development of quantum theory on a torus has a long history, and can be traced back to the 1920s, with the attempts by Nordström, Kaluza and Klein to define a fourth spatial dimension with a finite size, being curved in the form of a torus, such that Einstein and Maxwell equations would be unified. Many developments were carried out considering cosmological problems in association with particle physics, leading to methods that are useful for areas of physics, in which size effects play an important role. This interest in finite size effect systems has been increasing rapidly over the last decades, due principally to experimental improvements. In this review, the foundations of compactified quantum field theory on a torus are presented in a unified way, in order to consider applications in particle and condensed matter physics. The theory on a torus Γ{sub D}{sup d}=(S{sup 1}){sup d}×R{sup D−d} is developed from a Lie-group representation and c{sup ∗}-algebra formalisms. As a first application, the quantum field theory at finite temperature, in its real- and imaginary-time versions, is addressed by focusing on its topological structure, the torus Γ{sub 4}{sup 1}. The toroidal quantum-field theory provides the basis for a consistent approach of spontaneous symmetry breaking driven by both temperature and spatial boundaries. Then the superconductivity in films, wires and grains are analyzed, leading to some results that are comparable with experiments. The Casimir effect is studied taking the electromagnetic and Dirac fields on a torus. In this case, the method of analysis is based on a generalized Bogoliubov transformation, that separates the Green function into two parts: one is associated with the empty space–time, while the other describes the impact of compactification. This provides a natural procedure for calculating the renormalized energy–momentum tensor. Self interacting four-fermion systems, described by the Gross–Neveu and Nambu
Open and Closed String field theory interpreted in classical Algebraic Topology
Sullivan, Dennis
2003-01-01
There is an interpretation of open string field theory in algebraic topology. An interpretation of closed string field theory can be deduced from this open string theory to obtain as well the interpretation of open and closed string field theory combined.
Jarvis, Frazer
2014-01-01
The technical difficulties of algebraic number theory often make this subject appear difficult to beginners. This undergraduate textbook provides a welcome solution to these problems as it provides an approachable and thorough introduction to the topic. Algebraic Number Theory takes the reader from unique factorisation in the integers through to the modern-day number field sieve. The first few chapters consider the importance of arithmetic in fields larger than the rational numbers. Whilst some results generalise well, the unique factorisation of the integers in these more general number fields often fail. Algebraic number theory aims to overcome this problem. Most examples are taken from quadratic fields, for which calculations are easy to perform. The middle section considers more general theory and results for number fields, and the book concludes with some topics which are more likely to be suitable for advanced students, namely, the analytic class number formula and the number field sieve. This is the fi...
International Nuclear Information System (INIS)
Several two dimensional quantum field theory models have more than one vacuum state. An investigation of super selection sectors in two dimensions from an axiomatic point of view suggests that there should be also states, called soliton or kink states, which interpolate different vacua. Familiar quantum field theory models, for which the existence of kink states have been proven, are the Sine-Gordon and the φ42-model. In order to establish the existence of kink states for a larger class of models, we investigate the following question: Which are sufficient conditions a pair of vacuum states has to fulfill, such that an interpolating kink state can be constructed? We discuss the problem in the framework of algebraic quantum field theory which includes, for example, the P(φ)2-models. We identify a large class of vacuum states, including the vacua of the P(φ)2-models, the Yukawa2-like models and special types of Wess-Zumino models, for which there is a natural way to construct an interpolating kink state. In two space-time dimensions, massive particle states are kink states. We apply the Haag-Ruelle collision theory to kink sectors in order to analyze the asymptotic scattering states. We show that for special configurations of n kinks the scattering states describe n freely moving non interacting particles. (orig.)
The theory of algebraic numbers
Pollard, Harry
1998-01-01
An excellent introduction to the basics of algebraic number theory, this concise, well-written volume examines Gaussian primes; polynomials over a field; algebraic number fields; and algebraic integers and integral bases. After establishing a firm introductory foundation, the text explores the uses of arithmetic in algebraic number fields; the fundamental theorem of ideal theory and its consequences; ideal classes and class numbers; and the Fermat conjecture. 1975 edition. References. List of Symbols. Index.
K-theory for ring C*-algebras - the case of number fields with higher roots of unity
Li, Xin
2012-01-01
We compute K-theory for ring C*-algebras in the case of higher roots of unity and thereby completely determine the K-theory for ring C*-algebras attached to rings of integers in arbitrary number fields.
Field algebra, Hilbert space and observables in two-dimensional higher-derivative field theories
International Nuclear Information System (INIS)
We discuss structural aspects related to the field algebra of two-dimensional higher-derivative quantum field theories. We present general selection criteria for the proper field subalgebra that generates Wightman functions satisfying the asymptotic factorization property and which define a semi-definite inner product Hilbert space. The positive definite inner product Hilbert space, which contains as a subspace of states the general Wightman functions of the corresponding standard canonical models, is a quotient space obtained by equivalence classes. For higher-derivative local gauge theories, besides the Lowenstein-Swieca condition, an additional condition must be imposed on the field algebra in order to obtain a physical subspace of states satisfying a reasonable set of physically meaningful axioms. (author)
Solutions in Bosonic String Field Theory and Higher Spin Algebras in AdS
Polyakov, Dimitri
2015-01-01
We find a class of analytic solutions in open bosonic string field theory, parametrized by the chiral copy of higher spin algebra in $AdS_3$. The solutions are expressed in terms of the generating function for the products of Bell polynomials in derivatives of bosonic space-time coordinates $X^m(z)$ of the open string, which form is determined in this work. The products of these polynomials form a natural operator algebra realizations of $W_\\infty$ (area-preserving diffeomorphisms), enveloping algebra of SU(2) and higher spin algebra in $AdS_3$. The class of SFT solutions found can, in turn, be interpreted as the "enveloping of enveloping", or the enveloping of $AdS_3$ higher spin algebra. We also discuss the extensions of this class of solutions to superstring theory and their relations to higher spin algebras in higher space-time dimensions.
Brouder, Christian
2002-01-01
The Laplace Hopf algebra created by Rota and coll. is generalized to provide an algebraic tool for combinatorial problems of quantum field theory. This framework encompasses commutation relations, normal products, time-ordered products and renormalisation. It considers the operator product and the time-ordered product as deformations of the normal product. In particular, it gives an algebraic meaning to Wick's theorem and it extends the concept of Laplace pairing to prove that the renormalise...
An algebraic approach towards the classification of 2 dimensional conformal field theories
International Nuclear Information System (INIS)
This thesis treats an algebraic method for the construction of 2-dimensional conformal field theories. The method consists of the study of the representation theory of the Virasoro algebra and suitable extensions of this. The classification of 2-dimensional conformal field theories is translated into the classification of combinations of representations which satisfy certain consistence conditions (unitarity and modular invariance). For a certain class of 2-dimensional field theories, namely the one with central charge c = 1 from the theory of Kac-Moody algebra's. there exist indications, but as yet mainly hope, that this construction will finally lead to a classification of 2-dimensional conformal field theories. 182 refs.; 2 figs.; 26 tabs
Application of Tomita-Takesaki theory in algebraic euclidean field theories
Schlingemann, D
1999-01-01
The construction of the known interacting quantum field theory models is mostly based on euclidean techniques. The expectation values of interesting quantities are usually given in terms of euclidean correlation functions from which one should be able to extract information about the behavior of the correlation functions of the Minkowskian counterpart. We think that the C*-algebraic approach to euclidean field theory gives an appropriate setup in order to study structural aspects model independently. A previous paper deals with a construction scheme which relates to each euclidean field theory a Poincaré covariant quantum field theory model in the sense of R. Haag and D. Kastler. Within the framework of R. Haag and D. Kastler, the physical concept of PCT symmetry and spin and statistics is related to the Tomita-Takesaki theory of von Neumann algebras and this important aspects has been studied by several authors. We express the PCT symmetry in terms of euclidean reflexions and we explicitly identify the corr...
Weiss, Edwin
1998-01-01
Careful organization and clear, detailed proofs characterize this methodical, self-contained exposition of basic results of classical algebraic number theory from a relatively modem point of view. This volume presents most of the number-theoretic prerequisites for a study of either class field theory (as formulated by Artin and Tate) or the contemporary treatment of analytical questions (as found, for example, in Tate's thesis).Although concerned exclusively with algebraic number fields, this treatment features axiomatic formulations with a considerable range of applications. Modem abstract te
Super Virasoro algebra and solvable supersymmetric quantum field theories
International Nuclear Information System (INIS)
Interesting and deep relationships between super Virasoro algebras and super soliton systems (super KdV, super mKdV and super sine-Gordon equations) are investigated at both classical and quantum levels. An infinite set of conserved quantities responsible for solvability is characterized by super Virasoro algebras only. Several members of the infinite set of conserved quantities are derived explicitly. (author)
Hopf algebras and the combinatorics of connected graphs in quantum field theory
Mestre, Angela; Oeckl, Robert
2008-01-01
In this talk, we are concerned with the formulation and understanding of the combinatorics of time-ordered n-point functions in terms of the Hopf algebra of field operators. Mathematically, this problem can be formulated as one in combinatorics or graph theory. It consists in finding a recursive algorithm that generates all connected graphs in their Hopf algebraic representation. This representation can be used directly and efficiently in evaluating Feynman graphs as contributions to the n-po...
Quantum exchange algebra and exact operator solution of A sub 2 -Toda field theory
Takimoto, Y; Kurokawa, H; Fujiwara, T
1999-01-01
Locality is analyzed for Toda field theories by noting novel chiral description in the conventional non-chiral formalism. It is shown that the canonicity of the interacting to free field mapping described by the classical solution is automatically guaranteed by the locality. Quantum Toda theories are investigated by applying the method of free field quantization. We give Toda exponential operators associated with fundamental weight vectors as bilinear forms of chiral fields satisfying characteristic quantum exchange algebra. It is shown that the locality leads to non-trivial relations among the R-matrix and the expansion coefficients of the exponential operators. The Toda exponentials are obtained for a A sub 2 -system by extending the algebraic method developed for the Liouville theory. The canonical commutation relations and the operatorial field equations are also examined.
Quantum Exchange Algebra and Exact Operator Solution of $A_{2}$-Toda Field Theory
Takimoto, Y; Kurokawa, H; Fujiwara, T
1999-01-01
Locality is analyzed for Toda field theories by noting novel chiral description in the conventional nonchiral formalism. It is shown that the canonicity of the interacting to free field mapping described by the classical solution is automatically guaranteed by the locality. Quantum Toda theories are investigated by applying the method of free field quantization. We give Toda exponential operators associated with fundamental weight vectors as bilinear forms of chiral fields satisfying characteristic quantum exchange algebra. It is shown that the locality leads to nontrivial relations among the ${\\cal R}$-matrix and the expansion coefficients of the exponential operators. The Toda exponentials are obtained for $A_2$-system by extending the algebraic method developed for Liouville theory. The canonical commutation relations and the operatorial field equations are also examined.
Quantum double actions on operator algebras and orbifold quantum field theories
International Nuclear Information System (INIS)
Starting from a local quantum field theory with an unbroken compact symmetry group G in 1+1 dimensional spacetime we construct disorder fields implementing gauge transformations on the fields (order variables) localized in a wedge region. Enlarging the local algebras by these disorder fields we obtain a nonlocal field theory, the fixpoint algebras of which under the appropriately extended action of the group G are shown to satisfy Haag duality in every simple sector. The specifically 1+1 dimensional phenomenon of violation of Haag duality of fixpoint nets is thereby clarified. In the case of a finite group G the extended theory is acted upon in a completely canonical way by the quantum double D(G) and satisfies R-matrix commutation relations as well as a Verlinde algebra. Furthermore, our methods are suitable for a concise and transparent approach to bosonization. The main technical ingredient is a strengthened version of the split property which should hold in all reasonable massive theories. In the appendices (part of) the results are extended to arbitary locally compact groups and our methods are adapted to chiral theories on the circle. (orig.)
Classical theory of algebraic numbers
Ribenboim, Paulo
2001-01-01
Gauss created the theory of binary quadratic forms in "Disquisitiones Arithmeticae" and Kummer invented ideals and the theory of cyclotomic fields in his attempt to prove Fermat's Last Theorem These were the starting points for the theory of algebraic numbers, developed in the classical papers of Dedekind, Dirichlet, Eisenstein, Hermite and many others This theory, enriched with more recent contributions, is of basic importance in the study of diophantine equations and arithmetic algebraic geometry, including methods in cryptography This book has a clear and thorough exposition of the classical theory of algebraic numbers, and contains a large number of exercises as well as worked out numerical examples The Introduction is a recapitulation of results about principal ideal domains, unique factorization domains and commutative fields Part One is devoted to residue classes and quadratic residues In Part Two one finds the study of algebraic integers, ideals, units, class numbers, the theory of decomposition, iner...
Rigidification of algebras over essentially algebraic theories
Rosicky, J
2012-01-01
Badzioch and Bergner proved a rigidification theorem saying that each homotopy simplicial algebra is weakly equivalent to a simplicial algebra. The question is whether this result can be extended from algebraic theories to finite limit theories and from simplicial sets to more general monoidal model categories. We will present some answers to this question.
The Casimir Effect from the Point of View of Algebraic Quantum Field Theory
Dappiaggi, Claudio; Nosari, Gabriele; Pinamonti, Nicola
2016-06-01
We consider a region of Minkowski spacetime bounded either by one or by two parallel, infinitely extended plates orthogonal to a spatial direction and a real Klein-Gordon field satisfying Dirichlet boundary conditions. We quantize these two systems within the algebraic approach to quantum field theory using the so-called functional formalism. As a first step we construct a suitable unital ∗-algebra of observables whose generating functionals are characterized by a labelling space which is at the same time optimal and separating and fulfils the F-locality property. Subsequently we give a definition for these systems of Hadamard states and we investigate explicit examples. In the case of a single plate, it turns out that one can build algebraic states via a pull-back of those on the whole Minkowski spacetime, moreover inheriting from them the Hadamard property. When we consider instead two plates, algebraic states can be put in correspondence with those on flat spacetime via the so-called method of images, which we translate to the algebraic setting. For a massless scalar field we show that this procedure works perfectly for a large class of quasi-free states including the Poincaré vacuum and KMS states. Eventually Wick polynomials are introduced. Contrary to the Minkowski case, the extended algebras, built in globally hyperbolic subregions can be collected in a global counterpart only after a suitable deformation which is expressed locally in terms of a *-isomorphism. As a last step, we construct explicitly the two-point function and the regularized energy density, showing, moreover, that the outcome is consistent with the standard results of the Casimir effect.
Wild Pfister forms over Henselian fields, K-theory, and conic division algebras
Garibaldi, Skip
2010-01-01
The epicenter of this paper concerns Pfister quadratic forms over a field $F$ with a Henselian discrete valuation. All characteristics are considered but we focus on the most complicated case where the residue field has characteristic 2 but $F$ does not. We also prove results about round quadratic forms, composition algebras, generalizations of composition algebras we call conic algebras, and central simple associative symbol algebras. Finally we give relationships between these objects and Kato's filtration on the Milnor $K$-groups of $F$.
Algebraic approach to quantum field theory on a class of noncommutative curved spacetimes
International Nuclear Information System (INIS)
In this talk (based on arXiv:0912.2252[hep-th]) we explain how to construct the quantum field theory of a free real scalar field on a class of noncommutative manifolds, obtained via deformation quantization using triangular Drinfel'd twists. We define action functionals in the framework of twist-deformed differential geometry, derive the associated equations of motion and solve them in terms of formal power series. In analogy to the commutative case, we can construct the Weyl algebra of field observables, which depends in general on the deformation of spacetime. We give an outlook to applications of our approach to noncommutative cosmology and black hole physics.
Computer algebra in quantum field theory integration, summation and special functions
Schneider, Carsten
2013-01-01
The book focuses on advanced computer algebra methods and special functions that have striking applications in the context of quantum field theory. It presents the state of the art and new methods for (infinite) multiple sums, multiple integrals, in particular Feynman integrals, difference and differential equations in the format of survey articles. The presented techniques emerge from interdisciplinary fields: mathematics, computer science and theoretical physics; the articles are written by mathematicians and physicists with the goal that both groups can learn from the other field, including
Wassermann, Antony
1998-01-01
Fusion of positive energy representations is defined using Connes' tensor product for bimodules over a von Neumann algebra. Fusion is computed using the analytic theory of primary fields and explicit solutions of the Knizhnik-Zamolodchikov equation.
String field representation of the Virasoro algebra
Mertes, Nicholas
2016-01-01
We construct a representation of the zero central charge Virasoro algebra using string fields in Witten's open bosonic string field theory. This construction is used to explore extensions of the KBc algebra and find novel algebraic solutions of open string field theory.
Distribution theory of algebraic numbers
Yang, Chung-Chun
2008-01-01
The book timely surveys new research results and related developments in Diophantine approximation, a division of number theory which deals with the approximation of real numbers by rational numbers. The book is appended with a list of challenging open problems and a comprehensive list of references. From the contents: Field extensions Algebraic numbers Algebraic geometry Height functions The abc-conjecture Roth''s theorem Subspace theorems Vojta''s conjectures L-functions.
On the stability of KMS states in perturbative algebraic quantum field theories
Drago, Nicolo; Pinamonti, Nicola
2016-01-01
We analyze the stability properties shown by KMS states for interacting massive scalar fields propagating over Minkowski spacetime, recently constructed in the framework of perturbative algebraic quantum field theories by Fredenhagen and Lindner \\cite{FredenhagenLindner}. In particular, we prove the validity of the return to equilibrium property when the interaction Lagrangean has compact spatial support. Surprisingly, this does not hold anymore, if the adiabatic limit is considered, namely when the interaction Lagrangean is invariant under spatial translations. Consequently, an equilibrium state under the adiabatic limit for a perturbative interacting theory evolved with the free dynamics does not converge anymore to the free equilibrium state. Actually, we show that its ergodic mean converges to a non equilibrium steady state for the free theory.
Algebra and Number Theory An Integrated Approach
Dixon, Martyn; Subbotin, Igor
2011-01-01
Explore the main algebraic structures and number systems that play a central role across the field of mathematics Algebra and number theory are two powerful branches of modern mathematics at the forefront of current mathematical research, and each plays an increasingly significant role in different branches of mathematics, from geometry and topology to computing and communications. Based on the authors' extensive experience within the field, Algebra and Number Theory has an innovative approach that integrates three disciplines-linear algebra, abstract algebra, and number theory-into one compr
Uniform Algebras Over Complete Valued Fields
Mason, Jonathan W
2012-01-01
UNIFORM algebras have been extensively investigated because of their importance in the theory of uniform approximation and as examples of complex Banach algebras. An interesting question is whether analogous algebras exist when a complete valued field other than the complex numbers is used as the underlying field of the algebra. In the Archimedean setting, this generalisation is given by the theory of real function algebras introduced by S. H. Kulkarni and B. V. Limaye in the 1980s. This thesis establishes a broader theory accommodating any complete valued field as the underlying field by involving Galois automorphisms and using non-Archimedean analysis. The approach taken keeps close to the original definitions from the Archimedean setting. Basic function algebras are defined and generalise real function algebras to all complete valued fields. Several examples are provided. Each basic function algebra is shown to have a lattice of basic extensions related to the field structure. In the non-Archimedean settin...
Directory of Open Access Journals (Sweden)
Ion C. Baianu
2009-04-01
Full Text Available A novel algebraic topology approach to supersymmetry (SUSY and symmetry breaking in quantum field and quantum gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic Jahn-Teller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non-Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier-Stieltjes transforms, and duality relations link, respectively, the quantum groups and quantum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasi-triangular, quasi-Hopf algebras, bialgebroids, Grassmann-Hopf algebras and higher dimensional algebra. On the one hand, this quantum algebraic approach is known to provide solutions to the quantum Yang-Baxter equation. On the other hand, our novel approach to extended quantum symmetries and their associated representations is shown to be relevant to locally covariant general relativity theories that are consistent with either nonlocal quantum field theories or local bosonic (spin models with the extended quantum symmetry of entangled, 'string-net condensed' (ground states.
Algebraic approach to quantum field theory on a class of noncommutative curved spacetimes
Energy Technology Data Exchange (ETDEWEB)
Ohl, Thorsten; Schenkel, Alexander [Lehrstuhl fuer Theoretische Physik II, Universitaet Wuerzburg (Germany)
2010-07-01
In this talk (based on arXiv:0912.2252[hep-th]) we explain how to construct the quantum field theory of a free real scalar field on a class of noncommutative manifolds, obtained via deformation quantization using triangular Drinfel'd twists. We define action functionals in the framework of twist-deformed differential geometry, derive the associated equations of motion and solve them in terms of formal power series. In analogy to the commutative case, we can construct the Weyl algebra of field observables, which depends in general on the deformation of spacetime. We give an outlook to applications of our approach to noncommutative cosmology and black hole physics.
Unified (p,q;α,γ,l)-deformation of oscillator algebra and two-dimensional conformal field theory
Energy Technology Data Exchange (ETDEWEB)
Burban, I.M., E-mail: burban@bitp.kiev.ua
2013-11-29
The unified (p,q;α,γ,l)-deformation of a number of well-known deformed oscillator algebras is introduced. The deformation is constructed by imputing new free parameters into the structure functions and by generalizing the defining relations of these algebras. The generalized Jordan–Schwinger and Holstein–Primakoff realizations of the U{sub pq}{sup αγl}(su(2)) algebra by the generalized (p,q;α,γ,l)-deformed operators are found. The generalized (p,q;α,γ,l)-deformation of the two-dimensional conformal field theory is established. By introducing the (p,q;α,γ,l)-operator product expansion (OPE) between the energy–momentum tensor and primary fields, we obtain the (p,q;α,γ,l)-deformed centerless Virasoro algebra. The two-point correlation function of the primary generalized (p,q;α,γ,l)-deformed fields is calculated.
Notes on the Theory of Algebraic Numbers
Wright, Steve
2015-01-01
A series of lecture notes on the elementary theory of algebraic numbers, using only knowledge of a first-semester graduate course in algebra (primarily groups and rings). No prerequisite knowledge of fields is required. Based primarily on the texts of E. Hecke, Lectures on the Theory of Algebraic Numbers, Springer-Verlag, 1981 (English translation by G. Brauer and J. Goldman) and D. Marcus, Number Fields, Springer, 1977.
Observable Algebra in Field Algebra of G-spin Models
Institute of Scientific and Technical Information of China (English)
蒋立宁
2003-01-01
Field algebra of G-spin models can provide the simplest examples of lattice field theory exhibiting quantum symmetry. Let D(G) be the double algebra of a finite group G and D(H), a sub-algebra of D(G) determined by subgroup H of G. This paper gives concrete generators and the structure of the observable algebra AH, which is a D(H)-invariant sub-algebra in the field algebra of G-spin models F, and shows that AH is a C*-algebra. The correspondence between H and AH is strictly monotonic. Finally, a duality between D(H) and AH is given via an irreducible vacuum C*-representation of F.
Transformation groups and algebraic K-theory
Lück, Wolfgang
1989-01-01
The book focuses on the relation between transformation groups and algebraic K-theory. The general pattern is to assign to a geometric problem an invariant in an algebraic K-group which determines the problem. The algebraic K-theory of modules over a category is studied extensively and appplied to the fundamental category of G-space. Basic details of the theory of transformation groups sometimes hard to find in the literature, are collected here (Chapter I) for the benefit of graduate students. Chapters II and III contain advanced new material of interest to researchers working in transformation groups, algebraic K-theory or related fields.
Gaiotto, Davide; Witten, Edward
2015-01-01
We introduce a "web-based formalism" for describing the category of half-supersymmetric boundary conditions in $1+1$ dimensional massive field theories with ${\\cal N}=(2,2)$ supersymmetry and unbroken $U(1)_R$ symmetry. We show that the category can be completely constructed from data available in the far infrared, namely, the vacua, the central charges of soliton sectors, and the spaces of soliton states on $\\mathbb{R}$, together with certain "interaction and boundary emission amplitudes". These amplitudes are shown to satisfy a system of algebraic constraints related to the theory of $A_\\infty$ and $L_\\infty$ algebras. The web-based formalism also gives a method of finding the BPS states for the theory on a half-line and on an interval. We investigate half-supersymmetric interfaces between theories and show that they have, in a certain sense, an associative "operator product." We derive a categorification of wall-crossing formulae. The example of Landau-Ginzburg theories is described in depth drawing on ide...
Indian Academy of Sciences (India)
Cătălin Ciupală
2005-02-01
In this paper we introduce non-commutative fields and forms on a new kind of non-commutative algebras: -algebras. We also define the Frölicher–Nijenhuis bracket in the non-commutative geometry on -algebras.
Topics in algebraic and topological K-theory
Baum, Paul Frank; Meyer, Ralf; Sánchez-García, Rubén; Schlichting, Marco; Toën, Bertrand
2011-01-01
This volume is an introductory textbook to K-theory, both algebraic and topological, and to various current research topics within the field, including Kasparov's bivariant K-theory, the Baum-Connes conjecture, the comparison between algebraic and topological K-theory of topological algebras, the K-theory of schemes, and the theory of dg-categories.
On the Formulation of Yang-Mills Theory with the Gauge Field Valued on the Octonionic Algebra
Restuccia, A
2014-01-01
We consider a formulation of Yang-Mills theory where the gauge field is valued on a non-associative algebra and the gauge transformation is the group of automorphisms of it. We show, under mild assumptions, that the only possible gauge formulation for the octonionic non-associative algebra are the usual $\\mathfrak{su}(2)$ or $\\mathfrak{u}(1)$ Yang-Mills theories. We also discuss the particular cases where the gauge transformations are the subalgebras $\\mathfrak{su}(3)$, $\\mathfrak{su}(2)$, or $\\mathfrak{u}(1)$ of the algebra $\\mathfrak{g}_2$, related to the corresponding subgroups of $G_2$, the group of automorphisms of the octonions.
Combinatorial Hopf algebra for the Ben Geloun-Rivasseau tensor field theory
Raasakka, Matti
2013-01-01
The Ben Geloun-Rivasseau quantum field theoretical model is the first tensor model shown to be perturbatively renormalizable. We define here an appropriate Hopf algebra describing the combinatorics of this new tensorial renormalization. The structure we propose is significantly different from the previously defined Connes-Kreimer combinatorial Hopf algebras due to the involved combinatorial and topological properties of the tensorial Feynman graphs. In particular, the 2- and 4-point function insertions must be defined to be non-trivial only if the superficial divergence degree of the associated Feynman integral is conserved.
Quantum fields, periods and algebraic geometry
Kreimer, Dirk
2014-01-01
We discuss how basic notions of graph theory and associated graph polynomials define questions for algebraic geometry, with an emphasis given to an analysis of the structure of Feynman rules as determined by those graph polynomials as well as algebraic structures of graphs. In particular, we discuss the appearance of renormalization scheme independent periods in quantum field theory.
Higher theories of algebraic structures
Matsuoka, Takuo
2016-01-01
The notion of (symmetric) coloured operad or "multicategory" can be obtained from the notion of commutative algebra through a certain general process which we call "theorization" (where our term comes from an analogy with William Lawvere's notion of algebraic theory). By exploiting the inductivity in the structure of higher associativity, we obtain the notion of "$n$-theory" for every integer $n\\ge 0$, which inductively "theorizes" $n$ times, the notion of commutative algebra. As a result, (c...
Field Theory on Noncommutative Space-Time and the Deformed Virasoro Algebra
Chaichian, M.; Demichev, A.; Presnajder, P.
2000-01-01
We consider a field theoretical model on the noncommutative cylinder which leads to a discrete-time evolution. Its Euclidean version is shown to be equivalent to a model on the complex $q$-plane. We reveal a direct link between the model on a noncommutative cylinder and the deformed Virasoro algebra constructed earlier on an abstract mathematical background. As it was shown, the deformed Virasoro generators necessarily carry a second index (in addition to the usual one), whose meaning, howeve...
Introduction to algebraic independence theory
Philippon, Patrice
2001-01-01
In the last five years there has been very significant progress in the development of transcendence theory. A new approach to the arithmetic properties of values of modular forms and theta-functions was found. The solution of the Mahler-Manin problem on values of modular function j(tau) and algebraic independence of numbers pi and e^(pi) are most impressive results of this breakthrough. The book presents these and other results on algebraic independence of numbers and further, a detailed exposition of methods created in last the 25 years, during which commutative algebra and algebraic geometry exerted strong catalytic influence on the development of the subject.
Algebraic K-theory of generalized schemes
DEFF Research Database (Denmark)
Anevski, Stella Victoria Desiree
Nikolai Durov has developed a generalization of conventional scheme theory in which commutative algebraic monads replace commutative unital rings as the basic algebraic objects. The resulting geometry is expressive enough to encompass conventional scheme theory, tropical algebraic geometry...... and geometry over the field with one element. It also permits the construction of important Arakelov theoretical objects, such as the completion \\Spec Z of Spec Z. In this thesis, we prove a projective bundle theorem for the eld with one element and compute the Chow rings of the generalized schemes Sp\\ec ZN......, appearing in the construction of \\Spec Z....
Foliation theory in algebraic geometry
McKernan, James; Pereira, Jorge
2016-01-01
Featuring a blend of original research papers and comprehensive surveys from an international team of leading researchers in the thriving fields of foliation theory, holomorphic foliations, and birational geometry, this book presents the proceedings of the conference "Foliation Theory in Algebraic Geometry," hosted by the Simons Foundation in New York City in September 2013. Topics covered include: Fano and del Pezzo foliations; the cone theorem and rank one foliations; the structure of symmetric differentials on a smooth complex surface and a local structure theorem for closed symmetric differentials of rank two; an overview of lifting symmetric differentials from varieties with canonical singularities and the applications to the classification of AT bundles on singular varieties; an overview of the powerful theory of the variety of minimal rational tangents introduced by Hwang and Mok; recent examples of varieties which are hyperbolic and yet the Green-Griffiths locus is the whole of X; and a classificati...
Elements of a theory of algebraic theories
Hyland, Martin
2013-01-01
Kleisli bicategories are a natural environment in which the combinatorics involved in various notions of algebraic theory can be handled in a uniform way. The setting allows a clear account of comparisons between such notions. Algebraic theories, symmetric operads and nonsymmetric operads are treated as examples.
Baianu, Ion C; Brown, Ronald; 10.3842/SIGMA.2009.051
2009-01-01
A novel algebraic topology approach to supersymmetry (SUSY) and symmetry breaking in quantum field and quantum gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic Jahn-Teller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non-Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier-Stieltjes transforms, and duality relations link, respectively, t...
C*-algebras and operator theory
Murphy, Gerald J
1990-01-01
This book constitutes a first- or second-year graduate course in operator theory. It is a field that has great importance for other areas of mathematics and physics, such as algebraic topology, differential geometry, and quantum mechanics. It assumes a basic knowledge in functional analysis but no prior acquaintance with operator theory is required.
Duality theories for Boolean algebras with operators
Givant, Steven
2014-01-01
In this new text, Steven Givant—the author of several acclaimed books, including works co-authored with Paul Halmos and Alfred Tarski—develops three theories of duality for Boolean algebras with operators. Givant addresses the two most recognized dualities (one algebraic and the other topological) and introduces a third duality, best understood as a hybrid of the first two. This text will be of interest to graduate students and researchers in the fields of mathematics, computer science, logic, and philosophy who are interested in exploring special or general classes of Boolean algebras with operators. Readers should be familiar with the basic arithmetic and theory of Boolean algebras, as well as the fundamentals of point-set topology.
Higher algebraic K-theory an overview
Lluis-Puebla, Emilio; Gillet, Henri; Soulé, Christophe; Snaith, Victor
1992-01-01
This book is a general introduction to Higher Algebraic K-groups of rings and algebraic varieties, which were first defined by Quillen at the beginning of the 70's. These K-groups happen to be useful in many different fields, including topology, algebraic geometry, algebra and number theory. The goal of this volume is to provide graduate students, teachers and researchers with basic definitions, concepts and results, and to give a sampling of current directions of research. Written by five specialists of different parts of the subject, each set of lectures reflects the particular perspective ofits author. As such, this volume can serve as a primer (if not as a technical basic textbook) for mathematicians from many different fields of interest.
Extende conformal field theories
Energy Technology Data Exchange (ETDEWEB)
Taormina, A. (Chicago Univ., IL (USA). Enrico Fermi Inst.)
1990-08-01
Some extended conformal field theories are briefly reviewed. They illustrate how non minimal models of the Virasoro algebra (c{ge}1) can become minimal with respect to a larger algebra. The accent is put on N-extended superconformal algebras, which are relevant in superstring compactification. (orig.).
Extended conformal field theories
Taormina, Anne
1990-08-01
Some extended conformal field theories are briefly reviewed. They illustrate how non minimal models of the Virasoro algebra (c≥1) can become minimal with respect to a larger algebra. The accent is put on N-extended superconformal algebras, which are relevant in superstring compactification.
A1-algebraic topology over a field
Morel, Fabien
2012-01-01
This text deals with A1-homotopy theory over a base field, i.e., with the natural homotopy theory associated to the category of smooth varieties over a field in which the affine line is imposed to be contractible. It is a natural sequel to the foundational paper on A1-homotopy theory written together with V. Voevodsky. Inspired by classical results in algebraic topology, we present new techniques, new results and applications related to the properties and computations of A1-homotopy sheaves, A1-homotogy sheaves, and sheaves with generalized transfers, as well as to algebraic vector bundles over affine smooth varieties.
Krichever-Novikov type algebras theory and applications
Schlichenmaier, Martin
2014-01-01
Krichever and Novikov introduced certain classes of infinite dimensionalLie algebrasto extend the Virasoro algebra and its related algebras to Riemann surfaces of higher genus. The author of this book generalized and extended them toa more general setting needed by the applications. Examples of applications are Conformal Field Theory, Wess-Zumino-Novikov-Witten models, moduli space problems, integrable systems, Lax operator algebras, and deformation theory of Lie algebra. Furthermore they constitute an important class of infinite dimensional Lie algebras which due to their geometric origin are
Algebraic and structural automata theory
Mikolajczak, B
1991-01-01
Automata Theory is part of computability theory which covers problems in computer systems, software, activity of nervous systems (neural networks), and processes of live organisms development.The result of over ten years of research, this book presents work in the following areas of Automata Theory: automata morphisms, time-varying automata, automata realizations and relationships between automata and semigroups.Aimed at those working in discrete mathematics and computer science, parts of the book are suitable for use in graduate courses in computer science, electronics, telecommunications, and control engineering. It is assumed that the reader is familiar with the basic concepts of algebra and graph theory.
Energy Technology Data Exchange (ETDEWEB)
Connes, A.; Kreimer, D. [Institut des Hautes Etudes Sci., Bures sur Yvette (France)
2000-03-01
This paper gives a complete selfcontained proof of our result (1999) showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann-Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra H which is commutative asan algebra. It is the dual Hopf algebra of the enveloping algebra of a Lie algebra G whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group G is the group of characters of H. We show then that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loop {gamma}(z) element of G, z element of C, where C is a small circle of complex dimensions around the integer dimension D of space-time. Our main result is that the renormalized theory is just the evaluation at z=D of the holomorphic part {gamma}{sub +} of the Birkhoff decomposition of {gamma}. We begin to analyse the group G and show that it is a semi-direct product of an easily understood abelian group by a highly non-trivial group closely tied up with groups of diffeomorphisms. (orig.)
Nonassociativity, Malcev algebras and string theory
International Nuclear Information System (INIS)
Nonassociative structures have appeared in the study of D-branes in curved backgrounds. In recent work, string theory backgrounds involving three-form fluxes, where such structures show up, have been studied in more detail. We point out that under certain assumptions these nonassociative structures coincide with nonassociative Malcev algebras which had appeared in the quantum mechanics of systems with non-vanishing three-cocycles, such as a point particle moving in the field of a magnetic charge. We generalize the corresponding Malcev algebras to include electric as well as magnetic charges. These structures find their classical counterpart in the theory of Poisson-Malcev algebras and their generalizations. We also study their connection to Stueckelberg's generalized Poisson brackets that do not obey the Jacobi identity and point out that nonassociative string theory with a fundamental length corresponds to a realization of his goal to find a non-linear extension of quantum mechanics with a fundamental length. Similar nonassociative structures are also known to appear in the cubic formulation of closed string field theory in terms of open string fields, leading us to conjecture a natural string-field theoretic generalization of the AdS/CFT-like (holographic) duality. (Copyright copyright 2013 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
International Conference on Semigroups, Algebras and Operator Theory
Meakin, John; Rajan, A
2015-01-01
This book discusses recent developments in semigroup theory and its applications in areas such as operator algebras, operator approximations and category theory. All contributing authors are eminent researchers in their respective fields, from across the world. Their papers, presented at the 2014 International Conference on Semigroups, Algebras and Operator Theory in Cochin, India, focus on recent developments in semigroup theory and operator algebras. They highlight current research activities on the structure theory of semigroups as well as the role of semigroup theoretic approaches to other areas such as rings and algebras. The deliberations and discussions at the conference point to future research directions in these areas. This book presents 16 unpublished, high-quality and peer-reviewed research papers on areas such as structure theory of semigroups, decidability vs. undecidability of word problems, regular von Neumann algebras, operator theory and operator approximations. Interested researchers will f...
Renormalization and Hopf Algebraic Structure of the 5-Dimensional Quartic Tensor Field Theory
Avohou, Remi Cocou; Tanasa, Adrian
2015-01-01
This paper is devoted to the study of renormalization of the quartic melonic tensor model in dimension (=rank) five. We review the perturbative renormalization and the computation of the one loop beta function, confirming the asymptotic freedom of the model. We then define the Connes-Kreimer-like Hopf algebra describing the combinatorics of the renormalization of this model and we analyze in detail, at one- and two-loop levels, the Hochschild cohomology allowing to write the combinatorial Dyson-Schwinger equations. Feynman tensor graph Hopf subalgebras are also exhibited.
Mixed motives and algebraic K-theory
Jannsen, Uwe
1990-01-01
The relations that could or should exist between algebraic cycles, algebraic K-theory, and the cohomology of - possibly singular - varieties, are the topic of investigation of this book. The author proceeds in an axiomatic way, combining the concepts of twisted Poincaré duality theories, weights, and tensor categories. One thus arrives at generalizations to arbitrary varieties of the Hodge and Tate conjectures to explicit conjectures on l-adic Chern characters for global fields and to certain counterexamples for more general fields. It is to be hoped that these relations ions will in due course be explained by a suitable tensor category of mixed motives. An approximation to this is constructed in the setting of absolute Hodge cycles, by extending this theory to arbitrary varieties. The book can serve both as a guide for the researcher, and as an introduction to these ideas for the non-expert, provided (s)he knows or is willing to learn about K-theory and the standard cohomology theories of algebraic varietie...
Multiple Schramm-Loewner evolutions for conformal field theories with Lie algebra symmetries
Energy Technology Data Exchange (ETDEWEB)
Sakai, Kazumitsu, E-mail: sakai@gokutan.c.u-tokyo.ac.jp [Institute of Physics, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8902 (Japan)
2013-02-11
We provide multiple Schramm-Loewner evolutions (SLEs) to describe the scaling limit of multiple interfaces in critical lattice models possessing Lie algebra symmetries. The critical behavior of the models is described by Wess-Zumino-Witten (WZW) models. Introducing a multiple Brownian motion on a Lie group as well as that on the real line, we construct the multiple SLE with additional Lie algebra symmetries. The connection between the resultant SLE and the WZW model can be understood via SLE martingales satisfied by the correlation functions in the WZW model. Due to interactions among SLE traces, these Brownian motions have drift terms which are determined by partition functions for the corresponding WZW model. As a concrete example, we apply the formula to the su{sup -hat} (2){sub k}-WZW model. Utilizing the fusion rules in the model, we conjecture that there exists a one-to-one correspondence between the partition functions and the topologically inequivalent configurations of the SLE traces. Furthermore, solving the Knizhnik-Zamolodchikov equation, we exactly compute the probabilities of occurrence for certain configurations (i.e. crossing probabilities) of traces for the triple SLE.
L∞-algebra models and higher Chern-Simons theories
Ritter, Patricia; Sämann, Christian
2016-10-01
We continue our study of zero-dimensional field theories in which the fields take values in a strong homotopy Lie algebra. In the first part, we review in detail how higher Chern-Simons theories arise in the AKSZ-formalism. These theories form a universal starting point for the construction of L∞-algebra models. We then show how to describe superconformal field theories and how to perform dimensional reductions in this context. In the second part, we demonstrate that Nambu-Poisson and multisymplectic manifolds are closely related via their Heisenberg algebras. As a byproduct of our discussion, we find central Lie p-algebra extensions of 𝔰𝔬(p + 2). Finally, we study a number of L∞-algebra models which are physically interesting and which exhibit quantized multisymplectic manifolds as vacuum solutions.
Division Algebras and Quantum Theory
Baez, John C
2011-01-01
Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the "three-fold way". It is perhaps easiest to see it in the study of irreducible unitary representations of groups on complex Hilbert spaces. These representations come in three kinds: those that are not isomorphic to their own dual (the truly "complex" representations), those that are self-dual thanks to a symmetric bilinear pairing (which are "real", in that they are the complexifications of representations on real Hilbert spaces), and those that are self-dual thanks to an antisymmetric bilinear pairing (which are...
Isomorphisms of Algebraic Number Fields
van Hoeij, Mark
2010-01-01
Let $\\mathbb{Q}(\\alpha)$ and $\\mathbb{Q}(\\beta)$ be algebraic number fields. We describe a new method to find (if they exist) all isomorphisms, $\\mathbb{Q}(\\beta) \\rightarrow \\mathbb{Q}(\\alpha)$. The algorithm is particularly efficient if the number of isomorphisms is one.
Derivations of the Moyal Algebra and Noncommutative Gauge Theories
Directory of Open Access Journals (Sweden)
Jean-Christophe Wallet
2009-01-01
Full Text Available The differential calculus based on the derivations of an associative algebra underlies most of the noncommutative field theories considered so far. We review the essential properties of this framework and the main features of noncommutative connections in the case of non graded associative unital algebras with involution. We extend this framework to the case of Z2-graded unital involutive algebras. We show, in the case of the Moyal algebra or some related Z2-graded version of it, that the derivation based differential calculus is a suitable framework to construct Yang-Mills-Higgs type models on Moyal (or related algebras, the covariant coordinates having in particular a natural interpretation as Higgs fields. We also exhibit, in one situation, a link between the renormalisable NC φ4-model with harmonic term and a gauge theory model. Some possible consequences of this are briefly discussed.
Algebraic differential calculus for gauge theories
International Nuclear Information System (INIS)
The guiding idea in this paper is that, from the point of view of physics, functions and fields are more important than the (space time) manifold over which they are defined. The line pursued in these notes belongs to the general framework of ideas that replaces the space M by the ring of functions on it. Our essential observation, underlying this work, is that much of mathematical physics requires only a few differential operators (Lie derivative, d, δ) operating on modules of sections of suitable bundles. A connection (=gauge potential) can be described by a lift of vector fields from the base to the total space of a principal bundle. Much of the information can be encoded in the lift without reference to the bundle structures. In this manner, one arrives at an 'algebraic differential calculus' and its graded generalization that we are going to discuss. We are going to give an exposition of 'algebraic gauge theory' in both ungraded and graded versions. We show how to deal with the essential features of electromagnetism, Dirac, Kaluza-Klein and 't Hooft-Polyakov monopoles. We also show how to break the symmetry from SU(2) to U(1) without Higgs field. We briefly show how to deal with tests particles in external fields and with the Lagrangian formulation of field theories. (orig./HSI)
Twisting theory for weak Hopf algebras
Institute of Scientific and Technical Information of China (English)
CHEN Ju-zhen; ZHANG Yan; WANG Shuan-hong
2008-01-01
The main aim of this paper is to study the twisting theory of weak Hopf algebras and give an equivalence between the (braided) monoidal categories of weak Hopf bimodules over the original and the twisted weak Hopf algebra to generalize the result from Oeckl (2000).
Function theory for a beltrami algebra
Directory of Open Access Journals (Sweden)
B. A. Case
1985-01-01
Full Text Available Complex functions are investigated which are solutions of an elliptic system of partial differential equations associated with a real parameter function. The functions f associated with a particualr parameter function g on a domain D form a Beltrami algebra denoted by the pair (D,g and a function theory is developed in this algebra. A strong conformality property holds for all functions in a (D,g algebra. For g≡|z|=r the algebra (D,r is that of the analytic functions.
Kleene Algebra with Products and Iteration Theories
Kozen, Dexter; Mamouras, Konstantinos
2013-01-01
We develop a typed equational system that subsumes both iteration theories and typed Kleene algebra in a common framework. Our approach is based on cartesian categories endowed with commutative strong monads to handle nondeterminism.
Arithmetic Deformation Theory of Lie Algebras
Rastegar, Arash
2012-01-01
This paper is devoted to deformation theory of graded Lie algebras over $\\Z$ or $\\Z_l$ with finite dimensional graded pieces. Such deformation problems naturally appear in number theory. In the first part of the paper, we use Schlessinger criteria for functors on Artin local rings in order to obtain universal deformation rings for deformations of graded Lie algebras and their graded representations. In the second part, we use a version of Schlessinger criteria for functors on the Artinian cat...
Multifractal vector fields and stochastic Clifford algebra.
Schertzer, Daniel; Tchiguirinskaia, Ioulia
2015-12-01
In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality.
Multifractal vector fields and stochastic Clifford algebra.
Schertzer, Daniel; Tchiguirinskaia, Ioulia
2015-12-01
In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality. PMID:26723166
Multifractal vector fields and stochastic Clifford algebra
Energy Technology Data Exchange (ETDEWEB)
Schertzer, Daniel, E-mail: Daniel.Schertzer@enpc.fr; Tchiguirinskaia, Ioulia, E-mail: Ioulia.Tchiguirinskaia@enpc.fr [University Paris-Est, Ecole des Ponts ParisTech, Hydrology Meteorology and Complexity HM& Co, Marne-la-Vallée (France)
2015-12-15
In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality.
Splitting full matrix algebras over algebraic number fields
Ivanyos, Gábor; Schicho, Joseph
2011-01-01
Let K be an algebraic number field of degree d and discriminant D over Q. Let A be an associative algebra over K given by structure constants such that A is siomorphic to the algebra M_n(K) of n by n matrices over K for some positive integer n. Suppose that d, n and D are bounded. Then an isomorphism of A with M_n(K) can be constructed by a polynomial time ff-algorithm. (An ff-algorithm is a deterministic procedure which is allowed to call oracles for factoring integers and factoring univariate polynomials over finite fields.) As a consequence, we obtain a polynomial time ff-algorithm to compute isomorphisms of central simple algebras of bounded degree over K.
Representation Theory of Algebraic Groups and Quantum Groups
Gyoja, A; Shinoda, K-I; Shoji, T; Tanisaki, Toshiyuki
2010-01-01
Invited articles by top notch expertsFocus is on topics in representation theory of algebraic groups and quantum groupsOf interest to graduate students and researchers in representation theory, group theory, algebraic geometry, quantum theory and math physics
Developments and retrospectives in Lie theory algebraic methods
Penkov, Ivan; Wolf, Joseph
2014-01-01
This volume reviews and updates a prominent series of workshops in representation/Lie theory, and reflects the widespread influence of those workshops in such areas as harmonic analysis, representation theory, differential geometry, algebraic geometry, and mathematical physics. Many of the contributors have had leading roles in both the classical and modern developments of Lie theory and its applications. This Work, entitled Developments and Retrospectives in Lie Theory, and comprising 26 articles, is organized in two volumes: Algebraic Methods and Geometric and Analytic Methods. This is the Algebraic Methods volume. The Lie Theory Workshop series, founded by Joe Wolf and Ivan Penkov and joined shortly thereafter by Geoff Mason, has been running for over two decades. Travel to the workshops has usually been supported by the NSF, and local universities have provided hospitality. The workshop talks have been seminal in describing new perspectives in the field covering broad areas of current research. Mos...
Wilson operator algebras and ground states for coupled BF theories
Tiwari, Apoorv; Chen, Xiao; Ryu, Shinsei
2016-01-01
The multi-flavor $BF$ theories in (3+1) dimensions with cubic or quartic coupling are the simplest topological quantum field theories that can describe fractional braiding statistics between loop-like topological excitations (three-loop or four-loop braiding statistics). In this paper, by canonically quantizing these theories, we study the algebra of Wilson loop and Wilson surface operators, and multiplets of ground states on three torus. In particular, by quantizing these coupled $BF$ theori...
Excision in algebraic K-theory and Karoubi's conjecture.
Suslin, A A; Wodzicki, M
1990-12-15
We prove that the property of excision in algebraic K-theory is for a Q-algebra A equivalent to the H-unitality of the latter. Our excision theorem, in particular, implies Karoubi's conjecture on the equality of algebraic and topological K-theory groups of stable C*-algebras. It also allows us to identify the algebraic K-theory of the symbol map in the theory of pseudodifferential operators. PMID:11607130
Contributions to the structure theory of non-simple C*-algebras
DEFF Research Database (Denmark)
Bentmann, Rasmus Moritz
This thesis is mainly concerned with classification results for non-simple purely ininite C*-algebras, specifically Cuntz-Krieger algebras and graph C*-algebras, and continuous fields of Kirchberg algebras. In Article A, we perform some computations concerning projective dimension in filtrated K-theory....... In joint work with Sara Arklint and Takeshi Katsura, we provide a range result complementing Gunnar Restor's classification theorem for Cuntz-Kieger algebras (Article B) and we investigate reduction of filtrated K-theory for C*-algebras of real rank zero, thereby obtaining a characterization of Cuntz...
Linear algebra meets Lie algebra: the Kostant-Wallach theory
Shomron, Noam; Parlett, Beresford N.
2008-01-01
In two languages, Linear Algebra and Lie Algebra, we describe the results of Kostant and Wallach on the fibre of matrices with prescribed eigenvalues of all leading principal submatrices. In addition, we present a brief introduction to basic notions in Algebraic Geometry, Integrable Systems, and Lie Algebra aimed at specialists in Linear Algebra.
Nicely semiramified division algebras over Henselian fields
Directory of Open Access Journals (Sweden)
Karim Mounirh
2005-01-01
Full Text Available This paper deals with the structure of nicely semiramified valued division algebras. We prove that any defectless finite-dimensional central division algebra over a Henselian field E with an inertial maximal subfield and a totally ramified maximal subfield (not necessarily of radical type (resp., split by inertial and totally ramified field extensions of E is nicely semiramified.
Understanding geometric algebra for electromagnetic theory
Arthur, John W
2011-01-01
"This book aims to disseminate geometric algebra as a straightforward mathematical tool set for working with and understanding classical electromagnetic theory. It's target readership is anyone who has some knowledge of electromagnetic theory, predominantly ordinary scientists and engineers who use it in the course of their work, or postgraduate students and senior undergraduates who are seeking to broaden their knowledge and increase their understanding of the subject. It is assumed that the reader is not a mathematical specialist and is neither familiar with geometric algebra or its application to electromagnetic theory. The modern approach, geometric algebra, is the mathematical tool set we should all have started out with and once the reader has a grasp of the subject, he or she cannot fail to realize that traditional vector analysis is really awkward and even misleading by comparison"--Provided by publisher.
Complete algebraic vector fields on affine surfaces
Kaliman, Shulim; Kutzschebauch, Frank; Leuenberger, Matthias
2014-01-01
Let $\\AAutH (X)$ be the subgroup of the group $\\AutH (X)$ of holomorphic automorphisms of a normal affine algebraic surface $X$ generated by elements of flows associated with complete algebraic vector fields. Our main result is a classification of all normal affine algebraic surfaces $X$ quasi-homogeneous under $\\AAutH (X)$ in terms of the dual graphs of the boundaries $\\bX \\setminus X$ of their SNC-completions $\\bX$.
Valued Graphs and the Representation Theory of Lie Algebras
Directory of Open Access Journals (Sweden)
Joel Lemay
2012-07-01
Full Text Available Quivers (directed graphs, species (a generalization of quivers and their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this paper, we discuss the most important results in the representation theory of species, such as Dlab and Ringel’s extension of Gabriel’s theorem, which classifies all species of finite and tame representation type. We also explain the link between species and K-species (where K is a field. Namely, we show that the category of K -species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as “crushed” species, and we show that if K is a perfect field, then the tensor algebra of a K -species tensored with the algebraic closure of K is isomorphic to, or Morita equivalent to, the path algebra of a quiver.
Methods of algebraic geometry in control theory
Falb, Peter
1999-01-01
"Control theory represents an attempt to codify, in mathematical terms, the principles and techniques used in the analysis and design of control systems. Algebraic geometry may, in an elementary way, be viewed as the study of the structure and properties of the solutions of systems of algebraic equations. The aim of this book is to provide access to the methods of algebraic geometry for engineers and applied scientists through the motivated context of control theory" .* The development which culminated with this volume began over twenty-five years ago with a series of lectures at the control group of the Lund Institute of Technology in Sweden. I have sought throughout to strive for clarity, often using constructive methods and giving several proofs of a particular result as well as many examples. The first volume dealt with the simplest control systems (i.e., single input, single output linear time-invariant systems) and with the simplest algebraic geometry (i.e., affine algebraic geometry). While this is qui...
Real Algebraic Number Theory I: Diophantine Approximation Groups
Gendron, T. M.
2012-01-01
This is the first of three papers introducing a paradigm within which global algebraic number theory for the reals may be formulated so as to make possible the synthesis of algebraic and transcendental number theory into a coherent whole. We introduce diophantine approximation groups and their associated Kronecker foliations, using them to provide new algebraic and geometric characterizations of K-linear and algebraic dependence. As a consequence we find reformulations -- as algebraic and geo...
Introduction to the theory of abstract algebras
Pierce, Richard S
2014-01-01
Intended for beginning graduate-level courses, this text introduces various aspects of the theory of abstract algebra. The book is also suitable as independent reading for interested students at that level as well as a primary source for a one-semester course that an instructor may supplement to expand to a full year. Author Richard S. Pierce, a Professor of Mathematics at Seattle's University of Washington, places considerable emphasis on applications of the theory and focuses particularly on lattice theory.After a preliminary review of set theory, the treatment presents the basic definitions
The $K$-theory of real graph $C*$-algebras
Boersema, Jeffrey L.
2014-01-01
In this paper, we will introduce real graph algebras and develop the theory to the point of being able to calculate the $K$-theory of such algebras. The $K$-theory situation is significantly more complicated than in the case for complex graph algebras. To develop the long exact sequence to compute the $K$-theory of a real graph algebra, we need to develop a generalized theory of crossed products for real C*-algebras for groups with involution. We also need to deal with the additional algebrai...
Algebraic Theories and (Infinity,1)-Categories
Cranch, James
2010-11-01
We adapt the classical framework of algebraic theories to work in the setting of (infinity,1)-categories developed by Joyal and Lurie. This gives a suitable approach for describing highly structured objects from homotopy theory. A central example, treated at length, is the theory of E_infinity spaces: this has a tidy combinatorial description in terms of span diagrams of finite sets. We introduce a theory of distributive laws, allowing us to describe objects with two distributing E_infinity stuctures. From this we produce a theory of E_infinity ring spaces. We also study grouplike objects, and produce theories modelling infinite loop spaces (or connective spectra), and infinite loop spaces with coherent multiplicative structure (or connective ring spectra). We use this to construct the units of a grouplike E_infinity ring space in a natural manner. Lastly we provide a speculative pleasant description of the K-theory of monoidal quasicategories and quasicategories with ring-like structures.
Karpilovsky, G
1989-01-01
This monograph gives a systematic account of certain important topics pertaining to field theory, including the central ideas, basic results and fundamental methods.Avoiding excessive technical detail, the book is intended for the student who has completed the equivalent of a standard first-year graduate algebra course. Thus it is assumed that the reader is familiar with basic ring-theoretic and group-theoretic concepts. A chapter on algebraic preliminaries is included, as well as a fairly large bibliography of works which are either directly relevant to the text or offer supplementary material of interest.
Introduction to the representation theory of algebras
Barot, Michael
2015-01-01
This book gives a general introduction to the theory of representations of algebras. It starts with examples of classification problems of matrices under linear transformations and explains the three common setups: representation of quivers, modules over algebras and additive functors over certain categories. The main part is devoted to (i) module categories, presenting the unicity of the decomposition into indecomposable modules, the Auslander–Reiten theory and the technique of knitting; (ii) the use of combinatorial tools such as dimension vectors and integral quadratic forms; and (iii) deeper theorems such as Gabriel‘s Theorem, the trichotomy and the Theorem of Kac – all accompanied by further examples. Each section includes exercises to facilitate understanding. By keeping the proofs as basic and comprehensible as possible and introducing the three languages at the beginning, this book is suitable for readers from the advanced undergraduate level onwards and enables them to consult related, specifi...
On the Theory of Algebraic Numbers with Elements of Small Height
Göral, Haydar
2015-01-01
In this paper, we study the field of algebraic numbers with elements of small height.We first show that the nonstandard algebraic numbers whose logarithmic height isinfinitesimal has the Mann property, and exploiting this we obtain a diophantine approximation result in number theory.Then, we prove that the theory of algebraic numbers with elements of small height is not simple and has the independence property.We also relate the simplicity of a certain pair with Lehmer's conjecture and obtain...
Algebraic model theory for languages without equality
Elgueta Montó, Raimon
1994-01-01
In our opinion, it is fair to distinguish two separate branches in the origins of model theory. The first one, the model theory of first-order logic, can be traced back to the pioneering work of L. Lowenheim, T. Skolem, K. Gödel, A. Tarski and A.I. MaI 'cev, published before the mid 30's. This branch was put forward during the 40s' and 50s’ by several authors, including A. Tarski, L. Henkin, A. Robinson, J. Los. Their contribution, however, was rather influenced by modern algebra, a disciplin...
Algebraic K-theory, K-regularity, and -duality of -stable C ∗-algebras
Mahanta, Snigdhayan
2015-12-01
We develop an algebraic formalism for topological -duality. More precisely, we show that topological -duality actually induces an isomorphism between noncommutative motives that in turn implements the well-known isomorphism between twisted K-theories (up to a shift). In order to establish this result we model topological K-theory by algebraic K-theory. We also construct an E ∞ -operad starting from any strongly self-absorbing C ∗-algebra . Then we show that there is a functorial topological K-theory symmetric spectrum construction on the category of separable C ∗-algebras, such that is an algebra over this operad; moreover, is a module over this algebra. Along the way we obtain a new symmetric spectra valued functorial model for the (connective) topological K-theory of C ∗-algebras. We also show that -stable C ∗-algebras are K-regular providing evidence for a conjecture of Rosenberg. We conclude with an explicit description of the algebraic K-theory of a x+ b-semigroup C ∗-algebras coming from number theory and that of -stabilized noncommutative tori.
On the Theory of Generalized Algebraic Transformations
Strecka, Jozef
2010-01-01
This book deals with the theory of generalized algebraic transformations, which is elaborated with the aim to provide a relatively simple theoretical tool that enables an exact treatment of diverse more complex lattice-statistical models. In addition to a brief historical account on the developments of this exact mapping method, the versatility of generalized algebraic transformations will be convincingly evidenced when providing exact results for two different families of exactly solvable models. The family of exactly solved Ising models brings a deeper insight into various aspects closely associated especially with phase transitions and critical phenomena. The second class of exactly solved Ising-Heisenberg models sheds light on striking quantum manifestations of spontaneously long-range ordered systems, which are closely connected with a mutual interplay between quantum and cooperative phenomena.
Institute of Scientific and Technical Information of China (English)
无
2004-01-01
Through most of Greek history, mathematicians concentrated on geometry, although Euclid considered the theory of numbers. The Greek mathematician Diophantus (3rd century),however, presented problems that had to be solved by what we would today call algebra. His book is thus the first algebra text.
Duality and products in algebraic (co)homology theories
Kowalzig, N.; Kraehmer, U.
2008-01-01
The origin and interplay of products and dualities in algebraic (co)homology theories is ascribed to a ×A-Hopf algebra structure on the relevant universal enveloping algebra. This provides a unified treatment for example of results by Van den Bergh about Hochschild (co)homology and by Huebschmann about Lie–Rinehart (co)homology.
Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure
Directory of Open Access Journals (Sweden)
Kenny De Commer
2013-12-01
Full Text Available Let g be a compact simple Lie algebra. We modify the quantized enveloping ∗-algebra associated to g by a real-valued character on the positive part of the root lattice. We study the ensuing Verma module theory, and the associated quotients of these modified quantized enveloping ∗-algebras. Restricting to the locally finite part by means of a natural adjoint action, we obtain in particular examples of quantum homogeneous spaces in the operator algebraic setting.
Elementary number theory an algebraic approach
Bolker, Ethan D
2007-01-01
This text uses the concepts usually taught in the first semester of a modern abstract algebra course to illuminate classical number theory: theorems on primitive roots, quadratic Diophantine equations, and the Fermat conjecture for exponents three and four. The text contains abundant numerical examples and a particularly helpful collection of exercises, many of which are small research problems requiring substantial study or outside reading. Some problems call for new proofs for theorems already covered or for inductive explorations and proofs of theorems found in later chapters.Ethan D. Bolke
Algebraic Graph Theory Morphisms, Monoids and Matrices
Knauer, Ulrich
2011-01-01
This is a highly self-contained book about algebraic graph theory which iswritten with a view to keep the lively and unconventional atmosphere of a spoken text to communicate the enthusiasm the author feels about this subject. The focus is on homomorphisms and endomorphisms, matrices and eigenvalues. Graph models are extremely useful for almost all applications and applicators as they play an important role as structuring tools. They allow to model net structures -like roads, computers, telephones -instances of abstract data structures -likelists, stacks, trees -and functional or object orient
Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination
Cohen, Cyril; Mahboubi, Assia
2012-01-01
International audience This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic properties. The theory of real algebraic numbers and more generally of semi-algebraic varieties is at the core of a number of effective methods in real analysis, including decision procedures for non linear arithmetic or optimization ...
Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination
Mahboubi, Assia; Cohen, Cyril
2012-01-01
This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic properties. The theory of real algebraic numbers and more generally of semi-algebraic varieties is at the core of a number of effective methods in real analysis, including decision procedures for non linear arithmetic or optimization methods for real valued fu...
Real Algebraic Number Theory I: Diophantine Approximation Groups
Gendron, T M
2012-01-01
The is the first of three papers introducing a paradigm within which global algebraic number theory for the reals may be formulated so as to make possible the synthesis of algebraic and transcendental number theory into a coherent whole. We introduce diophantine approximation groups and their associated Kronecker foliations, using them to provide new algebraic and geometric characterizations of K-linear and algebraic dependence. As a consequence we find reformulations -- as algebraic and geometric (graph) rigidities -- of the Theorems of Baker and Lindemann-Weierstrass, the Logarithm Conjecture and the Schanuel Conjecture. There is an Appendix describing diophantine approximation groups as model theoretic types.
Feynman-diagram evaluation in the electroweak theory with computer algebra
Weiglein, G.
2001-01-01
The evaluation of quantum corrections in the theory of the electroweak and strong interactions via higher-order Feynman diagrams requires complicated and laborious calculations, which however can be structured in a strictly algorithmic way. These calculations are ideally suited for the application of computer algebra systems, and computer algebra has proven to be a very valuable tool in this field already over several decades. It is sketched how computer algebra is presently applied in evalua...
Realisation of a Lorentz algebra in Lorentz violating theory
Energy Technology Data Exchange (ETDEWEB)
Ganguly, Oindrila [S. N. Bose National Centre for Basic Sciences, Kolkata (India)
2012-11-15
A Lorentz non-invariant higher derivative effective action in flat spacetime, characterised by a constant vector, can be made invariant under infinitesimal Lorentz transformations by restricting the allowed field configurations. These restricted fields are defined as functions of the background vector in such a way that background dependence of the dynamics of the physical system is no longer manifest. We show here that they also provide a field basis for the realisation of a Lorentz algebra and allow the construction of a Poincare invariant symplectic two-form on the covariant phase space of the theory. (orig.)
Polylogarithm identities, cluster algebras and the N=4 supersymmetric theory
Vergu, C
2015-01-01
Scattering amplitudes in N = 4 super-Yang Mills theory can be computed to higher perturbative orders than in any other four-dimensional quantum field theory. The results are interesting transcendental functions. By a hidden symmetry (dual conformal symmetry) the arguments of these functions have a geometric interpretation in terms of configurations of points in CP^3 and they turn out to be cluster coordinates. We briefly introduce cluster algebras and discuss their Poisson structure and the Sklyanin bracket. Finally, we present a 40-term trilogarithm identity which was discovered by accident while studying the physical results.
Algebraic structure and Poisson's theory of mechanico-electrical systems
Institute of Scientific and Technical Information of China (English)
Liu Hong-Ji; Tang Yi-Fa; Fu Jing-Li
2006-01-01
The algebraic structure and Poisson's integral theory of mechanico-electrical systems are studied.The Hamilton canonical equations and generalized Hamilton canonical equations and their the contravariant algebraic forms for mechanico-electrical systems are obtained.The Lie algebraic structure and the Poisson's integral theory of Lagrange mechanico-electrical systems are derived.The Lie algebraic structure admitted and Poisson's integral theory of the Lagrange-Maxwell mechanico-electrical systems are presented.Two examples are presented to illustrate these results.
Wilson operator algebras and ground states for coupled BF theories
Tiwari, Apoorv; Ryu, Shinsei
2016-01-01
The multi-flavor $BF$ theories in (3+1) dimensions with cubic or quartic coupling are the simplest topological quantum field theories that can describe fractional braiding statistics between loop-like topological excitations (three-loop or four-loop braiding statistics). In this paper, by canonically quantizing these theories, we study the algebra of Wilson loop and Wilson surface operators, and multiplets of ground states on three torus. In particular, by quantizing these coupled $BF$ theories on the three-torus, we explicitly calculate the $\\mathcal{S}$- and $\\mathcal{T}$-matrices, which encode fractional braiding statistics and topological spin of loop-like excitations, respectively. In the coupled $BF$ theories with cubic and quartic coupling, the Hopf link and Borromean ring of loop excitations, together with point-like excitations, form composite particles.
Algebraic discrete Morse theory for the hull resolution
Norén, Patrik
2015-01-01
We study how powerful algebraic discrete Morse theory is when applied to hull resolutions. The main result describes all cases when the hull resolution of the edge ideal of the complement of a triangle-free graph can be made minimal using algebraic discrete Morse theory.
K-theory of Continuous Deformations of C*-algebras
Institute of Scientific and Technical Information of China (English)
Takahiro SUDO
2007-01-01
We study K-theory of continuous deformations of C*-algebras to obtain that their K-theory is the same as that of the fiber at zero. We also consider continuous or discontinuous deformations of Cuntz and Toeplitz algebras.
Optical potentials in algebraic scattering theory
Energy Technology Data Exchange (ETDEWEB)
Levay, Peter [Institute of Theoretical Physics, Technical University of Budapest, Budapest (Hungary)
1999-02-12
Using the theory of induced representations new realizations for the Lie algebras of the groups SO(2, 1), SO(2, 2), SO(3, 2) are found. The eigenvalue problem of the Casimir operators yield Schroedinger equations with non-Hermitian interaction terms (i.e. optical potentials). For the group SO(2, 2) we have a two-parameter family of (matrix-valued) potentials containing terms of Poeschl-Teller and Gendenshtein type. We calculate the S-matrices for special values of this two-parameter family. In particular we also include a derivation of the S-matrix for the two-dimensional scattering problem on a complex Gendenshtein potential. The canonically transformed realization results in a non-local optical potential. (author)
Decomposition Theory in the Teaching of Elementary Linear Algebra.
London, R. R.; Rogosinski, H. P.
1990-01-01
Described is a decomposition theory from which the Cayley-Hamilton theorem, the diagonalizability of complex square matrices, and functional calculus can be developed. The theory and its applications are based on elementary polynomial algebra. (KR)
Algebraic number theory and code design for Rayleigh fading channels
Oggier, F
2014-01-01
Algebraic number theory is having an increasing impact in code design for many different coding applications, such as single antenna fading channels and more recently, MIMO systems.Extended work has been done on single antenna fading channels, and algebraic lattice codes have been proven to be an effective tool. The general framework has been settled in the last ten years and many explicit code constructions based on algebraic number theory are now available. The aim of this work is to provide both an overview on algebraic lattice code designs for Rayleigh fading channels, as well as a tutoria
Valued Graphs and the Representation Theory of Lie Algebras
Lemay, Joel
2011-01-01
Quivers (directed graphs) and species (a generalization of quivers) and their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this paper, we discuss the most important results in the representation theory of species, such as Dlab and Ringel's extension of Gabriel's theorem, which classifies all species of finite and tame representation type. We also explain the link between species and K-species (where K is a field). Namely, we show that the category of K-species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles that do not appear in the literature. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as "crushed" species...
Algebraic K-theory of strict ring spectra
Rognes, John
2014-01-01
We view strict ring spectra as generalized rings. The study of their algebraic K-theory is motivated by its applications to the automorphism groups of compact manifolds. Partial calculations of algebraic K-theory for the sphere spectrum are available at regular primes, but we seek more conceptual answers in terms of localization and descent properties. Calculations for ring spectra related to topological K-theory suggest the existence of a motivic cohomology theory for strictly commutative ri...
Space-time algebra for the generalization of gravitational field equations
Indian Academy of Sciences (India)
Süleyman Demir
2013-05-01
The Maxwell–Proca-like field equations of gravitolectromagnetism are formulated using space-time algebra (STA). The gravitational wave equation with massive gravitons and gravitomagnetic monopoles has been derived in terms of this algebra. Using space-time algebra, the most generalized form of gravitoelectromagnetic Klein–Gordon equation has been obtained. Finally, the analogy in formulation between massive gravitational theory and electromagnetism has been discussed.
Central simple Poisson algebras
Institute of Scientific and Technical Information of China (English)
SU; Yucai; XU; Xiaoping
2004-01-01
Poisson algebras are fundamental algebraic structures in physics and symplectic geometry. However, the structure theory of Poisson algebras has not been well developed. In this paper, we determine the structure of the central simple Poisson algebras related to locally finite derivations, over an algebraically closed field of characteristic zero.The Lie algebra structures of these Poisson algebras are in general not finitely-graded.
Algebraic K-theory of crystallographic groups the three-dimensional splitting case
Farley, Daniel Scott
2014-01-01
The Farrell-Jones isomorphism conjecture in algebraic K-theory offers a description of the algebraic K-theory of a group using a generalized homology theory. In cases where the conjecture is known to be a theorem, it gives a powerful method for computing the lower algebraic K-theory of a group. This book contains a computation of the lower algebraic K-theory of the split three-dimensional crystallographic groups, a geometrically important class of three-dimensional crystallographic group, representing a third of the total number. The book leads the reader through all aspects of the calculation. The first chapters describe the split crystallographic groups and their classifying spaces. Later chapters assemble the techniques that are needed to apply the isomorphism theorem. The result is a useful starting point for researchers who are interested in the computational side of the Farrell-Jones isomorphism conjecture, and a contribution to the growing literature in the field.
Bollhöfer, Matthias; Kressner, Daniel; Mehl, Christian; Stykel, Tatjana
2015-01-01
This edited volume highlights the scientific contributions of Volker Mehrmann, a leading expert in the area of numerical (linear) algebra, matrix theory, differential-algebraic equations and control theory. These mathematical research areas are strongly related and often occur in the same real-world applications. The main areas where such applications emerge are computational engineering and sciences, but increasingly also social sciences and economics. This book also reflects some of Volker Mehrmann's major career stages. Starting out working in the areas of numerical linear algebra (his first full professorship at TU Chemnitz was in "Numerical Algebra," hence the title of the book) and matrix theory, Volker Mehrmann has made significant contributions to these areas ever since. The highlights of these are discussed in Parts I and II of the present book. Often the development of new algorithms in numerical linear algebra is motivated by problems in system and control theory. These and his later major work on ...
Higher order Fourier analysis as an algebraic theory I
Szegedy, Balazs
2009-01-01
Ergodic theory, Higher order Fourier analysis and the hyper graph regularity method are three possible approaches to Szemer\\'edi type theorems in abelian groups. In this paper we develop an algebraic theory that creates a connection between these approaches. Our main method is to take the ultra product of abelian groups and to develop a precise algebraic theory of higher order characters on it. These results then can be turned back into approximative statements about finite Abelian groups.
Lectures on Iwahori-Hecke Algebras and their Representation Theory
Cherednik, Ivan; Howe, Roger; Lusztig, George
2002-01-01
Two basic problems of representation theory are to classify irreducible representations and decompose representations occuring naturally in some other context. Algebras of Iwahori-Hecke type are one of the tools and were, probably, first considered in the context of representation theory of finite groups of Lie type. This volume consists of notes of the courses on Iwahori-Hecke algebras and their representation theory, given during the CIME summer school which took place in 1999 in Martina Franca, Italy.
Singularity Theory for W-Algebra Potentials
J Gaite
1994-01-01
The Landau potentials of W3-algebra models are analyzed with algebraic-geometric methods. The number of ground states and the number of independent perturbations of every potential coincide and can be computed. This number agrees with the structure of ground states obtained in a previous paper, name
Fourier theory and C∗-algebras
Bédos, Erik; Conti, Roberto
2016-07-01
We discuss a number of results concerning the Fourier series of elements in reduced twisted group C∗-algebras of discrete groups, and, more generally, in reduced crossed products associated to twisted actions of discrete groups on unital C∗-algebras. A major part of the article gives a review of our previous work on this topic, but some new results are also included.
Evolution algebras and their applications
Tian, Jianjun Paul
2008-01-01
Behind genetics and Markov chains, there is an intrinsic algebraic structure. It is defined as a type of new algebra: as evolution algebra. This concept lies between algebras and dynamical systems. Algebraically, evolution algebras are non-associative Banach algebras; dynamically, they represent discrete dynamical systems. Evolution algebras have many connections with other mathematical fields including graph theory, group theory, stochastic processes, dynamical systems, knot theory, 3-manifolds, and the study of the Ihara-Selberg zeta function. In this volume the foundation of evolution algebra theory and applications in non-Mendelian genetics and Markov chains is developed, with pointers to some further research topics.
A universal characterization of higher algebraic K-theory
Blumberg, Andrew J; Tabuada, Goncalo
2010-01-01
We establish a universal characterization of the higher algebraic $K$-theory of stable infinity categories. Specifically, we prove that the (connective) algebraic $K$-theory spectrum construction is the universal functor, with values in a stable presentable infinity category, which inverts Morita equivalences, preserves filtered colimits, and satisfies Waldhausen's additivity theorem. In order to prove these results, we construct and study a suitable localization of the category of presheaves of spectra on small stable infinity categories. In this category, Waldhausen's S.-construction corresponds to the suspension functor and the algebraic K-theory spectrum becomes co-representable. This latter result leads to a complete classification of all natural transformations from algebraic K-theory to topological Hochschild homology (THH) and topological cyclic homology (TC). In particular, we obtain a canonical universal description of the cyclotomic trace map.
On the Isomorphism Conjecture in algebraic K-theory
Bartels, Arthur; Farrell, Tom; Jones, Lowell; Reich, Holger
2001-01-01
The Isomorphism Conjecture is a conceptional approach towards a calculation of the algebraic K-theory of a group ring RG, where G is an infinite group. In this paper we prove the conjecture in dimensions n
Cryptography by means of linear algebra and number theory
Elfadel, Ajaeb
2014-01-01
ABSTRACT: This thesis focuses on the techniques of cryptography in linear algebra and number theory. We first give the necessary review on modular arithmetic. Under Linear Algebra, Hill cipher cryptographic technique and its variations are studied. Under number theory, on the other hand, the definition of Euler function, and some important theorems in this regard are given. The cryptographic techniques such as the Caesar cipher, Exponential transformations and the Public key cryptographic tec...
Polymer Parametrised Field Theory
Laddha, Alok
2008-01-01
Free scalar field theory on 2 dimensional flat spacetime, cast in diffeomorphism invariant guise by treating the inertial coordinates of the spacetime as dynamical variables, is quantized using LQG type `polymer' representations for the matter field and the inertial variables. The quantum constraints are solved via group averaging techniques and, analogous to the case of spatial geometry in LQG, the smooth (flat) spacetime geometry is replaced by a discrete quantum structure. An overcomplete set of Dirac observables, consisting of (a) (exponentials of) the standard free scalar field creation- annihilation modes and (b) canonical transformations corresponding to conformal isometries, are represented as operators on the physical Hilbert space. None of these constructions suffer from any of the `triangulation' dependent choices which arise in treatments of LQG. In contrast to the standard Fock quantization, the non- Fock nature of the representation ensures that the algebra of conformal isometries as well as tha...
Singularity theory for W-algebra potentials
Gaite, J C
1993-01-01
The Landau potentials of $W_3$-algebra models are analyzed with algebraic-geometric methods. The number of ground states and the number of independent perturbations of every potential coincide and can be computed. This number agrees with the structure of ground states obtained in a previous paper, namely, as the phase structure of the IRF models of Jimbo et al. The singularities associated to these potentials are identified.
Logarithmic conformal field theory
Gainutdinov, Azat; Ridout, David; Runkel, Ingo
2013-12-01
Conformal field theory (CFT) has proven to be one of the richest and deepest subjects of modern theoretical and mathematical physics research, especially as regards statistical mechanics and string theory. It has also stimulated an enormous amount of activity in mathematics, shaping and building bridges between seemingly disparate fields through the study of vertex operator algebras, a (partial) axiomatisation of a chiral CFT. One can add to this that the successes of CFT, particularly when applied to statistical lattice models, have also served as an inspiration for mathematicians to develop entirely new fields: the Schramm-Loewner evolution and Smirnov's discrete complex analysis being notable examples. When the energy operator fails to be diagonalisable on the quantum state space, the CFT is said to be logarithmic. Consequently, a logarithmic CFT is one whose quantum space of states is constructed from a collection of representations which includes reducible but indecomposable ones. This qualifier arises because of the consequence that certain correlation functions will possess logarithmic singularities, something that contrasts with the familiar case of power law singularities. While such logarithmic singularities and reducible representations were noted by Rozansky and Saleur in their study of the U (1|1) Wess-Zumino-Witten model in 1992, the link between the non-diagonalisability of the energy operator and logarithmic singularities in correlators is usually ascribed to Gurarie's 1993 article (his paper also contains the first usage of the term 'logarithmic conformal field theory'). The class of CFTs that were under control at this time was quite small. In particular, an enormous amount of work from the statistical mechanics and string theory communities had produced a fairly detailed understanding of the (so-called) rational CFTs. However, physicists from both camps were well aware that applications from many diverse fields required significantly more
The algebraic numbers definable in various exponential fields
Kirby, Jonathan; Onshuus, Alf
2011-01-01
We prove the following theorems: Theorem 1: For any E-field with cyclic kernel, in particular $\\mathbb C$ or the Zilber fields, all real abelian algebraic numbers are pointwise definable. Theorem 2: For the Zilber fields, the only pointwise definable algebraic numbers are the real abelian numbers.
Algebraic approach to form factors in the complex sinh-Gordon theory
Lashkevich, Michael
2016-01-01
We study form factors of the quantum complex sinh-Gordon theory in the algebraic approach. In the case of exponential fields the form factors can be obtained from the known form factors of the $Z_N$-symmetric Ising model. The algebraic construction also provides an Ansatz for form factors of descendant operators. We obtain generating functions of such form factors and establish their main properties: the cluster factorization and reflection equations.
Gravity, Gauge Theories and Geometric Algebra
Lasenby, A; Gull, S F; Lasenby, Anthony; Doran, Chris; Gull, Stephen
1998-01-01
A new gauge theory of gravity is presented. The theory is constructed in a flat background spacetime and employs gauge fields to ensure that all relations between physical quantities are independent of the positions and orientations of the matter fields. In this manner all properties of the background spacetime are removed from physics, and what remains are a set of `intrinsic' relations between physical fields. The properties of the gravitational gauge fields are derived from both classical and quantum viewpoints. Field equations are then derived from an action principle, and consistency with the minimal coupling procedure selects an action that is unique up to the possible inclusion of a cosmological constant. This in turn singles out a unique form of spin-torsion interaction. A new method for solving the field equations is outlined and applied to the case of a time-dependent, spherically-symmetric perfect fluid. A gauge is found which reduces the physics to a set of essentially Newtonian equations. These e...
Homotopy Classification of Bosonic String Field Theory
Muenster, Korbinian; Sachs, Ivo
2012-01-01
We prove the decomposition theorem for the loop homotopy algebra of quantum closed string field theory and use it to show that closed string field theory is unique up to gauge transformations on a given string background and given S-matrix. For the theory of open and closed strings we use results in open-closed homotopy algebra to show that the space of inequivalent open string field theories is isomorphic to the space of classical closed string backgrounds. As a further application of the op...
Supersymmetry Algebra in Super Yang-Mills Theories
Yokoyama, Shuichi
2015-01-01
We compute supersymmetry algebra (superalgebra) in supersymmetric Yang-Mills theories (SYM) consisting of a vector multiplet including fermionic contribution in six dimensions. We show that the contribution of fermion is given by boundary terms. From six dimensional results we determine superalgebras of five and four dimensional SYM by dimensional reduction. In five dimensional superalgebra the Kaluza-Klein momentum and the instanton particle charge are not the same but algebraically indistinguishable. We also extend this calculation including a hyper multiplet and for maximally SYM. We derive extended supersymmetry algebras in those four dimensional SYM with the holomorphic coupling constant given in hep-th/9408099.
Algebraic Formulation of the Operatorial Perturbation Theory; 1
Müller, A H; Müller, Ary W. Espinosa; Vásquez, Adelio R. Matamala
1996-01-01
A new totally algebraic formalism based on general, abstract ladder operators has been proposed. This approach heavily grounds in the superoperator formalism of Primas. However it is necessary to introduce many improvements in his formalism. In this regard, it has been introduced a new set of superoperators featured by their algebraic structure. Also, two lemmas and one theorem have been developed in order to algebraically reformulate the theory on more rigorous grounds. Finally, we have been able to build a coherent and self-contained formalism independent on any matricial representation , removing in this way the degeneracy problem .
Function algebras on finite sets basic course on many-valued logic and clone theory
Lau, Dietlinde
2006-01-01
Gives an introduction to the theory of function algebras. This book gives the general concepts of the Universal Algebra in order to familiarize the reader from the beginning on with the algebraic side of function algebras. It is a source on function algebras for students and researchers in mathematical logic and theoretical computer science.
Supersymmetry algebra in super Yang-Mills theories
Yokoyama, Shuichi
2015-01-01
We compute supersymmetry algebra (superalgebra) in supersymmetric Yang-Mills theories (SYM) consisting of a vector multiplet including fermionic contribution in six dimensions. We show that the contribution of fermion is given by boundary terms. From six dimensional results we determine superalgebras of five and four dimensional SYM by dimensional reduction. In five dimensional superalgebra the Kaluza-Klein momentum and the instanton particle charge are not the same but algebraically indistin...
A Workshop on Algebraic Design Theory and Hadamard Matrices
2015-01-01
This volume develops the depth and breadth of the mathematics underlying the construction and analysis of Hadamard matrices and their use in the construction of combinatorial designs. At the same time, it pursues current research in their numerous applications in security and cryptography, quantum information, and communications. Bridges among diverse mathematical threads and extensive applications make this an invaluable source for understanding both the current state of the art and future directions. The existence of Hadamard matrices remains one of the most challenging open questions in combinatorics. Substantial progress on their existence has resulted from advances in algebraic design theory using deep connections with linear algebra, abstract algebra, finite geometry, number theory, and combinatorics. Hadamard matrices arise in a very diverse set of applications. Starting with applications in experimental design theory and the theory of error-correcting codes, they have found unexpected and important ap...
Monotonic Property in Field Algebra of G-Spin Model
Institute of Scientific and Technical Information of China (English)
蒋立宁
2003-01-01
Let F be the field algebra of G-spin model, D(G) the double algebra of a finite group G and D(H) the sub-Hopf algerba of D(G) determined by the subgroup H of G. The paper builds a correspondence between D(H) and the D(H)-invariant sub-C*-algebra AH in F, and proves that the correspondence is strictly monotonic.
Kac-Moody algebras in gravity and M-theories
Houart, Laurent
2005-01-01
The formulation of gravity and M-theories as very-extended Kac-Moody invariant theories is reviewed. Exact solutions describing intersecting extremal brane configurations smeared in all directions but one are presented. The intersection rules characterising these solutions are neatly encoded in the algebra. The existence of dualities for all G+++ and their group theoretical-origin are discussed.
Higher Gauge Theories from Lie n-algebras and Off-Shell Covariantization
Carow-Watamura, Ursula; Ikeda, Noriaki; Kaneko, Yukio; Watamura, Satoshi
2016-01-01
We analyze higher gauge theories in various dimensions using a supergeometric method based on a differential graded symplectic manifold, called a QP-manifold, which is closely related to the BRST-BV formalism in gauge theories. Extensions of the Lie 2-algebra gauge structure are formulated within the Lie n-algebra induced by the QP-structure. We find that in 5 and 6 dimensions there are special extensions of the gauge algebra. In these cases, a restriction of the gauge symmetry by imposing constraints on the auxiliary gauge fields leads to a covariantized theory. As an example we show that we can obtain an off-shell covariantized higher gauge theory in 5 dimensions, which is similar to the one proposed in [1] (arxiv:1206.5643).
Matrix Representation of Renormalization in Perturbative Quantum Field Theory
Ebrahimi-Fard, K.; Guo, L.
2005-01-01
We formulate the Hopf algebraic approach of Connes and Kreimer to renormalization in perturbative quantum field theory using triangular matrix representation. We give a Rota-Baxter anti-homomorphism from general regularized functionals on the Feynman graph Hopf algebra to triangular matrices with entries in a Rota-Baxter algebra. For characters mapping to the group of unipotent triangular matrices we derive the algebraic Birkhoff decomposition for matrices using Spitzer's identity. This simpl...
Symmetries of topological field theories in the BV-framework
Gieres, F.; Grimstrup, J. M.; Nieder, H.; Pisar, T.; Schweda, M.
2001-01-01
Topological field theories of Schwarz-type generally admit symmetries whose algebra does not close off-shell, e.g. the basic symmetries of BF models or vector supersymmetry of the gauge-fixed action for Chern-Simons theory (this symmetry being at the origin of the perturbative finiteness of the theory). We present a detailed discussion of all these symmetries within the algebraic approach to the Batalin-Vilkovisky formalism. Moreover, we discuss the general algebraic construction of topologic...
Schaum's outline of theory and problems of linear algebra
Lipschutz, Seymour
2001-01-01
This third edition of the successful outline in linear algebra--which sold more than 400,000 copies in its past two editions--has been thoroughly updated to increase its applicability to the fields in which linear algebra is now essential: computer science, engineering, mathematics, physics, and quantitative analysis. Revised coverage includes new problems relevant to computer science and a revised chapter on linear equations.
Twining characters and Picard groups in rational conformal field theory
Fuchs, Jurgen; Runkel, Ingo; Schweigert, Christoph
2006-01-01
Picard groups of tensor categories play an important role in rational conformal field theory. The Picard group of the representation category C of a rational vertex algebra can be used to construct examples of (symmetric special) Frobenius algebras in C. Such an algebra A encodes all data needed to ensure the existence of correlators of a local conformal field theory. The Picard group of the category of A-bimodules has a physical interpretation, too: it describes internal symmetries of the co...
Coends in conformal field theory
Fuchs, Jürgen
2016-01-01
The idea of "summing over all intermediate states" that is central for implementing locality in quantum systems can be realized by coend constructions. In the concrete case of systems of conformal blocks for a certain class of conformal vertex algebras, one deals with coends in functor categories. Working with these coends involves quite a few subtleties which, even though they have in principle already been understood twenty years ago, have not been sufficiently appreciated by the conformal field theory community.
Algebra of formal vector fields on the line and Buchstaber's conjecture
Millionschikov, Dmitri
2008-01-01
Let L_1 denotes the Lie algebra of formal vector fields on the line which vanish at the origin together with their first derivatives. Buchstaber and Shokurov have shown that the universal enveloping algebra U(L_1) is isomorphic to the tensor product of the Landweber-Novikov algebra S in complex cobordism theory by reals. The cohomology H*(L_1) has trivial multiplication. Buchstaber conjectured that H*(L_1) is generated with respect to non-trivial Massey products by H^1(L_1). Feigin, Fuchs and...
Indian Academy of Sciences (India)
Subhash J Bhatt
2006-05-01
Given an -tempered strongly continuous action of $\\mathbb{R}$ by continuous $∗$-automorphisms of a Frechet $∗$-algebra , it is shown that the enveloping -*-algebra $E(S(\\mathbb{R},A^∞,))$ of the smooth Schwartz crossed product $S(\\mathbb{R},A^∞,)$ of the Frechet algebra $A^∞$ of $C^∞$-elements of is isomorphic to the -*-crossed product $C^∗(\\mathbb{R}, E(A), )$ of the enveloping -*-algebra () of by the induced action. When is a hermitian $\\mathcal{Q}$-algebra, one gets -theory isomorphism $R K_∗(S(\\mathbb{R},A^∞,))=K_∗(C^∗(\\mathbb{R}, E(A),)$ for the representable -theory of Frechet algebras. An application to the differential structure of a *-algebra defined by densely defined differential seminorms is given.
Spin structures on algebraic curves and their applications in string theories
International Nuclear Information System (INIS)
The free fields on a Riemann surface carrying spin structures live on an unramified r-covering of the surface itself. When the surface is represented as an algebraic curve related to the vanishing of the Weierstrass polynomial, its r-coverings are algebraic curves as well. We construct explicitly the Weierstrass polynomial associated to the r-coverings of an algebraic curve. Using standard techniques of algebraic geometry it is then possible to solve the inverse Jacobi problem for the odd spin structures. As an application we derive the partition functions of bosonic string theories in many examples, including two general curves of genus three and four. The partition functions are explicitly expressed in terms of branch points apart from a factor which is essentially a theta constant. 53 refs., 4 figs. (Author)
Mean-Field Dynamical Semigroups on C*-ALGEBRAS
Duffield, N. G.; Werner, R. F.
We study a notion of the mean-field limit of a sequence of dynamical semigroups on the n-fold tensor products of a C*-algebra { A} with itself. In analogy with the theory of semigroups on Banach spaces we give abstract conditions for the existence of these limits. These conditions are verified in the case of semigroups whose generators are determined by the successive resymmetrizations of a fixed operator, as well as generators which can be approximated by generators of this type. This includes the time evolutions of the mean-field versions of quantum lattice systems. In these cases the limiting dynamical semigroup is given by a continuous flow on the state space of { A}. For a class of such flows we show stability by constructing a Liapunov function. We also give examples where the limiting evolution is given by a diffusion, rather than a flow on the state space of { A}.
Meromorphic c=24 Conformal Field Theories
Schellekens, Adrian Norbert
1993-01-01
Modular invariant conformal field theories with just one primary field and central charge $c=24$ are considered. It has been shown previously that if the chiral algebra of such a theory contains spin-1 currents, it is either the Leech lattice CFT, or it contains a Kac-Moody sub-algebra with total central charge 24. In this paper all meromorphic modular invariant combinations of the allowed Kac-Moody combinations are obtained. The result suggests the existence of 71 meromorphic $c=24$ theories, including the 41 that were already known.
Embedding of exact $C^{*}$-algebras and continuous fields in the Cuntz algebra $O_2$
Kirchberg, E; Kirchberg, Eberhard
1997-01-01
We prove that any separable exact C*-algebra is isomorphic to a subalgebra of the Cuntz algebra ${\\cal O}_2.$ We further prove that if $A$ is a simple separable unital nuclear C*-algebra, then ${\\cal O}_2 \\otimes A \\cong {\\cal O}_2,$ and if, in addition, $A$ is purely infinite, then ${\\cal O}_{\\infty} The embedding of exact C*-algebras in $ØA{2}$ is continuous in the following sense. If $A$ is a continuous field of C*-algebras over a compact manifold or finite CW complex $X$ with fiber $A (x)$ over $x \\in X,$ such that the algebra of continuous sections of $A$ is separable and exact, then there is a family of injective homomorphisms $\\phi_x : A (x) \\to {\\cal O}_2$ such that for every continuous section $a$ of $A$ the function $x \\mapsto \\phi_x (a (x))$ is continuous. Moreover, one can say something about the modulus of continuity of the functions $x \\mapsto \\phi_x (a (x))$ in terms of the structure of the continuous field. In particular, we show that the continuous field $\\theta and $v (\\theta)$ are mapped t...
Partial Fractions in Calculus, Number Theory, and Algebra
Yackel, C. A.; Denny, J. K.
2007-01-01
This paper explores the development of the method of partial fraction decomposition from elementary number theory through calculus to its abstraction in modern algebra. This unusual perspective makes the topic accessible and relevant to readers from high school through seasoned calculus instructors.
Algebraic K-theory of discrete subgroups of Lie groups.
Farrell, F T; Jones, L E
1987-05-01
Let G be a Lie group (with finitely many connected components) and Gamma be a discrete, cocompact, torsion-free subgroup of G. We rationally calculate the algebraic K-theory of the integral group ring ZGamma in terms of the homology of Gamma with trivial rational coefficients. PMID:16593834
Algebraic formulation of quantum theory, particle identity and entanglement
Govindarajan, T. R.
2016-08-01
Quantum theory as formulated in conventional framework using statevectors in Hilbert spaces misses the statistical nature of the underlying quantum physics. Formulation using operators 𝒞∗ algebra and density matrices appropriately captures this feature in addition leading to the correct formulation of particle identity. In this framework, Hilbert space is an emergent concept. Problems related to anomalies and quantum epistemology are discussed.
Pure L-functions from algebraic geometry over finite fields
Wan, D
2000-01-01
This is an expository paper which gives a simple arithmetic introduction to the conjectures of Weil and Dwork concerning zeta functions of algebraic varieties over finite fields. A number of further open questions are raised.
Extended Supersymmetric BMS$_3$ algebras and Their Free Field Realisations
Banerjee, Nabamita; Lodato, Ivano; Mukhi, Sunil; Neogi, Turmoli
2016-01-01
We study $N=(2,4,8)$ supersymmetric extensions of the three dimensional BMS algebra (BMS$_3$) with most generic possible central extensions. We find that $N$-extended supersymmetric BMS$_3$ algebras can be derived by a suitable contraction of two copies of the extended superconformal algebras. Extended algebras from all the consistent contractions are obtained by scaling left-moving and right-moving supersymmetry generators symmetrically, while Virasoro and R-symmetry generators are scaled asymmetrically. On the way, we find that the BMS/GCA correspondence does not in general hold for supersymmetric systems. Using the $\\beta$-$\\gamma$ and the ${\\mathfrak b}$-${\\mathfrak c}$ systems, we construct free field realisations of all the extended super-BMS$_3$ algebras.
Infinite Conformal Algebras in Supersymmetric Theories on Four Manifolds
Johansen, Andrei
1994-01-01
We study a supersymmetric theory twisted on a K\\"ahler four manifold $M=\\Sigma_1 \\times \\Sigma_2 ,$ where $\\Sigma_{1,2}$ are 2D Riemann surfaces. We demonstrate that it possesses a "left-moving" conformal stress tensor on $\\Sigma_1$ ($\\Sigma_2$) in a BRST cohomology, which generates the Virasoro algebra with the conventional commutation relations. The central charge of the Virasoro algebra has a purely geometric origin and is proportional to the Euler characteristic $\\c$ of the $\\Sigma_2$ ($\\...
Algebraic Graph Theory (a short course for postgraduate students and researchers)
Shokrollahi, Arman
2008-01-01
Algebraic graph theory is the branch of mathematics that studies graphs by using algebraic properties of associated matrices. More in particular, spectral graph theory studies the relation between graph properties and the spectrum of the adjacency matrix or Laplace matrix. And the theory of association schemes and coherent configurations studies the algebra generated by associated matrices.
The Nonlinear Field Space Theory
Mielczarek, Jakub; Trześniewski, Tomasz
2016-08-01
In recent years the idea that not only the configuration space of particles, i.e. spacetime, but also the corresponding momentum space may have nontrivial geometry has attracted significant attention, especially in the context of quantum gravity. The aim of this letter is to extend this concept to the domain of field theories, by introducing field spaces (i.e. phase spaces of field values) that are not affine spaces. After discussing the motivation and general aspects of our approach we present a detailed analysis of the prototype (quantum) Nonlinear Field Space Theory of a scalar field on the Minkowski background. We show that the nonlinear structure of a field space leads to numerous interesting predictions, including: non-locality, generalization of the uncertainty relations, algebra deformations, constraining of the maximal occupation number, shifting of the vacuum energy and renormalization of the charge and speed of propagation of field excitations. Furthermore, a compact field space is a natural way to implement the "Principle of finiteness" of physical theories, which once motivated the Born-Infeld theory. Thus the presented framework has a variety of potential applications in the theories of fundamental interactions (e.g. quantum gravity), as well as in condensed matter physics (e.g. continuous spin chains), and can shed new light on the issue of divergences in quantum field theories.
Many rational points coding theory and algebraic geometry
Hurt, Norman E
2003-01-01
This monograph presents a comprehensive treatment of recent results on algebraic geometry as they apply to coding theory and cryptography, with the goal the study of algebraic curves and varieties with many rational points. They book surveys recent developments on abelian varieties, in particular the classification of abelian surfaces, hyperelliptic curves, modular towers, Kloosterman curves and codes, Shimura curves and modular jacobian surfaces. Applications of abelian varieties to cryptography are presented including a discussion of hyperelliptic curve cryptosystems. The inter-relationship of codes and curves is developed building on Goppa's results on algebraic-geometry cods. The volume provides a source book of examples with relationships to advanced topics regarding Sato-Tate conjectures, Eichler-Selberg trace formula, Katz-Sarnak conjectures and Hecke operators.
Algebraic formulation of the operatorial perturbation theory; 2, applications
Espinosa-Müller, A W
1996-01-01
The algebraic approach to operator perturbation method has been applied to two quantum--mechanical systems ``The Stark Effect in the Harmonic Oscillator'' and ``The Generalized Zeeman Effect''. To that end, two realizations of the superoperators involved in the formalism have been carried out. The first of them has been based on the Heisenberg--Dirac algebra of \\hat{a}^\\dagger, \\hat{a}, \\hat{1} operators, the second one has been based in the angular momemtum algebra of \\hat{L}_+, \\hat{L}_- and \\hat{L}_0 operators. The successful results achieved in predicting the discrete spectra of both systems have put in evidence the reliability and accuracy of the theory.
Guerra, Francesco
2005-01-01
A coincise review about Euclidean (Quantum) Field Theory is presented. It deals with the general structural properties, the connections with Quantum Field Theory, the exploitation in Constructive Quantum Field Theory, and the physical interpretation.
Energy Technology Data Exchange (ETDEWEB)
Dattoli, Giuseppe; Torre, Amalia [ENEA, Centro Ricerche Frascati, Rome (Italy). Dipt. Innovazione; Ottaviani, Pier Luigi [ENEA, Centro Ricerche Bologna (Italy); Vasquez, Luis [Madris, Univ. Complutense (Spain). Dept. de Matemateca Aplicado
1997-10-01
The finite-difference based integration method for evolution-line equations is discussed in detail and framed within the general context of the evolution operator picture. Exact analytical methods are described to solve evolution-like equations in a quite general physical context. The numerical technique based on the factorization formulae of exponential operator is then illustrated and applied to the evolution-operator in both classical and quantum framework. Finally, the general view to the finite differencing schemes is provided, displaying the wide range of applications from the classical Newton equation of motion to the quantum field theory.
Picard groups in rational conformal field theory
Fröhlich, J.; Fuchs, J.; Runkel, I.; Schweigert, C.
2005-01-01
Algebra and representation theory in modular tensor categories can be combined with tools from topological field theory to obtain a deeper understanding of rational conformal field theories in two dimensions: It allows us to establish the existence of sets of consistent correlation functions, to demonstrate some of their properties in a model-independent manner, and to derive explicit expressions for OPE coefficients and coefficients of partition functions in terms of invariants of links in t...
Double affine Hecke algebras and 2-dimensional local fields
Kapranov, M.
1998-01-01
We give an interpretation of the double affine Hecke algebra of Cherednik as the (suitably regularized) algebra of double cosets of a group G by a subgroup J, extending the well known interpretations of finite and affine Hecke algebras. In this interpretation, G consists of K-points of a split reductive group where K is a 2-dimensional local field such as Q_p((t)) or F_q((t_1))((t_2)), and J is a certain analog of the Iwahori subgroup.
Dual number coefficient octonion algebra, field equations and conservation laws
Chanyal, B. C.; Chanyal, S. K.
2016-08-01
Starting with octonion algebra, we develop the dual number coefficient octonion (DNCO) algebra having sixteen components. DNCO forms of generalized potential, field and current equations are discussed in consistent manner. We have made an attempt to write the DNCO form of generalized Dirac-Maxwell's equations in presence of electric and magnetic charges (dyons). Accordingly, we demonstrate the work-energy theorem of classical mechanics reproducing the continuity equation for dyons in terms of DNCO algebra. Further, we discuss the DNCO form of linear momentum conservation law for dyons.
Currents in supersymmetric field theories
Derendinger, Jean-Pierre
2016-01-01
A general formalism to construct and improve supercurrents and source or anomaly superfields in two-derivative N=1 supersymmetric theories is presented. It includes arbitrary gauge and chiral superfields and a linear superfield coupled to gauge fields. These families of supercurrent structures are characterized by their energy-momentum tensors and R currents and they display a specific relation to the dilatation current of the theory. The linear superfield is introduced in order to describe the gauge coupling as a background (or propagating) field. Supersymmetry does not constrain the dependence on this gauge coupling field of gauge kinetic terms and holomorphicity restrictions are absent. Applying these results to an effective (Wilson) description of super-Yang-Mills theory, matching or cancellation of anomalies leads to an algebraic derivation of the all-order NSVZ beta function.
Vortex lattice theory: A linear algebra approach
Chamoun, George C.
Vortex lattices are prevalent in a large class of physical settings that are characterized by different mathematical models. We present a coherent and generalized Hamiltonian fluid mechanics-based formulation that reduces all vortex lattices into a classic problem in linear algebra for a non-normal matrix A. Via Singular Value Decomposition (SVD), the solution lies in the null space of the matrix (i.e., we require nullity( A) > 0) as well as the distribution of its singular values. We demonstrate that this approach provides a good model for various types of vortex lattices, and makes it possible to extract a rich amount of information on them. The contributions of this thesis can be classified into four main points. The first is asymmetric equilibria. A 'Brownian ratchet' construct was used which converged to asymmetric equilibria via a random walk scheme that utilized the smallest singular value of A. Distances between configurations and equilibria were measured using the Frobenius norm ||·||F and 2-norm ||·||2, and conclusions were made on the density of equilibria within the general configuration space. The second contribution used Shannon Entropy, which we interpret as a scalar measure of the robustness, or likelihood of lattices to occur in a physical setting. Third, an analytic model was produced for vortex street patterns on the sphere by using SVD in conjunction with expressions for the center of vorticity vector and angular velocity. Equilibrium curves within the configuration space were presented as a function of the geometry, and pole vortices were shown to have a critical role in the formation and destruction of vortex streets. The fourth contribution entailed a more complete perspective of the streamline topology of vortex streets, linking the bifurcations to critical points on the equilibrium curves.
Algebra of Complex Vectors and Applications in Electromagnetic Theory and Quantum Mechanics
Directory of Open Access Journals (Sweden)
Kundeti Muralidhar
2015-08-01
Full Text Available A complex vector is a sum of a vector and a bivector and forms a natural extension of a vector. The complex vectors have certain special geometric properties and considered as algebraic entities. These represent rotations along with specified orientation and direction in space. It has been shown that the association of complex vector with its conjugate generates complex vector space and the corresponding basis elements defined from the complex vector and its conjugate form a closed complex four dimensional linear space. The complexification process in complex vector space allows the generation of higher n-dimensional geometric algebra from (n — 1-dimensional algebra by considering the unit pseudoscalar identification with square root of minus one. The spacetime algebra can be generated from the geometric algebra by considering a vector equal to square root of plus one. The applications of complex vector algebra are discussed mainly in the electromagnetic theory and in the dynamics of an elementary particle with extended structure. Complex vector formalism simplifies the expressions and elucidates geometrical understanding of the basic concepts. The analysis shows that the existence of spin transforms a classical oscillator into a quantum oscillator. In conclusion the classical mechanics combined with zeropoint field leads to quantum mechanics.
Application of Fuzzy Algebra in Automata theory
Directory of Open Access Journals (Sweden)
Kharatti Lal
2016-06-01
Full Text Available In our first application we consider strings of fuzzy singletons as input to a fuzzy finite state machine. The notion of fuzzy automata was introduced in [58]. There has been considerable growth in the area [18]. In this section present a theory of free fuzzy monoids and apply the results to the area of fuzzy automata. In fuzzy automata, the set of strings of input symbols can be considered to be a free monoid. We introduced the motion of fuzzy strings of input symbols, where the fuzzy strings from free fuzzy submonoids of the free monoids of input strings. We show that fuzzy automata with fuzzy input are equivalent to fuzzy automata with crisp input. Hence the result of fuzzy automata theory can be immediately applied to those of fuzzy automata theory with fuzzy input. The result are taken from [7] and [34].
Inverse Determinant Sums and Connections Between Fading Channel Information Theory and Algebra
Vehkalahti, Roope
2011-01-01
Since the invention of space-time coding numerous algebraic methods have been applied to code design. In particular algebraic number theory and central simple algebras have been at the forefront of the research. In the first part of the paper we will push this direction further and show how the error probability of algebraic codes is tied to some central aspects of algebraic number theory and central simple algebras. In particular we prove how the error probability of several algebraic codes is tied to the corresponding zeta functions and unit groups. In the second part of this paper we turn to study what information theory can say about algebra. We will first derive some corollaries from the diversity-multiplexing gain tradeoff (DMT) Zheng and Tse and later show how these results can be used to analyze the unit group of orders of certain division algebras.
Algebraic field descriptions in three-dimensional Euclidean space
Salingaros, Nikos; Ilamed, Yehiel
1984-08-01
In this paper, we use the differential forms of three-dimensional Euclidean space to realize a Clifford algebra which is isomorphic to the algebra of the Pauli matrices or the complex quaternions. This is an associative vector-antisymmetric tensor algebra with division: We provide the algebraic inverse of an eight-component spinor field which is the sum of a scalar + vector + pseudovector + pseudoscalar. A surface of singularities is defined naturally by the inverse of an eight-component spinor and corresponds to a generalized Minkowski “double” light cone in the parameter space. A general description of finite spatial rotations, which utilizes the Baker-Campbell-Hausdorff formula, generalizes the usual infinitesimal treatments of the rotation group. We derive an explicit expression for the angle corresponding to two successive finite rotations in any direction. We also discuss Lorentz transformations and duality rotations of the electromagnetic field and exhibit a relationship between the algebraic inverse and a duality rotated field. Using a combined transformation, one can always transform an arbitrary electromagnetic field ( E≠0) into a pure electric field, but never into a pure magnetic field.
Higher order Fourier analysis as an algebraic theory II
Szegedy, Balazs
2009-01-01
Our approach to higher order Fourier analysis is to study the ultra product of finite (or compact) Abelian groups on which a new algebraic theory appears. This theory has consequences on finite (or compact) groups usually in the form of approximative statements. The present paper is the second part of a paper in which higher order characters and decompositions were introduced. We generalize the concept of the Pontrjagin dual group and introduce higher order versions of it. We study the algebraic structure of the higher order dual groups. We prove a simple formula for the Gowers uniformity norms in terms of higher order decompositions. We present a simple spectral algorithm to produce higher order decompositions. We briefly study a multi linear version of Fourier analysis. Along these lines we obtain new inverse theorems for Gowers's norms.
Index maps in the K-theory of graph algebras
DEFF Research Database (Denmark)
Meier Carlsen, Toke; Eilers, Søren; Tomforde, Mark
2012-01-01
Let C*(E) be the graph C*-algebra associated to a graph E and let J be a gauge-invariant ideal in C*(E). We compute the cyclic six-term exact sequence in K-theory associated to the extension in terms of the adjacency matrix associated to E. The ordered six-term exact sequence is a complete stable...
On logical, algebraic, and probabilistic aspects of fuzzy set theory
Mesiar, Radko
2016-01-01
The book is a collection of contributions by leading experts, developed around traditional themes discussed at the annual Linz Seminars on Fuzzy Set Theory. The different chapters have been written by former PhD students, colleagues, co-authors and friends of Peter Klement, a leading researcher and the organizer of the Linz Seminars on Fuzzy Set Theory. The book also includes advanced findings on topics inspired by Klement’s research activities, concerning copulas, measures and integrals, as well as aggregation problems. Some of the chapters reflect personal views and controversial aspects of traditional topics, while others deal with deep mathematical theories, such as the algebraic and logical foundations of fuzzy set theory and fuzzy logic. Originally thought as an homage to Peter Klement, the book also represents an advanced reference guide to the mathematical theories related to fuzzy logic and fuzzy set theory with the potential to stimulate important discussions on new research directions in the fiel...
Algebraic and combinatorial Brill-Noether theory
Caporaso, Lucia
2011-01-01
The interplay between algebro-geometric and combinatorial Brill-Noether theory is studied. The Brill-Noether variety of a graph shown to be non-empty if the Brill-Noether number is non-negative, as a consequence of the analogous fact for smooth projective curves. Similarly, the existence of a graph for which the Brill-Noether variety is empty implies the emptiness of the corresponding Brill-Noether variety for a general curve. The main tool is a refinement of Baker's Specialization Lemma.
K theoretical approach to the fusion rules of conformal quantum field theories
International Nuclear Information System (INIS)
Conformally invariant quantum field theories are investigated using concepts of the algebraic approach to quantum field theory as well as techniques from the theory of operator algebras. Arguments from the study of statistical lattice models in one and two dimensions, from recent developments in algebraic quantum field theory, and from other sources suggest that there exists and intimate connection between conformal field theories and a special class of C*-algebras, the so-called AF-algebras. For a series of Virasoro minimal models, this correspondence is made explicit by constructing path representations of the irreducible highest weight modules. We then focus on the K0-invariant of these path AF-algebras and show how its functorial properties allow to exploit the abstract theory of superselection sectors in order to derive the fusion rules of the W-algebras hidden in the Virasoro minimal models. (orig.)
Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations
Gavrylenko, P.; Marshakov, A.
2016-02-01
We consider the conformal blocks in the theories with extended conformal W-symmetry for the integer Virasoro central charges. We show that these blocks for the generalized twist fields on sphere can be computed exactly in terms of the free field theory on the covering Riemann surface, even for a non-abelian monodromy group. The generalized twist fields are identified with particular primary fields of the W-algebra, and we propose a straightforward way to compute their W-charges. We demonstrate how these exact conformal blocks can be effectively computed using the technique arisen from the gauge theory/CFT correspondence. We discuss also their direct relation with the isomonodromic tau-function for the quasipermutation monodromy data, which can be an encouraging step on the way of definition of generic conformal blocks for W-algebra using the isomonodromy/CFT correspondence.
Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations
Gavrylenko, P
2015-01-01
We consider the conformal blocks in the theories with extended conformal W-symmetry for the integer Virasoro central charges. We show that these blocks for the generalized twist fields on sphere can be computed exactly in terms of the free field theory on the covering Riemann surface, even for a non-abelian monodromy group. The generalized twist fields are identified with particular primary fields of the W-algebra, and we propose a straightforward way to compute their W-charges. We demonstrate how these exact conformal blocks can be effectively computed using the technique arisen from the gauge theory/CFT correspondence. We discuss also their direct relation with the isomonodromic tau-function for the quasipermutation monodromy data, which can be an encouraging step on the way of definition of generic conformal blocks for W-algebra using the isomonodromy/CFT correspondence.
A note on the "logarithmic-W_3" octuplet algebra and its Nichols algebra
Semikhatov, A M
2013-01-01
We describe a Nichols-algebra-motivated construction of an octuplet chiral algebra that is a "W_3-counterpart" of the triplet algebra of (p,1) logarithmic models of two-dimensional conformal field theory.
Class field theory. The Bonn lectures
International Nuclear Information System (INIS)
Clear presentation. Quick and immediate access to the subject. A classic (established and prominent German original). The present manuscript is an improved edition of a text that first appeared under the same title in Bonner Mathematische Schriften, no.26, and originated from a series of lectures given by the author in 1965/66 in Wolfgang Krull's seminar in Bonn. Its main goal is to provide the reader, acquainted with the basics of algebraic number theory, a quick and immediate access to class field theory. This script consists of three parts, the first of which discusses the cohomology of finite groups. The second part discusses local class field theory, and the third part concerns the class field theory of finite algebraic number fields.
Three dimensional maximally supersymmetric field theory revisited
International Nuclear Information System (INIS)
The field theoretical realization of maximally extended supersymmetry algebra in 3 dimensions is revisited here. The existence of an interacting field theory which also manifests the full automorphism demands the existence of extra global symmetry transformations. They form the infinite-dimensional group of volume-preserving diffeomorphisms of an internal 3-dimensional space. Upon regularization, it will be truncated to a finite-dimensional unitary group. This extra symmetry together with multiplicity of component fields allow one to consistently introduce self-interactions through Nambu brackets or matrix 4-commutators. It turns out that there exists a conformal field theory which is invariant under OSp(4 | 8) x U(N) groups of global transformations. Relaxing full automorphism to its biggest subgroup, one can consistently mass-deforms the theory. In addition to ordinary associative algebraic structures, a non-associative structure also underlies this theory. (author)
Bilinear covariants and spinor fields duality in quantum Clifford algebras
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Abłamowicz, Rafał, E-mail: rablamowicz@tntech.edu [Department of Mathematics, Box 5054, Tennessee Technological University, Cookeville, Tennessee 38505 (United States); Gonçalves, Icaro, E-mail: icaro.goncalves@ufabc.edu.br [Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, 05508-090, São Paulo, SP (Brazil); Centro de Matemática, Computação e Cognição, Universidade Federal do ABC, 09210-170 Santo André, SP (Brazil); Rocha, Roldão da, E-mail: roldao.rocha@ufabc.edu.br [Centro de Matemática, Computação e Cognição, Universidade Federal do ABC, 09210-170 Santo André, SP (Brazil); International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste (Italy)
2014-10-15
Classification of quantum spinor fields according to quantum bilinear covariants is introduced in a context of quantum Clifford algebras on Minkowski spacetime. Once the bilinear covariants are expressed in terms of algebraic spinor fields, the duality between spinor and quantum spinor fields can be discussed. Thus, by endowing the underlying spacetime with an arbitrary bilinear form with an antisymmetric part in addition to a symmetric spacetime metric, quantum algebraic spinor fields and deformed bilinear covariants can be constructed. They are thus compared to the classical (non quantum) ones. Classes of quantum spinor fields classes are introduced and compared with Lounesto's spinor field classification. A physical interpretation of the deformed parts and the underlying Z-grading is proposed. The existence of an arbitrary bilinear form endowing the spacetime already has been explored in the literature in the context of quantum gravity [S. W. Hawking, “The unpredictability of quantum gravity,” Commun. Math. Phys. 87, 395 (1982)]. Here, it is shown further to play a prominent role in the structure of Dirac, Weyl, and Majorana spinor fields, besides the most general flagpoles and flag-dipoles. We introduce a new duality between the standard and the quantum spinor fields, by showing that when Clifford algebras over vector spaces endowed with an arbitrary bilinear form are taken into account, a mixture among the classes does occur. Consequently, novel features regarding the spinor fields can be derived.
Vector fields and nilpotent Lie algebras
Grayson, Matthew; Grossman, Robert
1987-01-01
An infinite-dimensional family of flows E is described with the property that the associated dynamical system: x(t) = E(x(t)), where x(0) is a member of the set R to the Nth power, is explicitly integrable in closed form. These flows E are of the form E = E1 + E2, where E1 and E2 are the generators of a nilpotent Lie algebra, which is either free, or satisfies some relations at a point. These flows can then be used to approximate the flows of more general types of dynamical systems.
Bursa, Francis; Kroyter, Michael
2010-01-01
String field theory is a candidate for a full non-perturbative definition of string theory. We aim to define string field theory on a space-time lattice to investigate its behaviour at the quantum level. Specifically, we look at string field theory in a one dimensional linear dilaton background. We report the first results of our simulations.
Algebraic perturbation theory for dense liquids with discrete potentials
Adib, Artur B.
2006-01-01
A simple theory for the leading-order correction g_1(r) to the structure of a hard-sphere liquid with discrete (e.g. square-well) potential perturbations is proposed. The theory makes use of a general approximation that effectively eliminates four-particle correlations from g_1(r) with good accuracy at high densities. For the particular case of discrete perturbations, the remaining three-particle correlations can be modeled with a simple volume-exclusion argument, resulting in an algebraic an...
Huang, Yu-tin; Johansson, Henrik
2013-04-26
We show that three-dimensional supergravity amplitudes can be obtained as double copies of either three-algebra super-Chern-Simons matter theory or two-algebra super-Yang-Mills theory when either theory is organized to display the color-kinematics duality. We prove that only helicity-conserving four-dimensional gravity amplitudes have nonvanishing descendants when reduced to three dimensions, implying the vanishing of odd-multiplicity S-matrix elements, in agreement with Chern-Simons matter theory. We explicitly verify the double-copy correspondence at four and six points for N = 12,10,8 supergravity theories and discuss its validity for all multiplicity.
An Algebraic PT-Symmetric Quantum Theory with a Maximal Mass
Rodionov, V N
2016-01-01
In this paper we draw attention to the fact that the studies by V. G. Kadyshevsky devoted to the creation of the which \\emph{\\emph{to the geometric quantum field theory with a fundamental mass}} containing non-Hermitian mass extensions. It is important that these ideas recently received a powerful development in the form of construction of the non-Hermitian algebraic approach. The central point of these theories is the construction of new scalar products in which the average values of non-Hermitian Hamiltonians are valid. Among numerous works on this subject may be to allocate as purely mathematical and containing a discussion of experimental results. In this regard, we consider as the development of algebraic relativistic pseudo-Hermitian quantum theory with a maximal mass and experimentally significant investigations are discussed
Tabak, John
2004-01-01
Looking closely at algebra, its historical development, and its many useful applications, Algebra examines in detail the question of why this type of math is so important that it arose in different cultures at different times. The book also discusses the relationship between algebra and geometry, shows the progress of thought throughout the centuries, and offers biographical data on the key figures. Concise and comprehensive text accompanied by many illustrations presents the ideas and historical development of algebra, showcasing the relevance and evolution of this branch of mathematics.
Fusion rules in conformal field theory
International Nuclear Information System (INIS)
Several aspects of fusion rings and fusion rule algebras, and of their manifestations in two-dimensional (conformal) field theory, are described: diagonalization and the connection with modular invariance; the presentation in terms of quotients of polynomial rings; fusion graphs; various strategies that allow for a partial classification; and the role of the fusion rules in the conformal bootstrap programme. (orig.)
Covariant Noncommutative Field Theory
International Nuclear Information System (INIS)
The covariant approach to noncommutative field and gauge theories is revisited. In the process the formalism is applied to field theories invariant under diffeomorphisms. Local differentiable forms are defined in this context. The lagrangian and hamiltonian formalism is consistently introduced
Brane Topological Field Theories and Hurwitz numbers for CW-complexes
Natanzon, Sergey M.
2009-01-01
We expand Topological Field Theory on some special CW-complexes (brane complexes). This Brane Topological Field Theory one-to-one corresponds to infinite dimensional Frobenius Algebras, graduated by CW-complexes of lesser dimension. We define general and regular Hurwitz numbers of brane complexes and prove that they generate Brane Topological Field Theories. For general Hurwitz numbers corresponding algebra is an algebra of coverings of lesser dimension. For regular Hurwitz numbers the Froben...
C*-algebras of Toeplitz type associated with algebraic number fields
Cuntz, Joachim; Laca, Marcelo
2011-01-01
We associate with the ring $R$ of algebraic integers in a number field a C*-algebra $\\mathfrak T[R]$. It is an extension of the C*-algebra $\\mathfrak A[R]$ studied previously in \\cite{Cun}, \\cite{CuLi}, \\cite{CuLi2} and, in contrast to $\\mathfrak A[R]$, it is functorial under homomorphisms of rings. It can also be defined using the left regular representation of the $ax+b$-semigroup $R\\rtimes R^\\times$ on $\\ell^2 (R\\rtimes R^\\times)$. The algebra $\\mathfrak T[R]$ carries a natural one-parameter automorphism group $(\\sigma_t)_{t\\in\\mathbb R}$. We determine its KMS-structure. The technical difficulties that we encounter are due to the presence of the class group in the case where $R$ is not a principal ideal domain. In that case, for a fixed large inverse temperature, the simplex of KMS-states splits over the class group. The "partition functions" are partial Dedekind $\\zeta$-functions. We prove a result characterizing the asymptotic behavior of quotients of such partial $\\zeta$-functions, which we then use to ...
SUBGROUPS OF CLASS GROUPS OF ALGEBRAIC QUADRATIC FUNCTION FIELDS
Institute of Scientific and Technical Information of China (English)
王鲲鹏; 张贤科
2003-01-01
Ideal class groups H(K) of algebraic quadratic function fields K are studied. Necessaryand sufficient condition is given for the class group H(K) to contain a cyclic subgroup of anyorder n, which holds true for both real and imaginary fields K. Then several series of functionfields K, including real, inertia imaginary, and ramified imaginary quadratic function fields, aregiven, for which the class groups H(K) are proved to contain cyclic subgroups of order n.
Local aspects of free open bosonic string field theory
Tomassini, Luca
2008-01-01
We show that strictly local observables with arbitrarily small support in space-time exist in covariant free open bosonic string field theory. The main ingredient of the proof is a modified version of the well known DDF operators, which we rigourously define. This result allows in principle the definition of a net of local observable algebras and should be considered as a first step towards the application of algebraic quantum field theory to this case
Won, Chang-Hee; Michel, Anthony N
2008-01-01
This volume - dedicated to Michael K. Sain on the occasion of his seventieth birthday - is a collection of chapters covering recent advances in stochastic optimal control theory and algebraic systems theory. Written by experts in their respective fields, the chapters are thematically organized into four parts: Part I focuses on statistical control theory, where the cost function is viewed as a random variable and performance is shaped through cost cumulants. In this respect, statistical control generalizes linear-quadratic-Gaussian and H-infinity control. Part II addresses algebraic systems th
Diagrammatic Kazhdan-Lusztig theory for the (walled) Brauer algebra
Cox, Anton
2010-01-01
We determine the decomposition numbers for the Brauer and walled Brauer algebra in characteristic zero in terms of certain polynomials associated to cap and curl diagrams (recovering a result of Martin in the Brauer case). We consider a second family of polynomials associated to such diagrams, and use these to determine projective resolutions of the standard modules. We then relate these two families of polynomials to Kazhdan-Lusztig theory via the work of Lascoux-Sch\\"utzenberger and Boe, inspired by work of Brundan and Stroppel in the cap diagram case.
Flanders, Harley
1975-01-01
Algebra presents the essentials of algebra with some applications. The emphasis is on practical skills, problem solving, and computational techniques. Topics covered range from equations and inequalities to functions and graphs, polynomial and rational functions, and exponentials and logarithms. Trigonometric functions and complex numbers are also considered, together with exponentials and logarithms.Comprised of eight chapters, this book begins with a discussion on the fundamentals of algebra, each topic explained, illustrated, and accompanied by an ample set of exercises. The proper use of a
Some Properties of Intuitionistic Fuzzy Lie Algebras over a Fuzzy Field
Antony, P. L.; Lilly, P. L.
2011-01-01
The concept of intuitionistic fuzzy Lie algebra over a fuzzy field is introduced. We study the "necessity" and "possibility" operators on intuitionistic fuzzy Lie algebra over a fuzzy field and give some properties of homomorphic images.
Conformal field theory and 2D critical phenomena. Part 1
International Nuclear Information System (INIS)
Review of the recent developments in the two-dimensional conformal field theory and especially its applications to the physics of 2D critical phenomena is given. It includes the Ising model, the Potts model. Minimal models, corresponding to theories invariant under higher symmetries, such as superconformal theories, parafermionic theories and theories with current and W-algebras are also discussed. Non-hamiltonian approach to two-dimensional field theory is formulated. 126 refs
On the algebraic K-theory of the complex K-theory spectrum
Ausoni, Christian
2010-03-01
Let p>3 be a prime, let ku be the connective complex K-theory spectrum, and let K(ku) be the algebraic K-theory spectrum of ku. We study the p-primary homotopy type of the spectrum K(ku) by computing its mod (p,v_1) homotopy groups. We show that up to a finite summand, these groups form a finitely generated free module over a polynomial algebra F_p[b], where b is a class of degree 2p+2 defined as a higher Bott element.
Algebraic K-theory and abstract homotopy theory
Blumberg, Andrew J
2007-01-01
We decompose the K-theory space of a Waldhausen category in terms of its Dwyer-Kan simplicial localization. This leads to a criterion for functors to induce equivalences of K-theory spectra that generalizes and explains many of the criteria appearing in the literature.
Mathematical aspects of quantum field theories
Strobl, Thomas
2015-01-01
Despite its long history and stunning experimental successes, the mathematical foundation of perturbative quantum field theory is still a subject of ongoing research. This book aims at presenting some of the most recent advances in the field, and at reflecting the diversity of approaches and tools invented and currently employed. Both leading experts and comparative newcomers to the field present their latest findings, helping readers to gain a better understanding of not only quantum but also classical field theories. Though the book offers a valuable resource for mathematicians and physicists alike, the focus is more on mathematical developments. This volume consists of four parts: The first Part covers local aspects of perturbative quantum field theory, with an emphasis on the axiomatization of the algebra behind the operator product expansion. The second Part highlights Chern-Simons gauge theories, while the third examines (semi-)classical field theories. In closing, Part 4 addresses factorization homolo...
Certain number-theoretic episodes in algebra
Sivaramakrishnan, R
2006-01-01
Many basic ideas of algebra and number theory intertwine, making it ideal to explore both at the same time. Certain Number-Theoretic Episodes in Algebra focuses on some important aspects of interconnections between number theory and commutative algebra. Using a pedagogical approach, the author presents the conceptual foundations of commutative algebra arising from number theory. Self-contained, the book examines situations where explicit algebraic analogues of theorems of number theory are available. Coverage is divided into four parts, beginning with elements of number theory and algebra such as theorems of Euler, Fermat, and Lagrange, Euclidean domains, and finite groups. In the second part, the book details ordered fields, fields with valuation, and other algebraic structures. This is followed by a review of fundamentals of algebraic number theory in the third part. The final part explores links with ring theory, finite dimensional algebras, and the Goldbach problem.
Combinatorics and field theory
Bender, Carl M.; Brody, Dorje C.; Meister, Bernhard K.
2006-01-01
For any given sequence of integers there exists a quantum field theory whose Feynman rules produce that sequence. An example is illustrated for the Stirling numbers. The method employed here offers a new direction in combinatorics and graph theory.
Observable currents in lattice field theories
Zapata, José A
2016-01-01
Observable currents are spacetime local objects that induce physical observables when integrated on an auxiliary codimension one surface. Since the resulting observables are independent of local deformations of the integration surface, the currents themselves carry most of the information about the induced physical observables. I study observable currents in a multisymplectic framework for Lagrangian field theory over discrete spacetime. A weak version of observable currents preserves many of their properties, while inducing a family of observables capable of separating points in the space of physically distinct solutions. A Poisson bracket gives the space of observable currents the structure of a Lie algebra. Peierls bracket for bulk observables gives an algebra homomorphism mapping equivalence classes of bulk observables to weak observable currents. The study covers scalar fields, nonlinear sigma models and gauge theories (including gauge theory formulations of general relativity) on the lattice. Even when ...
Garrett, Paul B
2007-01-01
Designed for an advanced undergraduate- or graduate-level course, Abstract Algebra provides an example-oriented, less heavily symbolic approach to abstract algebra. The text emphasizes specifics such as basic number theory, polynomials, finite fields, as well as linear and multilinear algebra. This classroom-tested, how-to manual takes a more narrative approach than the stiff formalism of many other textbooks, presenting coherent storylines to convey crucial ideas in a student-friendly, accessible manner. An unusual feature of the text is the systematic characterization of objects by universal
On the algebraic structure of self-dual gauge fields and sigma models
International Nuclear Information System (INIS)
An extensive and detailed analysis of self-dual Gauge Fields, in particular with axial symmetry, is presented, culminating in a purely algebraic procedure to generate solutions. The method which is particularly suited for the construction of multimonopole solutions for a theory with arbitrary G, is also applicable to a wide class of nonlinear sigma models. The relevant symmetries as well as the associated linear problems which underly the exact solubility of the problem, are constructed and discussed in detail. (author)
Lower bounds on the class number of algebraic function fields defined over any finite field
Ballet, Stéphane
2011-01-01
We give lower bounds on the number of effective divisors of degree $\\leq g-1$ with respect to the number of places of certain degrees of an algebraic function field of genus $g$ defined over a finite field. We deduce lower bounds and asymptotics for the class number, depending mainly on the number of places of a certain degree. We give examples of towers of algebraic function fields having a large class number.
Categorical Algebra and its Applications
1988-01-01
Categorical algebra and its applications contain several fundamental papers on general category theory, by the top specialists in the field, and many interesting papers on the applications of category theory in functional analysis, algebraic topology, algebraic geometry, general topology, ring theory, cohomology, differential geometry, group theory, mathematical logic and computer sciences. The volume contains 28 carefully selected and refereed papers, out of 96 talks delivered, and illustrates the usefulness of category theory today as a powerful tool of investigation in many other areas.
Topological insulators and C∗-algebras: Theory and numerical practice
Hastings, Matthew B.; Loring, Terry A.
2011-07-01
We apply ideas from C∗-algebra to the study of disordered topological insulators. We extract certain almost commuting matrices from the free Fermi Hamiltonian, describing band projected coordinate matrices. By considering topological obstructions to approximating these matrices by exactly commuting matrices, we are able to compute invariants quantifying different topological phases. We generalize previous two dimensional results to higher dimensions; we give a general expression for the topological invariants for arbitrary dimension and several symmetry classes, including chiral symmetry classes, and we present a detailed K-theory treatment of this expression for time reversal invariant three dimensional systems. We can use these results to show non-existence of localized Wannier functions for these systems. We use this approach to calculate the index for time-reversal invariant systems with spin-orbit scattering in three dimensions, on sizes up to 12 3, averaging over a large number of samples. The results show an interesting separation between the localization transition and the point at which the average index (which can be viewed as an "order parameter" for the topological insulator) begins to fluctuate from sample to sample, implying the existence of an unsuspected quantum phase transition separating two different delocalized phases in this system. One of the particular advantages of the C∗-algebraic technique that we present is that it is significantly faster in practice than other methods of computing the index, allowing the study of larger systems. In this paper, we present a detailed discussion of numerical implementation of our method.
K-theory of the chair tiling via AF-algebras
Julien, Antoine; Savinien, Jean
2016-08-01
We compute the K-theory groups of the groupoid C∗-algebra of the chair tiling, using a new method. We use exact sequences of Putnam to compute these groups from the K-theory groups of the AF-algebras of the substitution and the induced lower dimensional substitutions on edges and vertices.
New symbolic tools for differential geometry, gravitation, and field theory
Anderson, I. M.; Torre, C. G.
2012-01-01
DifferentialGeometry is a Maple software package which symbolically performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, spinor calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the variational calculus. These capabilities, combined with dramatic recent improvements in symbolic approaches to solving algebraic and differential equations, have allowed for development of powerful new tools for solving research problems in gravitation and field theory. The purpose of this paper is to describe some of these new tools and present some advanced applications involving: Killing vector fields and isometry groups, Killing tensors, algebraic classification of solutions of the Einstein equations, and symmetry reduction of field equations.
DEFF Research Database (Denmark)
Geil, Hans Olav; Matsumoto, Ryutaroh
2009-01-01
We present a new bound on the number of Fq -rational places in an algebraic function field. It uses information about the generators of the Weierstrass semigroup related to a rational place. As we demonstrate, the bound has implications to the theory of towers of function fields....
Geometric Algebras and Extensors
Fernandez, V. V.; Moya, A. M.; Rodrigues Jr., W. A.
2007-01-01
This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of geometrical theories of the gravitational field. In this first paper we introduce the key algebraic tools for the development of our program, namely the euclidean geometrical algebra of multivectors Cl(V,G_{E}) and the theory of its deformations leading to met...
Superspace conformal field theory
Energy Technology Data Exchange (ETDEWEB)
Quella, Thomas [Koeln Univ. (Germany). Inst. fuer Theoretische Physik; Schomerus, Volker [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)
2013-07-15
Conformal sigma models and WZW models on coset superspaces provide important examples of logarithmic conformal field theories. They possess many applications to problems in string and condensed matter theory. We review recent results and developments, including the general construction of WZW models on type I supergroups, the classification of conformal sigma models and their embedding into string theory.
Superspace conformal field theory
International Nuclear Information System (INIS)
Conformal sigma models and WZW models on coset superspaces provide important examples of logarithmic conformal field theories. They possess many applications to problems in string and condensed matter theory. We review recent results and developments, including the general construction of WZW models on type I supergroups, the classification of conformal sigma models and their embedding into string theory.
Supergeometry in locally covariant quantum field theory
Hack, Thomas-Paul; Schenkel, Alexander
2015-01-01
In this paper we analyze supergeometric locally covariant quantum field theories. We develop suitable categories SLoc of super-Cartan supermanifolds, which generalize Lorentz manifolds in ordinary quantum field theory, and show that, starting from a few representation theoretic and geometric data, one can construct a functor A : SLoc --> S*Alg to the category of super-*-algebras which can be interpreted as a non-interacting super-quantum field theory. This construction turns out to disregard supersymmetry transformations as the morphism sets in the above categories are too small. We then solve this problem by using techniques from enriched category theory, which allows us to replace the morphism sets by suitable morphism supersets that contain supersymmetry transformations as their higher superpoints. We construct super-quantum field theories in terms of enriched functors eA : eSLoc --> eS*Alg between the enriched categories and show that supersymmetry transformations are appropriately described within the en...
The Clifford algebra of physical space and Dirac theory
Vaz, Jayme, Jr.
2016-09-01
The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term β \\psi in the usual Dirac factorization of the Klein-Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation.
The Clifford algebra of physical space and Dirac theory
Vaz, Jayme, Jr.
2016-09-01
The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term β \\psi in the usual Dirac factorization of the Klein–Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation.
Light-Front quantization of field theory
Srivastava, P P
1996-01-01
Some basic topics in Light-Front (LF) quantized field theory are reviewed. Poincarè algebra and the LF Spin operator are discussed. The local scalar field theory of the conventional framework is shown to correspond to a non-local Hamiltonian theory on the LF in view of the constraint equations on the phase space, which relate the bosonic condensates to the non-zero modes. This new ingredient is useful to describe the spontaneous symmetry breaking on the LF. The instability of the symmetric phase in two dimensional scalar theory when the coupling constant grows is shown in the LF theory renormalized to one loop order. Chern-Simons gauge theory regarded to describe excitations with fractional statistics, is quantized in the light-cone gauge and a simple LF Hamiltonian obtained which may allow us to construct renormalized theory of anyons.
Light-front quantization of field theory
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Srivastava, Prem P. [Universidade do Estado, Rio de Janeiro, RJ (Brazil). Inst. de Fisica]|[Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, RJ (Brazil)
1996-07-01
Some basic topics in Light-Front (LF) quantized field theory are reviewed. Poincare algebra and the LF spin operator are discussed. The local scalar field theory of the conventional framework is shown to correspond to a non-local Hamiltonian theory on the LF in view of the constraint equations on the phase space, which relate the bosonic condensates to the non-zero modes. This new ingredient is useful to describe the spontaneous symmetry breaking on the LF. The instability of the symmetric phase in two dimensional scalar theory when the coupling constant grows is shown in the LF theory renormalized to one loop order. Chern-Simons gauge theory, regarded to describe excitations with fractional statistics, is quantized in the light-cone gauge and a simple LF Hamiltonian obtained which may allow us to construct renormalized theory of anyons. (author). 20 refs.
Jorgensen, PET
1987-01-01
Historically, operator theory and representation theory both originated with the advent of quantum mechanics. The interplay between the subjects has been and still is active in a variety of areas.This volume focuses on representations of the universal enveloping algebra, covariant representations in general, and infinite-dimensional Lie algebras in particular. It also provides new applications of recent results on integrability of finite-dimensional Lie algebras. As a central theme, it is shown that a number of recent developments in operator algebras may be handled in a particularly e
Shafarevich, I
1994-01-01
This book, the first printing of which was published as volume 38 of the Encyclopaedia of Mathematical Sciences, presents a modern approach to homological algebra, based on the systematic use of the terminology and ideas of derived categories and derived functors. The book contains applications of homological algebra to the theory of sheaves on topological spaces, to Hodge theory, and to the theory of modules over rings of algebraic differential operators (algebraic D-modules). The authors Gelfand and Manin explain all the main ideas of the theory of derived categories. Both authors are well-known researchers and the second, Manin, is famous for his work in algebraic geometry and mathematical physics. The book is an excellent reference for graduate students and researchers in mathematics and also for physicists who use methods from algebraic geometry and algebraic topology.
Bergshoeff, Eric A; Penas, Victor A; Riccioni, Fabio
2016-01-01
We present the dual formulation of double field theory at the linearized level. This is a classically equivalent theory describing the duals of the dilaton, the Kalb-Ramond field and the graviton in a T-duality or O(D,D) covariant way. In agreement with previous proposals, the resulting theory encodes fields in mixed Young-tableau representations, combining them into an antisymmetric 4-tensor under O(D,D). In contrast to previous proposals, the theory also requires an antisymmetric 2-tensor and a singlet, which are not all pure gauge. The need for these additional fields is analogous to a similar phenomenon for "exotic" dualizations, and we clarify this by comparing with the dualizations of the component fields. We close with some speculative remarks on the significance of these observations for the full non-linear theory yet to be constructed.
Elements of mathematics algebra
Bourbaki, Nicolas
2003-01-01
This is a softcover reprint of the English translation of 1990 of the revised and expanded version of Bourbaki's, Algèbre, Chapters 4 to 7 (1981). This completes Algebra, 1 to 3, by establishing the theories of commutative fields and modules over a principal ideal domain. Chapter 4 deals with polynomials, rational fractions and power series. A section on symmetric tensors and polynomial mappings between modules, and a final one on symmetric functions, have been added. Chapter 5 was entirely rewritten. After the basic theory of extensions (prime fields, algebraic, algebraically closed, radical extension), separable algebraic extensions are investigated, giving way to a section on Galois theory. Galois theory is in turn applied to finite fields and abelian extensions. The chapter then proceeds to the study of general non-algebraic extensions which cannot usually be found in textbooks: p-bases, transcendental extensions, separability criterions, regular extensions. Chapter 6 treats ordered groups and fields and...
Combinatorial Algebra for second-quantized Quantum Theory
Blasiak, P; Solomon, A I; Horzela, A; Penson, K A
2010-01-01
We describe an algebra G of diagrams which faithfully gives a diagrammatic representation of the structures of both the Heisenberg-Weyl algebra H - the associative algebra of the creation and annihilation operators of quantum mechanics - and U(L_H), the enveloping algebra of the Heisenberg Lie algebra L_H. We show explicitly how G may be endowed with the structure of a Hopf algebra, which is also mirrored in the structure of U(L_H). While both H and U(L_H) are images of G, the algebra G has a richer structure and therefore embodies a finer combinatorial realization of the creation-annihilation system, of which it provides a concrete model.
Wu, Ning
1998-01-01
In this paper, we will construct a gauge field model, in which the masses of gauge fields are non-zero and the local gauge symmetry is strictly preserved. A SU(N) gauge field model is discussed in details in this paper. In the limit $\\alpha \\longrightarrow 0$ or $\\alpha \\longrightarrow \\infty$, the gauge field model discussed in this paper will return to Yang-Mills gauge field model. This theory could be regarded as theoretical development of Yang-Mills gauge field theory.
Baden Fuller, A J
2014-01-01
Engineering Field Theory focuses on the applications of field theory in gravitation, electrostatics, magnetism, electric current flow, conductive heat transfer, fluid flow, and seepage.The manuscript first ponders on electric flux, electrical materials, and flux function. Discussions focus on field intensity at the surface of a conductor, force on a charged surface, atomic properties, doublet and uniform field, flux tube and flux line, line charge and line sink, field of a surface charge, field intensity, flux density, permittivity, and Coulomb's law. The text then takes a look at gravitation
Dynamical origin of the $\\star_\\theta$-noncommutativity in field theory from quantum mechanics
Rosenbaum, Marcos; Vergara, J. David; Juárez, L. Román
2006-01-01
We show that introducing an extended Heisenberg algebra in the context of the Weyl-Wigner-Groenewold-Moyal formalism leads to a deformed product of the classical dynamical variables that is inherited to the level of quantum field theory, and that allows us to relate the operator space noncommutativity in quantum mechanics to the quantum group inspired algebra deformation noncommutativity in field theory.
Homotopy Theory of Probability Spaces I: Classical independence and homotopy Lie algebras
Park, Jae-Suk
2015-01-01
This is the first installment of a series of papers whose aim is to lay a foundation for homotopy probability theory by establishing its basic principles and practices. The notion of a homotopy probability space is an enrichment of the notion of an algebraic probability space with ideas from algebraic homotopy theory. This enrichment uses a characterization of the laws of random variables in a probability space in terms of symmetries of the expectation. The laws of random variables are reinterpreted as invariants of the homotopy types of infinity morphisms between certain homotopy algebras. The relevant category of homotopy algebras is determined by the appropriate notion of independence for the underlying probability theory. This theory will be both a natural generalization and an effective computational tool for the study of classical algebraic probability spaces, while keeping the same central limit. This article is focused on the commutative case, where the laws of random variables are also described in t...
Algebraic equations an introduction to the theories of Lagrange and Galois
Dehn, Edgar
2004-01-01
Meticulous and complete, this presentation of Galois' theory of algebraic equations is geared toward upper-level undergraduate and graduate students. The theories of both Lagrange and Galois are developed in logical rather than historical form. And they are given a more thorough exposition than is customary. For this reason, and also because the author concentrates on concrete applications of algebraic theory, Algebraic Equations is an excellent supplementary text, offering students a concrete introduction to the abstract principles of Galois theory. Of further value are the many numerical ex
Symmetries in perturbative quantum field theory
International Nuclear Information System (INIS)
The basic point to be developed in this report amounts to prove that general properties of renormalizable lagrangian field theories can be studied only relying on general theorems of renormalization theory, without any reference to a given renormalization scheme. Moreover, most renormalization problems are thus reduced to purely algebraic ones. The first part of this report is concerned with a general introduction to renormalization theory. General theorems, nammely the quantum action principles, are stated there. In the second part, a few explicit problems are treated in order to exhibit the general techniques needed to get all the results stated in the last part
Benschop, Nico F
2009-01-01
""Associative Digital Network Theory"" is intended for researchers at industrial laboratories, teachers and students at technical universities, in electrical engineering, computer science and applied mathematics departments, interested in new developments of modeling and designing digital networks (DN: state machines, sequential and combinational logic) in general, as a combined math/engineering discipline. As background an undergraduate level of modern applied algebra (Birkhoff-Bartee: ""Modern Applied Algebra"" - 1970, and Hartmanis-Stearns: ""Algebraic Structure of Sequential Machines"" - 1
Mandl, Franz
2010-01-01
Following on from the successful first (1984) and revised (1993) editions, this extended and revised text is designed as a short and simple introduction to quantum field theory for final year physics students and for postgraduate students beginning research in theoretical and experimental particle physics. The three main objectives of the book are to: Explain the basic physics and formalism of quantum field theory To make the reader proficient in theory calculations using Feynman diagrams To introduce the reader to gauge theories, which play a central role in elementary particle physic
Algebraic structures, physics and geometry from a Unified Field Theoretical framework
Cirilo-Lombardo, Diego Julio
2014-01-01
Starting from a Unified Field Theory (UFT) proposed previously by the authors, the possible fermionic representations arising from the same spacetime are considered from the algebraic and geometrical viewpoint. We specifically demonstrate in this UFT general context that the underlying basis of the single geometrical structure P (G,M) (the principal fiber bundle over the real spacetime manifold M with structural group G) reflecting the symmetries of the different fields carry naturally a biquaternionic structure instead of a complex one. This fact allows us to analyze algebraically and to interpret physically in a straighforward way the Majorana and Dirac representations and the relation of such structures with the spacetime signature and non-hermitian (CP) dynamic operators. Also, from the underlying structure of the tangent space, the existence of hidden (super) symmetries and the possibility of supersymmetric extensions of these UFT models are given showing that Rothstein's theorem is incomplete for that d...
Strings - Links between conformal field theory, gauge theory and gravity
International Nuclear Information System (INIS)
String theory is a candidate framework for unifying the gauge theories of interacting elementary particles with a quantum theory of gravity. The last years we have made considerable progress in understanding non-perturbative aspects of string theory, and in bringing string theory closer to experiment, via the search for the Standard Model within string theory, but also via phenomenological models inspired by the physics of strings. Despite these advances, many deep problems remain, amongst which a non-perturbative definition of string theory, a better understanding of holography, and the cosmological constant problem. My research has concentrated on various theoretical aspects of quantum theories of gravity, including holography, black holes physics and cosmology. In this Habilitation thesis I have laid bare many more links between conformal field theory, gauge theory and gravity. Most contributions were motivated by string theory, like the analysis of supersymmetry preserving states in compactified gauge theories and their relation to affine algebras, time-dependent aspects of the holographic map between quantum gravity in anti-de-Sitter space and conformal field theories in the bulk, the direct quantization of strings on black hole backgrounds, the embedding of the no-boundary proposal for a wave-function of the universe in string theory, a non-rational Verlinde formula and the construction of non-geometric solutions to supergravity
Arithmetic of Calabi-Yau Varieties and Rational Conformal Field Theory
Schimmrigk, R
2003-01-01
It is proposed that certain techniques from arithmetic algebraic geometry provide a framework which is useful to formulate a direct and intrinsic link between the geometry of Calabi-Yau manifolds and the underlying conformal field theory. Specifically it is pointed out how the algebraic number field determined by the fusion rules of the conformal field theory can be derived from the number theoretic structure of the cohomological Hasse-Weil L-function determined by Artin's congruent zeta function of the algebraic variety. In this context a natural number theoretic characterization arises for the quantum dimensions in this geometrically determined algebraic number field.
Field Equations and Lagrangian for the Kaluza Metric Evaluated with Tensor Algebra Software
Directory of Open Access Journals (Sweden)
L. L. Williams
2015-01-01
Full Text Available This paper calculates the Kaluza field equations with the aid of a computer package for tensor algebra, xAct. The xAct file is provided with this paper. We find that Thiry’s field equations are correct, but only under limited circumstances. The full five-dimensional field equations under the cylinder condition are provided here, and we see that most of the other references miss at least some terms from them. We go on to establish the remarkable Kaluza Lagrangian, and verify that the field equations calculated from it match those calculated with xAct, thereby demonstrating self-consistency of these results. Many of these results can be found scattered throughout the literature, and we provide some pointers for historical purposes. But our intent is to provide a definitive exposition of the field equations of the classical, five-dimensional metric ansatz of Kaluza, along with the computer algebra data file to verify them, and then to recover the unique Lagrangian for the theory. In common terms, the Kaluza theory is an “ω=0” scalar field theory, but with unique electrodynamic couplings.
Forward error correction based on algebraic-geometric theory
A Alzubi, Jafar; M Chen, Thomas
2014-01-01
This book covers the design, construction, and implementation of algebraic-geometric codes from Hermitian curves. Matlab simulations of algebraic-geometric codes and Reed-Solomon codes compare their bit error rate using different modulation schemes over additive white Gaussian noise channel model. Simulation results of Algebraic-geometric codes bit error rate performance using quadrature amplitude modulation (16QAM and 64QAM) are presented for the first time and shown to outperform Reed-Solomon codes at various code rates and channel models. The book proposes algebraic-geometric block turbo codes. It also presents simulation results that show an improved bit error rate performance at the cost of high system complexity due to using algebraic-geometric codes and Chase-Pyndiah’s algorithm simultaneously. The book proposes algebraic-geometric irregular block turbo codes (AG-IBTC) to reduce system complexity. Simulation results for AG-IBTCs are presented for the first time.
Logarithmic conformal field theory: beyond an introduction
Creutzig, Thomas; Ridout, David
2013-12-01
This article aims to review a selection of central topics and examples in logarithmic conformal field theory. It begins with the remarkable observation of Cardy that the horizontal crossing probability of critical percolation may be computed analytically within the formalism of boundary conformal field theory. Cardy’s derivation relies on certain implicit assumptions which are shown to lead inexorably to indecomposable modules and logarithmic singularities in correlators. For this, a short introduction to the fusion algorithm of Nahm, Gaberdiel and Kausch is provided. While the percolation logarithmic conformal field theory is still not completely understood, there are several examples for which the formalism familiar from rational conformal field theory, including bulk partition functions, correlation functions, modular transformations, fusion rules and the Verlinde formula, has been successfully generalized. This is illustrated for three examples: the singlet model \\mathfrak {M} (1,2), related to the triplet model \\mathfrak {W} (1,2), symplectic fermions and the fermionic bc ghost system; the fractional level Wess-Zumino-Witten model based on \\widehat{\\mathfrak {sl}} \\left( 2 \\right) at k=-\\frac{1}{2}, related to the bosonic βγ ghost system; and the Wess-Zumino-Witten model for the Lie supergroup \\mathsf {GL} \\left( 1 {\\mid} 1 \\right), related to \\mathsf {SL} \\left( 2 {\\mid} 1 \\right) at k=-\\frac{1}{2} and 1, the Bershadsky-Polyakov algebra W_3^{(2)} and the Feigin-Semikhatov algebras W_n^{(2)}. These examples have been chosen because they represent the most accessible, and most useful, members of the three best-understood families of logarithmic conformal field theories. The logarithmic minimal models \\mathfrak {W} (q,p), the fractional level Wess-Zumino-Witten models, and the Wess-Zumino-Witten models on Lie supergroups (excluding \\mathsf {OSP} \\left( 1 {\\mid} 2n \\right)). In this review, the emphasis lies on the representation theory
Logarithmic conformal field theory: beyond an introduction
International Nuclear Information System (INIS)
This article aims to review a selection of central topics and examples in logarithmic conformal field theory. It begins with the remarkable observation of Cardy that the horizontal crossing probability of critical percolation may be computed analytically within the formalism of boundary conformal field theory. Cardy’s derivation relies on certain implicit assumptions which are shown to lead inexorably to indecomposable modules and logarithmic singularities in correlators. For this, a short introduction to the fusion algorithm of Nahm, Gaberdiel and Kausch is provided. While the percolation logarithmic conformal field theory is still not completely understood, there are several examples for which the formalism familiar from rational conformal field theory, including bulk partition functions, correlation functions, modular transformations, fusion rules and the Verlinde formula, has been successfully generalized. This is illustrated for three examples: the singlet model M(1,2), related to the triplet model W(1,2), symplectic fermions and the fermionic bc ghost system; the fractional level Wess–Zumino–Witten model based on sl-hat (2) at k=−(1/2), related to the bosonic βγ ghost system; and the Wess–Zumino–Witten model for the Lie supergroup GL(1∣1), related to SL(2∣1) at k=−(1/2) and 1, the Bershadsky–Polyakov algebra W3(2) and the Feigin–Semikhatov algebras Wn(2). These examples have been chosen because they represent the most accessible, and most useful, members of the three best-understood families of logarithmic conformal field theories. The logarithmic minimal models W(q,p), the fractional level Wess–Zumino–Witten models, and the Wess–Zumino–Witten models on Lie supergroups (excluding OSP(1∣2n)). In this review, the emphasis lies on the representation theory of the underlying chiral algebra and the modular data pertaining to the characters of the representations. Each of the archetypal logarithmic conformal field theories is
Effective quantum field theories
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Certain dimensional parameters play a crucial role in the understanding of weak and strong interactions based on SU(2) x U(1) and SU(3) symmetry group theories and of grand unified theories (GUT's) based on SU(5). These parameters are the confinement scale of quantum chromodynamics and the breaking scales of SU(2) x U(1) and SU(5). The concepts of effective quantum field theories and renormalisability are discussed with reference to the economics and ethics of research. (U.K.)
Supersymmetric gauge theories with a free algebra of invariants
Dotti, Gustavo; Manohar, Aneesh V.(Department of Physics, University of California at San Diego, La Jolla, CA 92093, United States); Skiba, Witold
1998-01-01
We study the low-energy dynamics of all N=1 supersymmetric gauge theories whose basic gauge invariant fields are unconstrained. This set includes all theories whose matter Dynkin index is less than the index of the adjoint representation. We study the dynamically generated superpotential in these theories, and show that there is a W=0 branch if and only if anomaly matching is satisfied at the origin. An interesting example studied in detail is SO(13) with a spinor, a theory with a dynamically...
Filtrated K-theory for real rank zero C*-algebras
DEFF Research Database (Denmark)
Arklint, Sara Esther; Restorff, Gunnar; Ruiz, Efren
2012-01-01
The smallest primitive ideal spaces for which there exist counterexamples to the classification of non-simple, purely infinite, nuclear, separable C*-algebras using filtrated K-theory, are four-point spaces. In this article, we therefore restrict to real rank zero C*-algebras with four-point prim...
Matrix algebra and sampling theory : The case of the Horvitz-Thompson estimator
Dol, W.; Steerneman, A.G.M.; Wansbeek, T.J.
1996-01-01
Matrix algebra is a tool not commonly employed in sampling theory. The intention of this paper is to help change this situation by showing, in the context of the Horvitz-Thompson (HT) estimator, the convenience of the use of a number of matrix-algebra results. Sufficient conditions for the consisten
Fundamental problems of gauge field theory
International Nuclear Information System (INIS)
As a result of the experimental and theoretical developments of the last two decades, gauge field theory, in one form or another, now provides the standard language for the description of Nature; QCD and the standard model of the electroweak interactions illustrate this point. It is a basic task of mathematical physics to provide a solid foundation for these developments by putting the theory in a physically transparent and mathematically rigorous form. The lecture notes collected in this volume concentrate on the many unsolved problems which arise here, and on the general ideas and methods which have been proposed for their solution. In particular, the use of rigorous renormalization group methods to obtain control over the continuum limit of lattice gauge field theories, the exploration of the extraordinary enigmatic connections between Kac-Moody-Virasoro algebras and string theory, and the systematic use of the theory of local algebras and indefinite metric spaces to classify the charged C* states in gauge field theories are mentioned
Non-local potentials with LS terms in algebraic scattering theory
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Levay, Peter [Department of Theoretical Physics, Institute of Physics, Technical University of Budapest, Budapest (Hungary)
1997-10-21
The group theoretical analysis of Coulomb scattering based on the SO(3,1) group is revisited. Using matrix-valued differential operators, modifying the angular momentum and the Runge-Lenz vector used hitherto for the realization of the so(3,1) (Lorentz) algebra, we obtain a three-dimensional solvable two-channel scattering problem. The interaction term besides the Coulomb potential contains a non-local potential of LS-type. Using the momentum representation the S-matrix can be calculated analytically. By employing a canonical transformation, another solvable three-dimensional scattering problem is found, in agreement with the expectations of algebraic scattering theory. The potential in this case is of Poeschl-Teller type with an LS term. It is also pointed out that our matrix-valued realization of the so(3,1) algebra can be cast to an instructive form with the help of su(2) gauge fields. An interesting connection between gauge transformations and supersymmetry transformations of supersymmetric quantum mechanics is also observed. These results enable us to construct other solvable scattering problems by using su(2) gauge transformations. (author)
Geometric representation of interval exchange maps over algebraic number fields
Poggiaspalla, G.; Lowenstein, J. H.; Vivaldi, F.
2008-01-01
This paper is concerned with the restriction of interval exchange transformations (IETs) to algebraic number fields, which leads to maps on lattices. We characterize renormalizability arithmetically and study its relationships with a geometrical quantity that we call the drift vector. We exhibit some examples of renormalizable IETs with zero and non-zero drift vectors and carry out some investigations of their properties. In particular we look for evidence of the finite decomposition property on a family of IETs extending the example studied in Lowenstein et al (2007 Dyn. Syst. 22 73-106).
Finite temperature field theory
Das, Ashok
1997-01-01
This book discusses all three formalisms used in the study of finite temperature field theory, namely the imaginary time formalism, the closed time formalism and thermofield dynamics. Applications of the formalisms are worked out in detail. Gauge field theories and symmetry restoration at finite temperature are among the practical examples discussed in depth. The question of gauge dependence of the effective potential and the Nielsen identities are explained. The nonrestoration of some symmetries at high temperature (such as supersymmetry) and theories on nonsimply connected space-times are al
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The superconformal algebras of meromorphic vector fields with multipoles, the central extension and the relevant abelian differential of the third kind on the super-Riemann sphere are constructed. The background of our theory concerns with the interaction of closed superstrings. (orig.)
Hopf algebra structures in particle physics
Weinzierl, Stefan
2004-01-01
In the recent years, Hopf algebras have been introduced to describe certain combinatorial properties of quantum field theories. I will give a basic introduction to these algebras and review some occurrences in particle physics.
Wentzel, Gregor
2003-01-01
A prominent figure in twentieth-century physics, Gregor Wentzel made major contributions to the development of quantum field theory, first in Europe and later at the University of Chicago. His Quantum Theory of Fields offers a knowledgeable view of the original literature of elementary quantum mechanics and helps make these works accessible to interested readers.An introductory volume rather than an all-inclusive account, the text opens with an examination of general principles, without specification of the field equations of the Lagrange function. The following chapters deal with particular
K-theory of locally finite graph C∗-algebras
Iyudu, Natalia
2013-09-01
We calculate the K-theory of the Cuntz-Krieger algebra OE associated with an infinite, locally finite graph, via the Bass-Hashimoto operator. The formulae we get express the Grothendieck group and the Whitehead group in purely graph theoretic terms. We consider the category of finite (black-and-white, bi-directed) subgraphs with certain graph homomorphisms and construct a continuous functor to abelian groups. In this category K0 is an inductive limit of K-groups of finite graphs, which were calculated in Cornelissen et al. (2008) [3]. In the case of an infinite graph with the finite Betti number we obtain the formula for the Grothendieck group K0(OE)=Z, where β(E) is the first Betti number and γ(E) is the valency number of the graph E. We note that in the infinite case the torsion part of K0, which is present in the case of a finite graph, vanishes. The Whitehead group depends only on the first Betti number: K1(OE)=Z. These allow us to provide a counterexample to the fact, which holds for finite graphs, that K1(OE) is the torsion free part of K0(OE).
Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras
Davison, Ben
2016-01-01
This paper is a companion paper to 1512.08898, on the general definition of Donaldson--Thomas invariants for Jacobi algebras, or equivalently, the integrality conjecture for such algebras. In this paper we concentrate on the Hodge-theoretic aspects of the theory, and explore the structure of the Cohomological Hall algebra associated to a quiver and potential, introduced by Kontsevich and Soibelman. Via a study of the representation theory of these algebras, we introduce a perverse filtration on them, and prove that they are quantum enveloping algebras, for which the integrality theorem, and the wall crossing theorem relating DT invariants for different Bridgeland stability conditions, are a K-theoretic shadow of the existence of PBW bases.
Transformations among large c conformal field theories
Jankiewicz, M; Jankiewicz, Marcin; Kephart, Thomas W.
2005-01-01
We show that there is a set of transformations that relates all of the 24 dimensional even self-dual (Niemeier) lattices, and also leads to non-lattice objects that cannot be used as a compactification torus. We extend our observations to higher dimensional conformal field theories where we generate c=24k theories with spectra decomposable into the irreducible representations of the Fischer-Griess Monster. We observe interesting periodicities in the coefficients of of extremal partition functions and characters of the extremal vertex operator algebras.
Energy Technology Data Exchange (ETDEWEB)
Foussats, A.; Laura, R.; Zandron, O.
1986-09-01
Supergravity theories on a free differential algebra are examined and the factorisation condition is imposed leading to factorised solutions. H-gauge transformations for the pseudo-connections and pseudo-curvatures are also deduced.
International Nuclear Information System (INIS)
Supergravity theories on a free differential algebra are examined and the factorisation condition is imposed leading to factorised solutions. H-gauge transformations for the pseudo-connections and pseudo-curvatures are also deduced. (author)
Perturbative quantization of Yang-Mills theory with classical double as gauge algebra
Energy Technology Data Exchange (ETDEWEB)
Ruiz Ruiz, F. [Universidad Complutense de Madrid, Departamento de Fisica Teorica I, Madrid (Spain)
2016-02-15
Perturbative quantization of Yang-Mills theory with a gauge algebra given by the classical double of a semisimple Lie algebra is considered. The classical double of a real Lie algebra is a nonsemisimple real Lie algebra that admits a nonpositive definite invariant metric, the indefiniteness of the metric suggesting an apparent lack of unitarity. It is shown that the theory is UV divergent at one loop and that there are no radiative corrections at higher loops. One-loop UV divergences are removed through renormalization of the coupling constant, thus introducing a renormalization scale. The terms in the classical action that would spoil unitarity are proved to be cohomologically trivial with respect to the Slavnov-Taylor operator that controls gauge invariance for the quantum theory. Hence they do not contribute gauge invariant radiative corrections to the quantum effective action and the theory is unitary. (orig.)
Extensions of conformal symmetry in two-dimensional quantum field theory
International Nuclear Information System (INIS)
Conformal symmetry extensions in a two-dimensional quantum field theory are the main theme of the work presented in this thesis. After a brief exposition of the formalism for conformal field theory, the motivation for studying extended symmetries in conformal field theory is presented in some detail. Supersymmetric extensions of conformal symmetry are introduced. An overview of the algebraic superconformal symmetry is given. The relevance of higher-spin bosonic extensions of the Virasoro algebra in relation to the classification program for so-called rational conformal theories is explained. The construction of a large class of bosonic extended algebras, the so-called Casimir algebras, are presented. The representation theory of these algebras is discussed and a large class of new unitary models is identified. The superspace formalism for O(N)-extended superconformal quantum field theory is presented. It is shown that such theories exist for N ≤ 4. Special attention is paid to the case N = 4 and it is shown that the allowed central charges are c(n+,n-) = 6n+n-/(n+,n-), where n+ and n- are positive integers. A different class of so(N)-extended superconformal algebras is analyzed. The representation theory is studied and it is established that certain free field theories provide realizations of the algebras with level S = 1. Finally the so-called BRST construction for extended conformal algebras is considered. A nilpotent BRST charge is constructed for a large class of algebras, which contains quadratically nonlinear algebras that fall outside the traditional class if finitely generated Lie (super)algebras. The results are especially relevant for the construction of string models based on extended conformal symmetry. (author). 118 refs.; 7 tabs
Algebraic K-theory and derived equivalences suggested by T-duality for torus orientifolds
Rosenberg, Jonathan
2016-01-01
We show that certain isomorphisms of (twisted) KR-groups that underlie T-dualities of torus orientifold string theories have purely algebraic analogues in terms of algebraic K-theory of real varieties and equivalences of derived categories of (twisted) coherent sheaves. The most interesting conclusion is a kind of Mukai duality in which the "dual abelian variety" to a smooth projective genus-1 curve over R with no real points is (mildly) noncommutative.
Solomon, Alan D
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Boolean Algebra includes set theory, sentential calculus, fundamental ideas of Boolean algebras, lattices, rings and Boolean algebras, the structure of a Boolean algebra, and Boolean
Superconformal field theories and cyclic homology
Eager, Richard
2015-01-01
One of the predictions of the AdS/CFT correspondence is the matching of protected operators between a superconformal field theory and its holographic dual. We review the spectrum of protected operators in quiver gauge theories that flow to superconformal field theories at low energies. The spectrum is determined by the cyclic homology of an algebra associated to the quiver gauge theory. Identifying the spectrum of operators with cyclic homology allows us to apply the Hochschild-Kostant-Rosenberg theorem to relate the cyclic homology groups to deRham cohomology groups. The map from cyclic homology to deRham cohomology can be viewed as a mathematical avatar of the passage from open to closed strings under the AdS/CFT correspondence.
Picard–Vessiot extensions of artinian simple module algebras
Amano, Katsutoshi; Masuoka, Akira
2004-01-01
This paper pursues Takeuchi's Hopf algebraic approach [M. Takeuchi, A Hopf algebraic approach to the Picard–Vessiot theory, J. Algebra 122 (1989) 481–509] to the Picard–Vessiot (PV) theory for differential equations, to involve the PV extensions of difference equations. Differential fields and total difference rings in the standard PV theory are unified here by artinian simple (AS) module algebras over a cocommutative, pointed smooth Hopf algebra.
A Jacobi identity for intertwining operator algebras
Huang, Y Z
1997-01-01
We find a Jacobi identity for intertwining operator algebras. Most of the main properties of genus-zero conformal field theories, including the main properties of vertex operator algebras, modules, intertwining operators, Verlinde algebras, and fusing and braiding matrices, are incorporated into this identity. We prove that intertwining operators for a suitable vertex operator algebra satisfy this Jacobi identity. Two equivalent definitions of intertwining operator algebra in terms of this Jacobi identity are given.
You, Setthivoine
2015-11-01
A new canonical field theory has been developed to help interpret the interaction between plasma flows and magnetic fields. The theory augments the Lagrangian of general dynamical systems to rigourously demonstrate that canonical helicity transport is valid across single particle, kinetic and fluid regimes, on scales ranging from classical to general relativistic. The Lagrangian is augmented with two extra terms that represent the interaction between the motion of matter and electromagnetic fields. The dynamical equations can then be re-formulated as a canonical form of Maxwell's equations or a canonical form of Ohm's law valid across all non-quantum regimes. The field theory rigourously shows that helicity can be preserved in kinetic regimes and not only fluid regimes, that helicity transfer between species governs the formation of flows or magnetic fields, and that helicity changes little compared to total energy only if density gradients are shallow. The theory suggests a possible interpretation of particle energization partitioning during magnetic reconnection as canonical wave interactions. This work is supported by US DOE Grant DE-SC0010340.
Higher spin double field theory: a proposal
Bekaert, Xavier; Park, Jeong-Hyuck
2016-07-01
We construct a double field theory coupled to the fields present in Vasiliev's equations. Employing the "semi-covariant" differential geometry, we spell a functional in which each term is completely covariant with respect to O(4, 4) T-duality, doubled diffeomorphisms, Spin(1, 3) local Lorentz symmetry and, separately, HS(4) higher spin gauge symmetry. We identify a minimal set of BPS-like conditions whose solutions automatically satisfy the full Euler-Lagrange equations. As such a solution, we derive a linear dilaton vacuum. With extra algebraic constraints further supplemented, the BPS-like conditions reduce to the bosonic Vasiliev equations.
An assessment of Evans' unified field theory II
Hehl, F W; Hehl, Friedrich W.; Obukhov, Yuri N.
2007-01-01
Evans developed a classical unified field theory of gravitation and electromagnetism on the background of a spacetime obeying a Riemann-Cartan geometry. In an accompanying paper I, we analyzed this theory and summarized it in nine equations. We now propose a variational principle for Evans' theory and show that it yields two field equations. The second field equation is algebraic in the torsion and we can resolve it with respect to the torsion. It turns out that for all physical cases the torsion vanishes and the first field equation, together with Evans' unified field theory, collapses to an ordinary Einstein equation.
Modular Structure and Duality in Conformal Quantum Field Theory
Brunetti, R; Longo, R
1993-01-01
Making use of a recent result of Borchers, an algebraic version of the Bisognano-Wichmann theorem is given for conformal quantum field theories, i.e. the Tomita-Takesaki modular group associated with the von Neumann algebra of a wedge region and the vacuum vector concides with the evolution given by the rescaled pure Lorentz transformations preserving the wedge. A similar geometric description is valid for the algebras associated with double cones. Moreover essential duality holds on the Minkowski space $M$, and Haag duality for double cones holds provided the net of local algebras is extended to a pre-cosheaf on the superworld $\\tilde M$, i.e. the universal covering of the Dirac-Weyl compactification of $M$. As a consequence a PCT symmetry exists for any algebraic conformal field theory in even space-time dimension. Analogous results hold for a Poincaré covariant theory provided the modular groups corresponding to wedge algebras have the expected geometrical meaning and the split property is satisfied. In p...
Modular structure and duality in conformal quantum field theory
International Nuclear Information System (INIS)
Making use of a recent result of Borchers, an algebraic version of the Bisognano-Wichmann theorem is given for conformal quantum field theories, i.e. the Tomita-Takesaki modular group associated with the von Neumann algebra of a wedge region and vacuum vector coincides with the evolution given by the rescaled pure Lorentz transformations preserving the wedge. A similar geometric description is valid for the algebras associated with double cones. Moreover essential duality holds on the Minkowski space M, and Haag duality for double cones holds provided the net of local algebras is extended to a pre-cosheaf on the superworld anti M, i.e. the universal covering of the Dirac-Weyl compactification of M. As a consequence a PCT symmetry exists for any algebraic conformal field theory in even spacetime dimension. Analogous results hold for a Poincare covariant theory provided the modular groups corresponding to wedge algebras have the expected geometrical meaning and the split property is satisfied. In particular the Poincare representation is unique in this case. (orig.)
Modular structure and duality in conformal quantum field theory
Energy Technology Data Exchange (ETDEWEB)
Brunetti, R. (Dipt. di Fisica, Univ. di Napoli ' ' Federico II' ' (Italy)); Guido, D. (Rome-2 Univ. (Italy). Dipt. di Matematica); Longo, R. (Rome-2 Univ. (Italy). Dipt. di Matematica)
1993-09-01
Making use of a recent result of Borchers, an algebraic version of the Bisognano-Wichmann theorem is given for conformal quantum field theories, i.e. the Tomita-Takesaki modular group associated with the von Neumann algebra of a wedge region and vacuum vector coincides with the evolution given by the rescaled pure Lorentz transformations preserving the wedge. A similar geometric description is valid for the algebras associated with double cones. Moreover essential duality holds on the Minkowski space M, and Haag duality for double cones holds provided the net of local algebras is extended to a pre-cosheaf on the superworld anti M, i.e. the universal covering of the Dirac-Weyl compactification of M. As a consequence a PCT symmetry exists for any algebraic conformal field theory in even spacetime dimension. Analogous results hold for a Poincare covariant theory provided the modular groups corresponding to wedge algebras have the expected geometrical meaning and the split property is satisfied. In particular the Poincare representation is unique in this case. (orig.)
Zeidler, Eberhard
This is the first volume of a modern introduction to quantum field theory which addresses both mathematicians and physicists ranging from advanced undergraduate students to professional scientists. The book tries to bridge the existing gap between the different languages used by mathematicians and physicists. For students of mathematics it is shown that detailed knowledge of the physical background helps to motivate the mathematical subjects and to discover interesting interrelationships between quite different mathematical topics. For students of physics, fairly advanced mathematics is presented, which is beyond the usual curriculum in physics. It is the author's goal to present the state of the art of realizing Einstein's dream of a unified theory for the four fundamental forces in the universe (gravitational, electromagnetic, strong, and weak interaction). From the reviews: "… Quantum field theory is one of the great intellectual edifices in the history of human thought. … This volume differs from othe...
Eringen, A Cemal
1999-01-01
Microcontinuum field theories constitute an extension of classical field theories -- of elastic bodies, deformations, electromagnetism, and the like -- to microscopic spaces and short time scales. Material bodies are here viewed as collections of large numbers of deformable particles, much as each volume element of a fluid in statistical mechanics is viewed as consisting of a large number of small particles for which statistical laws are valid. Classical continuum theories are valid when the characteristic length associated with external forces or stimuli is much larger than any internal scale of the body under consideration. When the characteristic lengths are comparable, however, the response of the individual constituents becomes important, for example, in considering the fluid or elastic properties of blood, porous media, polymers, liquid crystals, slurries, and composite materials. This volume is concerned with the kinematics of microcontinua. It begins with a discussion of strain, stress tensors, balanc...
Gurau, R; Rivasseau, V
2008-01-01
We propose a new formalism for quantum field theory which is neither based on functional integrals, nor on Feynman graphs, but on marked trees. This formalism is constructive, i.e. it computes correlation functions through convergent rather than divergent expansions. It applies both to Fermionic and Bosonic theories. It is compatible with the renormalization group, and it allows to define non-perturbatively {\\it differential} renormalization group equations. It accommodates any general stable polynomial Lagrangian. It can equally well treat noncommutative models or matrix models such as the Grosse-Wulkenhaar model. Perhaps most importantly it removes the space-time background from its central place in QFT, paving the way for a nonperturbative definition of field theory in noninteger dimension.
Gurau, Razvan
2009-01-01
Group field theories are higher dimensional generalizations of matrix models. Their Feynman graphs are fat and in addition to vertices, edges and faces, they also contain higher dimensional cells, called bubbles. In this paper, we propose a new, fermionic Group Field Theory, posessing a color symmetry, and take the first steps in a systematic study of the topological properties of its graphs. Unlike its bosonic counterpart, the bubbles of the Feynman graphs of this theory are well defined and readily identified. We prove that this graphs are combinatorial cellular complexes. We define and study the cellular homology of this graphs. Furthermore we define a homotopy transformation appropriate to this graphs. Finally, the amplitude of the Feynman graphs is shown to be related to the fundamental group of the cellular complex.
On the Algebraic Structure of Higher-Spin Field Equations and New Exact Solutions
Iazeolla, Carlo
2008-01-01
This Thesis reviews Vasiliev's approach to Higher-Spin Gauge Theory and contains some original results concerning new exact solutions of the Vasiliev equations and the representation theory of the higher-spin algebra. The review part covers the various formulations of the free theory as well as Vasiliev's full nonlinear equations, in particular focusing on their algebraic structure and on their properties in various space-time signatures. Then, the original results are presented. First, the 4D Vasiliev equations are formulated in space-times with signatures (4-p,p) and non-vanishing cosmological constant, and some new exact solutions are found, depending on continuous and discrete parameters: (a) an SO(4-p,p)-invariant family of solutions; (b) non-maximally symmetric solutions with vanishing Weyl tensors and higher-spin gauge fields, that differ from the maximally symmetric background solutions in the auxiliary field sector; and (c) solutions of the chiral models with an infinite tower of Weyl tensors proport...
Topological Field Theory and Matrix Product States
Kapustin, Anton; You, Minyoung
2016-01-01
It is believed that most (perhaps all) gapped phases of matter can be described at long distances by Topological Quantum Field Theory (TQFT). On the other hand, it has been rigorously established that in 1+1d ground states of gapped Hamiltonians can be approximated by Matrix Product States (MPS). We show that the state-sum construction of 2d TQFT naturally leads to MPS in their standard form. In the case of systems with a global symmetry G, this leads to a classification of gapped phases in 1+1d in terms of Morita-equivalence classes of G-equivariant algebras. Non-uniqueness of the MPS representation is traced to the freedom of choosing an algebra in a particular Morita class. In the case of Short-Range Entangled phases, we recover the group cohomology classification of SPT phases.
Polarization-free Quantization of Linear Field Theories
Lanéry, Suzanne
2016-01-01
It is well-known that there exist infinitely-many inequivalent representations of the canonical (anti)-commutation relations of Quantum Field Theory (QFT). A way out, suggested by Algebraic QFT, is to instead define the quantum theory as encompassing all possible (abstract) states. In the present paper, we describe a quantization scheme for general linear (aka. free) field theories that can be seen as intermediate between traditional Fock quantization and full Algebraic QFT, in the sense that: * it provides a constructive, explicit description of the resulting space of quantum states; * it does not require the choice of a polarization, aka. the splitting of classical solutions into positive vs. negative-frequency modes: in fact, any Fock representation corresponding to a "reasonable" choice of polarization is naturally embedded; * it supports the implementation of a "large enough" class of linear symplectomorphisms of the classical, infinite-dimensional phase space. The proposed quantization (like Algebraic Q...
International Nuclear Information System (INIS)
We describe the construction of a class of cubic gauge-invariant actions for superstring field theory, and the gauge-fixing of one representative. Fermion string fields are taken in the -1/2-picture and boson string fields in the 0-picture, which makes a picture-changing insertion carrying picture number -2 necessary. The construction of all such operators is outlined. We discuss the gauge b1 + b-1 = 0, in which the action formally linearizes. Nontrivial scattering amplitudes are obtained by approaching this gauge as a limit. 20 refs
Natural discretization in noncommutative field theory
Energy Technology Data Exchange (ETDEWEB)
Acatrinei, Ciprian Sorin, E-mail: acatrine@theory.nipne.ro [Department of Theoretical Physics, Horia Hulubei National Institute for Nuclear Physics, Bucharest (Romania)
2015-12-07
A discretization scheme for field theory is developed, in which the space time coordinates are assumed to be operators forming a noncommutative algebra. Generic waves without rotational symmetry are studied in (2+1) - dimensional scalar field theory with Heisenberg-type noncommutativity. In the representation chosen, the radial coordinate is naturally rendered discrete. Nonlocality along this coordinate, induced by noncommutativity, accounts for the angular dependence of the fields. A complete solution and the interpretation of its nonlocal features are given. The exact form of standing and propagating waves on such a discrete space is found in terms of finite series. A precise correspondence is established between the degree of nonlocality and the angular momentum of a field configuration. At small distance no classical singularities appear, even at the location of the sources. At large radius one recovers the usual commutative/continuum behaviour.
Izhakian, Zur; Rowen, Louis
2008-01-01
We develop the algebraic polynomial theory for "supertropical algebra," as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of "ghost elements," which also play the key role in our structure theory. Here, we work somewhat more generally over an ordered monoid, and develop a theory which contains the analogs of several basic theorems of classical commutative algebra. This structure enables one to develop a Zariski-type algebraic geomet...
On the general theory of quantized fields
International Nuclear Information System (INIS)
In my lecture I describe the present stage of the general theory of quantized fields on the example of 5 subjects. They are ordered in the direction from large to small distances. The first one is the by now classical problem of the structure of superselection sectors. It involves the behavior of the theory at spacelike infinity and is directly connected with particle statistics and internal symmetries. It has become popular in recent years by the discovery of a lot of nontrivial models in 2d conformal-field theory, by connections to integrable models and critical behavior in statistical mechanics and by the relations to the Jones' theory of subfactors in von Neumann algebras and to the corresponding geometrical objects (braids, knots, 3d manifolds, ...). At large timelike distances the by far most important feature of quantum field theory is the particle structure. This will be the second subject of my lecture. It follows the technically most involved part which is concerned with the behavior at finite distances. Two aspets, nuclearity which emphasizes the finite density of states in phase space, and the modular structure which relies on the infinite number of degrees of freedom present even locally, and their mutual relations will be treated. The next point, involving the structure at infinitesimal distances, is the connection between the Haag-Kastler framework of algebras of local and the framework of Wightman fields. Finally, problems in approaches to quantum gravity will be discussed, as far as they are accessible by the methods of the general theory of quantized fields. (orig.)
The algebraic structure of the Onsager algebra
DATE, ETSURO; Roan, Shi-shyr
2000-01-01
We study the Lie algebra structure of the Onsager algebra from the ideal theoretic point of view. A structure theorem of ideals in the Onsager algebra is obtained with the connection to the finite-dimensional representations. We also discuss the solvable algebra aspect of the Onsager algebra through the formal Lie algebra theory.
(α,β)-fuzzy Lie algebras over an (α,β)-fuzzy field
Antony, P. L.; Lilly, P. L.
2010-01-01
The concept of (α,β)-fuzzy Lie algebras over an (α,β)-fuzzy field is introduced. We provide characterizations of an $(\\in,\\in\\vee q)$-fuzzy Lie algebra over an $(\\in,\\in\\vee q)$-fuzzy field.
Quantum Algorithms for Problems in Number Theory, Algebraic Geometry, and Group Theory
van Dam, Wim; Sasaki, Yoshitaka
2013-09-01
Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same problem appears to be intractable on classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article will review the current state of quantum algorithms, focusing on algorithms for problems with an algebraic flavor that achieve an apparent superpolynomial speedup over classical computation.
Kim, S; Yee, H U; Kim, Seok; Lee, Ki-Myeong; Yee, Ho-Ung
2006-01-01
To a domain wall or string object, Noether charge and topological spatial objects can be attracted, forming a composite BPS (Bogomolny-Prasad-Sommerfield) object. We consider two field theories and derive a new BPS bound on composite linear solitons involving multiple charges. Among the BPS objects `supertubes' appear when the wall or string tension is canceled by the bound energy, and could take an arbitrary closed curve. In our theories, supertubes manifest as Chern-Simons solitons, dyonic instantons, charged semi-local vortices, and dyonic instantons on vortex flux sheet.
Holographic effective field theories
Martucci, Luca; Zaffaroni, Alberto
2016-06-01
We derive the four-dimensional low-energy effective field theory governing the moduli space of strongly coupled superconformal quiver gauge theories associated with D3-branes at Calabi-Yau conical singularities in the holographic regime of validity. We use the dual supergravity description provided by warped resolved conical geometries with mobile D3-branes. Information on the baryonic directions of the moduli space is also obtained by using wrapped Euclidean D3-branes. We illustrate our general results by discussing in detail their application to the Klebanov-Witten model.
Villarreal, Rafael
2015-01-01
The book stresses the interplay between several areas of pure and applied mathematics, emphasizing the central role of monomial algebras. It unifies the classical results of commutative algebra with central results and notions from graph theory, combinatorics, linear algebra, integer programming, and combinatorial optimization. The book introduces various methods to study monomial algebras and their presentation ideals, including Stanley-Reisner rings, subrings and blowup algebra-emphasizing square free quadratics, hypergraph clutters, and effective computational methods.
The inverse problem of differential Galois theory over the field R(z)
Dyckerhoff, Tobias
2008-01-01
We describe a Picard-Vessiot theory for differential fields with non algebraically closed fields of constants. As a technique for constructing and classifying Picard-Vessiot extensions, we develop a Galois descent theory. We utilize this theory to prove that every linear algebraic group $G$ over $\\mathbb{R}$ occurs as a differential Galois group over $\\mathbb{R}(z)$. The main ingredient of the proof is the Riemann-Hilbert correspondence for regular singular differential equations over $\\mathb...
The Galois Correspondence in Field Algebra of G-spin Model
Institute of Scientific and Technical Information of China (English)
Li Ning JIANG; Mao Zheng GUO
2005-01-01
Suppose that G is a finite group and D(G) the double algebra of G. For a given subgroup H of G, there is a sub-Hopf algebra D(G; H) of D(G). This paper gives the concrete construction of a D(G; H)-invariant subspace (A)H in field algebra of G-spin model and proves that if H is a normal subgroup of G, then (A)H is Galois closed.
Gravitational fields with a non Abelian bidimensional Lie algebra of symmetries
Sparano, G; Vinogradov, A M
2001-01-01
Vacuum gravitational fields invariant for a bidimensional non Abelian Lie algebra of Killing fields, are explicitly described. They are parameterized either by solutions of a transcendental equation (the tortoise equation) or by solutions of a linear second order differential equation on the plane. Gravitational fields determined via the tortoise equation, are invariant for a 3-dimensional Lie algebra of Killing fields with bidimensional leaves. Global gravitational fields out of local ones are also constructed.
Setare, M R
2016-01-01
In this paper we study the near horizon symmetry algebra of the non-extremal black hole solutions of the Chern-Simons-like theories of gravity, which are stationary but are not necessarily spherically symmetric. We define the extended off-shell ADT current which is an extension of the generalized ADT current. We use the extended off-shell ADT current to define quasi-local conserved charges such that they are conserved for Killing vectors and asymptotically Killing vectors which depend on dynamical fields of the considered theory. We apply this formalism to the Generalized Minimal Massive Gravity( GMMG) and obtain conserved charges of a spacetime which describes near horizon geometry of non-extremal black holes. Eventually, we find the algebra of conserved charges in Fourier modes. It is interesting that, similar to the Einstein gravity in the presence of negative cosmological constant, for the GMMG model also we obtain the Heisenberg algebra as the near horizon symmetry algebra of the black flower solutions. ...
Exploiting the Structure of Bipartite Graphs for Algebraic and Spectral Graph Theory Applications
Kunegis, Jérôme
2014-01-01
In this article, we extend several algebraic graph analysis methods to bipartite networks. In various areas of science, engineering and commerce, many types of information can be represented as networks, and thus the discipline of network analysis plays an important role in these domains. A powerful and widespread class of network analysis methods is based on algebraic graph theory, i.e., representing graphs as square adjacency matrices. However, many networks are of a very specific form that...
Taormina, Anne
1993-05-01
The representation theory of the doubly extended N=4 superconformal algebra is reviewed. The modular properties of the corresponding characters can be derived, using characters sumrules for coset realizations of these N=4 algebras. Some particular combinations of massless characters are shown to transform as affine SU(2) characters under S and T, a fact used to completely classify the massless sector of the partition function.
Kodera, Ryosuke
2016-01-01
We study quantized Coulomb branches of quiver gauge theories of Jordan type. We prove that the quantized Coulomb branch is isomorphic to the spherical graded Cherednik algebra in the unframed case, and is isomorphic to the spherical cyclotomic rational Cherednik algebra in the framed case. We also prove that the quantized Coulomb branch is a deformation of a subquotient of the Yangian of the affine $\\mathfrak{gl}(1)$.
Alvarez-Gaumé, Luís
1996-01-01
Quantum Field Theory provides the most fundamental language known to express the fundamental laws of Nature. It is the consequence of trying to describe physical phenomena within the conceptual framework of Quantum Mechanics and Special Relativity. The aim of these lectures will be to present a number of concepts and methods in the subject which many of us find difficult to understand. They may include (depending on time) : the need to introduce quantum fields, the realization of symmetries, the renormalization group, non-perturbative phenomena, infrared divergences and jets, etc. Some familiarity with the rudiments of Feynman diagrams and relativistic quantum mechanics will be appreciated.
KK-theory and Spectral Flow in von Neumann Algebras
DEFF Research Database (Denmark)
Kaad, Jens; Nest, Ryszard; Rennie, Adam
2007-01-01
We present a definition of spectral flow relative to any norm closed ideal J in any von Neumann algebra N. Given a path D(t) of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in K_0(J). In the case when N is semifinite, the numerical spectral flow...
KK -theory and spectral flow in von Neumann algebras
DEFF Research Database (Denmark)
Kaad, Jens; Nest, Ryszard; Rennie, Adam
2012-01-01
We present a definition of spectral flow for any norm closed ideal J in any von Neumann algebra N. Given a path of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in Ko (J). Given a semifinite spectral triple (A, H, D) relative to (N, t) with A separable...
Higher Spin Double Field Theory : A Proposal
Bekaert, Xavier
2016-01-01
We construct a double field theory of higher spin gravity. Employing "semi-covariant" differential geometry, we spell a functional in which each term is completely covariant with respect to $\\mathbf{O}(4,4)$ T-duality, doubled diffeomorphisms, $\\mathbf{Spin}(1,3)$ local Lorentz symmetry and, separately, $\\mathbf{HS}(4)$ higher spin gauge symmetry. We also propose a set of BPS-like conditions whose solutions automatically satisfy the full Euler-Lagrange equations. As such a solution, we derive a linear dilaton vacuum. With extra algebraic constraints further imposed, our BPS proposal reduces to the bosonic Vasiliev equations.
Supersymmetry in Open Superstring Field Theory
Erler, Theodore
2016-01-01
We realize the 16 unbroken supersymmetries on a BPS D-brane as invariances of the action of the corresponding open superstring field theory. We work in the small Hilbert space approach, where a symmetry of the action translates into a symmetry of the associated cyclic $A_\\infty$ structure. We compute the supersymmetry algebra, being careful to disentangle the components which produce a translation, a gauge transformation, and a symmetry transformation which vanishes on-shell. Via the minimal model theorem, we illustrate how supersymmetry of the action implies supersymmetry of the tree level open string scattering amplitudes.
Derived Koszul Duality and Involutions in the Algebraic K-Theory of Spaces
Blumberg, Andrew J
2009-01-01
We interpret different constructions of algebraic $K$-theory of spaces as an instance of derived Koszul (or bar) duality and also as an instance of Morita equivalence. We relate the interplay between these two descriptions to the homotopy involution. We define a geometric analog of the Swan theory $G^{\\bZ}(\\bZ[\\pi])$ in terms of $\\Sigma^{\\infty}_{+} \\Omega X$ and show that it is the algebraic $K$-theory of the $E_{\\infty}$ ring spectrum $DX=S^{X_{+}}$.
Putting Algebra Progress Monitoring into Practice: Insights from the Field
Foegen, Anne; Morrison, Candee
2010-01-01
Algebra progress monitoring is a research-based practice that extends a long history of research in curriculum-based measurement (CBM). This article describes the theoretical foundations and research evidence for algebra progress monitoring, along with critical features of the practice. A detailed description of one practitioner's implementation…
Equivariant K-theory and freeness of group actions on C*-algebras
Phillips, N Christopher
1987-01-01
Freeness of an action of a compact Lie group on a compact Hausdorff space is equivalent to a simple condition on the corresponding equivariant K-theory. This fact can be regarded as a theorem on actions on a commutative C*-algebra, namely the algebra of continuous complex-valued functions on the space. The successes of "noncommutative topology" suggest that one should try to generalize this result to actions on arbitrary C*-algebras. Lacking an appropriate definition of a free action on a C*-algebra, one is led instead to the study of actions satisfying conditions on equivariant K-theory - in the cases of spaces, simply freeness. The first third of this book is a detailed exposition of equivariant K-theory and KK-theory, assuming only a general knowledge of C*-algebras and some ordinary K-theory. It continues with the author's research on K-theoretic freeness of actions. It is shown that many properties of freeness generalize, while others do not, and that certain forms of K-theoretic freeness are related to ...
Schroedinger Invariance from Lifshitz Isometries in Holography and Field Theory
Hartong, Jelle; Obers, Niels A
2014-01-01
We study non-relativistic field theory coupled to a torsional Newton-Cartan geometry both directly as well as holographically. The latter involves gravity on asymptotically locally Lifshitz space-times. We define an energy-momentum tensor and a mass current and study the relation between conserved currents and conformal Killing vectors for flat Newton-Cartan backgrounds. It is shown that this involves two different copies of the Lifshitz algebra together with an equivalence relation that joins these two Lifshitz algebras into a larger Schroedinger algebra (without the central element). In the holographic setup this reveals a novel phenomenon in which a large bulk diffeomorphism is dual to a discrete gauge invariance of the boundary field theory.
Reduction of filtered k-theory and a characterization of Cuntz-Krieger algebras
DEFF Research Database (Denmark)
Arklint, Sara E.; Bentmann, Rasmus Moritz; Katsura, Takeshi
2014-01-01
We show that filtered K-theory is equivalent to a substantially smaller invariant for all real-rank-zero C*-algebras with certain primitive ideal spaces—including the infinitely many so-called accordion spaces for which filtered K-theory is known to be a complete invariant. As a consequence, we...
Simplicities and Automorphisms of a Sp ecial Infinite Dimensional Lie Algebra
Institute of Scientific and Technical Information of China (English)
YU De-min; LI Ai-hua
2013-01-01
In this paper, a special infinite dimensional Lie algebra is studied. The infinite dimensional Lie algebra appears in the fields of conformal theory, mathematical physics, statistic mechanics and Hamilton operator. The infinite dimensional Lie algebras is pop-ularized Virasoro-like Lie algebra. Isomorphisms, homomorphisms, ideals of the infinite dimensional Lie algebra are studied.
Super Lie n-algebra extensions, higher WZW models, and super p-branes with tensor multiplet fields
Fiorenza, Domenico; Schreiber, Urs
2013-01-01
We formalize higher dimensional and higher gauge WZW-type sigma-model local prequantum field theory, and discuss its rationalized/perturbative description in (super-)Lie n-algebra homotopy theory (the true home of the "FDA"-language used in the supergravity literature). We show generally how the intersection laws for such higher WZW-type sigma-model branes (open brane ending on background brane) are encoded precisely in (super-) L-infinity-extension theory and how the resulting "extended (super-)spacetimes" formalize spacetimes containing sigma model brane condensates. As an application we prove in Lie n-algebra homotopy theory that the complete super p-brane spectrum of superstring/M-theory is realized this way, including the pure sigma-model branes (the "old brane scan") but also the branes with tensor multiplet worldvolume fields, notably the D-branes and the M5-brane. For instance the degree-0 piece of the higher symmetry algebra of 11-dimensional spacetime with an M2-brane condensate turns out to be the ...
Characters in Conformal Field Theories from Thermodynamic Bethe Ansatz
Kuniba, A.; Nakanishi, T; Suzuki, J.
1993-01-01
We propose a new $q$-series formula for a character of parafermion conformal field theories associated to arbitrary non-twisted affine Lie algebra $\\widehat{g}$. We show its natural origin from a thermodynamic Bethe ansatz analysis including chemical potentials.
Periods and Hodge structures in perturbative quantum field theory
Weinzierl, Stefan
2013-01-01
There is a fruitful interplay between algebraic geometry on the one side and perturbative quantum field theory on the other side. I review the main relevant mathematical concepts of periods, Hodge structures and Picard-Fuchs equations and discuss the connection with Feynman integrals.
A superconformal algebra of meromorphic vector fields with three poles on the super Riemann sphere
International Nuclear Information System (INIS)
Based on the Riemann-Roch theorem, we construct a superconformal algebra of meromorphic vector fields with three poles and the relevant abelian differential of the third kind on the super Riemann sphere. The algebra includes two Ramond sectors as a subalgebra, and implies the picture of an interaction of three superstrings. (orig.)
Spectral theory of linear operators and spectral systems in Banach algebras
Müller, Vladimir
2003-01-01
This book is dedicated to the spectral theory of linear operators on Banach spaces and of elements in Banach algebras. It presents a survey of results concerning various types of spectra, both of single and n-tuples of elements. Typical examples are the one-sided spectra, the approximate point, essential, local and Taylor spectrum, and their variants. The theory is presented in a unified, axiomatic and elementary way. Many results appear here for the first time in a monograph. The material is self-contained. Only a basic knowledge of functional analysis, topology, and complex analysis is assumed. The monograph should appeal both to students who would like to learn about spectral theory and to experts in the field. It can also serve as a reference book. The present second edition contains a number of new results, in particular, concerning orbits and their relations to the invariant subspace problem. This book is dedicated to the spectral theory of linear operators on Banach spaces and of elements in Banach alg...
Genus Two Zhu Theory for Vertex Operator Algebras
Gilroy, Thomas
2015-01-01
We consider correlation functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We describe a generalisation of genus one Zhu recursion expressing an arbitrary genus two $n$--point correlation function in terms of $(n-1)$--point functions. We consider several applications including the correlation functions for the Heisenberg vertex operator algebra and its modules, Virasoro correlation functions and genus two Ward identities. We derive novel differential equations in terms of a differential operator on the genus two Siegel upper half plane for holomorphic $1$--forms, the normalised bidifferential of the second kind and the Heisenberg partition function. We also prove that the holomorphic mapping from the sewing parameter domain to the Siegel upper half plane is injective but not surjective.
On the algebraic structure of isotropic generalized elasticity theories
Auffray, Nicolas
2013-01-01
In this paper the algebraic structure of the isotropic nth-order gradient elasticity is investigated. In the classical isotropic elasticity it is well-known that the constitutive relation can be broken down into two uncoupled relations between elementary part of the strain and the stress tensors (deviatoric and spherical). In this paper we demonstrate that this result can not be generalized and since 2nd-order isotropic elasticity there exist couplings between elementary parts of higher-order strain and stress tensors. Therefore, and in certain way, nth-order isotropic elasticity have the same kind of algebraic structure as anisotropic classical elasticity. This structure is investigated in the case of 2nd-order isotropic elasticity, and moduli characterizing the behavior are provided.
Introduction to abstract algebra
Nicholson, W Keith
2012-01-01
Praise for the Third Edition ". . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . ."-Zentralblatt MATH The Fourth Edition of Introduction to Abstract Algebra continues to provide an accessible approach to the basic structures of abstract algebra: groups, rings, and fields. The book's unique presentation helps readers advance to abstract theory by presenting concrete examples of induction, number theory, integers modulo n, and permutations before the abstract structures are defined. Readers can immediately be
Double Field Theory on Group Manifolds (Thesis)
Hassler, Falk
2015-01-01
This thesis deals with Double Field Theory (DFT), an effective field theory capturing the low energy dynamics of closed strings on a torus. It renders T-duality on a torus manifest by adding $D$ winding coordinates in addition to the $D$ space time coordinates. An essential consistency constraint of the theory, the strong constraint, only allows for field configurations which depend on half of the coordinates of the arising doubled space. I derive DFT${}_\\mathrm{WZW}$, a generalization of the current formalism. It captures the low energy dynamics of a closed bosonic string propagating on a compact group manifold. Its classical action and the corresponding gauge transformations arise from Closed String Field Theory up to cubic order in the massless fields. These results are rewritten in terms of a generalized metric and extended to all orders in the fields. There is an explicit distinction between background and fluctuations. For the gauge algebra to close, the latter have to fulfill a modified strong constrai...
On the algebraic structure of isotropic generalized elasticity theories
AUFFRAY, Nicolas
2013-01-01
In this paper the algebraic structure of the isotropic nth-order gradient elasticity is investigated. In the classical isotropic elasticity it is well-known that the constitutive relation can be broken down into two uncoupled relations between elementary part of the strain and the stress tensors (deviatoric and spherical). In this paper we demonstrate that this result can not be generalized and since 2nd-order isotropic elasticity there exist couplings between elementary parts of higher-order...
On the algebraical structure of isotropic generalized elasticity theories
AUFFRAY, Nicolas
2013-01-01
International audience In this paper the algebraical structure of the isotropic nth-order gradient elasticity is investigated. In the classical isotropic elasticity it is well-known that the constitutive relation can be broken down into two uncoupled relations between elementary part of the strain and the stress tensors (deviatoric and spherical). In this paper we demonstrate that this result can not be generalized and since 2nd-order isotropic elasticity there exist couplings between elem...
Cluster Algebras from Dualities of 2d = (2, 2) Quiver Gauge Theories
Benini, Francesco; Park, Daniel S.; Zhao, Peng
2015-11-01
We interpret certain Seiberg-like dualities of two-dimensional = (2,2) quiver gauge theories with unitary groups as cluster mutations in cluster algebras, originally formulated by Fomin and Zelevinsky. In particular, we show how the complexified Fayet-Iliopoulos parameters of the gauge group factors transform under those dualities and observe that they are in fact related to the dual cluster variables of cluster algebras. This implies that there is an underlying cluster algebra structure in the quantum Kähler moduli space of manifolds constructed from the corresponding Kähler quotients. We study the S 2 partition function of the gauge theories, showing that it is invariant under dualities/mutations, up to an overall normalization factor, whose physical origin and consequences we spell out in detail. We also present similar dualities in = (2,2)* quiver gauge theories, which are related to dualities of quantum integrable spin chains.
Higgs Effective Field Theories
2016-01-01
The main focus of this meeting is to present new theoretical advancements related to effective field theories, evaluate the impact of initial results from the LHC Run2, and discuss proposals for data interpretation/presentation during Run2. A crucial role of the meeting is to bring together theorists from different backgrounds and with different viewpoints and to extend bridges towards the experimental community. To this end, we would like to achieve a good balance between senior and junior speakers, enhancing the visibility of younger scientists while keeping some overview talks.
Reverse Engineering Quantum Field Theory
Oeckl, Robert
2012-01-01
An approach to the foundations of quantum theory is advertised that proceeds by "reverse engineering" quantum field theory. As a concrete instance of this approach, the general boundary formulation of quantum theory is outlined.
Reverse engineering quantum field theory
Oeckl, Robert
2012-12-01
An approach to the foundations of quantum theory is advertised that proceeds by "reverse engineering" quantum field theory. As a concrete instance of this approach, the general boundary formulation of quantum theory is outlined.
Quantum Field Theory on Curved Backgrounds -- A Primer
Benini, Marco; Hack, Thomas-Paul
2013-01-01
Goal of this review is to introduce the algebraic approach to quantum field theory on curved backgrounds. Based on a set of axioms, first written down by Haag and Kastler, this method consists of a two-step procedure. In the first one, a suitable algebra of observables is assigned to a physical system, which is meant to encode all algebraic relations among observables, such as commutation relations, while, in the second step, one must select an algebraic state in order to recover the standard Hilbert space interpretation of a quantum system. As quantum field theories possess infinitely many degrees of freedom, many unitarily inequivalent Hilbert space representations exist and the power of such approach is the ability to treat them all in a coherent manner. We will discuss in detail the algebraic approach for free fields in order to give to the reader all necessary information to deal with the recent literature, which focuses on the applications to specific problems, mostly in cosmology.
Exotic Bbb R4 and quantum field theory
Asselmeyer-Maluga, Torsten; Mader, Roland
2012-02-01
Recent work on exotic smooth Bbb R4,s, i.e. topological Bbb R4 with exotic differential structure, shows the connection of 4-exotics with the codimension-1 foliations of S3, SU(2) WZW models and twisted K-theory KH(S3), H in H3(S3,Bbb Z). These results made it possible to explicate some physical effects of exotic 4-smoothness. Here we present a relation between exotic smooth Bbb R4 and operator algebras. The correspondence uses the leaf space of the codimension-1 foliation of S3 inducing a von Neumann algebra W(S3) as description. This algebra is a type III1 factor lying at the heart of any observable algebra of QFT. By using the relation to factor II, we showed that the algebra W(S3) can be interpreted as Drinfeld-Turaev deformation quantization of the space of flat SL(2, Bbb C) connections (or holonomies). Thus, we obtain a natural relation to quantum field theory. Finally we discuss the appearance of concrete action functionals for fermions or gauge fields and its connection to quantum-field-theoretical models like the Tree QFT of Rivasseau.
Studies in quantum field theory
International Nuclear Information System (INIS)
Washington University is currently conducting research in many areas of high energy theoretical and mathematical physics. These areas include: strong-coupling approximation; classical solutions of non-Abelian gauge theories; mean-field approximation in quantum field theory; path integral and coherent state representations in quantum field theory; lattice gauge calculations; the nature of perturbation theory in large orders; quark condensation in QCD; chiral symmetry breaking; the l/N expansion in quantum field theory; effective potential and action in quantum field theories, including QCD
Supergeometry in Locally Covariant Quantum Field Theory
Hack, Thomas-Paul; Hanisch, Florian; Schenkel, Alexander
2016-03-01
In this paper we analyze supergeometric locally covariant quantum field theories. We develop suitable categories SLoc of super-Cartan supermanifolds, which generalize Lorentz manifolds in ordinary quantum field theory, and show that, starting from a few representation theoretic and geometric data, one can construct a functor A : SLoc to S* Alg to the category of super-*-algebras, which can be interpreted as a non-interacting super-quantum field theory. This construction turns out to disregard supersymmetry transformations as the morphism sets in the above categories are too small. We then solve this problem by using techniques from enriched category theory, which allows us to replace the morphism sets by suitable morphism supersets that contain supersymmetry transformations as their higher superpoints. We construct super-quantum field theories in terms of enriched functors eA : eSLoc to eS* Alg between the enriched categories and show that supersymmetry transformations are appropriately described within the enriched framework. As examples we analyze the superparticle in 1|1-dimensions and the free Wess-Zumino model in 3|2-dimensions.
Thermo-Field Extension of Open String Field Theory
Cantcheff, M Botta
2015-01-01
We study the implementation of Thermo Field Dynamics (TFD) to the covariant formulation of Open String Field Theory (OSFT). In this paper, we extend the state space and fields according to the duplication rules of TFD and construct the corresponding classical action. The result is a theory whose fields would encode the statistical information of open strings and, noticeably, present degrees of freedom that could be identified as those of closed strings. The physical spectrum of the free theory is studied through the cohomology of the extended BRST charge, and, as a result, we get new fields in the spectrum. We also show, however, that their appearing in the action is directly related to the choice of the inner product in the extended algebra, so that many fields could be eliminated from the theory by choosing that product conveniently. Finally, we study the extension of the three-vertex interaction and provide a simple prescription for it whose results at tree-level amplitudes agree with those of the conventi...
Choy, Ting-Pong
One of the leading problems in condensed matter physics is what state of matter obtain when there is a strong Coulomb repulsion between the electrons. One of the exotic examples is the high temperature superconductivity which was discovered in copper-oxide ceramics (cuprates) over twenty years ago. Thus far, a satisfactory theory is absent. In particular, the nature of the electron state outside the superconducting phase remains controversial. In analogy with the BCS theory of a conventional superconductor, in which the metal is well known to be a Fermi liquid, a complete understanding of the normal state of cuprate is necessary prior to the study of the superconducting mechanism in the high temperature superconductors. In this thesis, we will provide a theory for these exotic normal state properties by studying the minimal microscopic model which captures the physics of strong electron correlation. Even in such a simple microscopic model, striking properties including charge localization and presence of a Luttinger surface resemble the normal state properties of cuprate. An exact low energy theory of a doped Mott insulator will be constructed by explicitly integrating (rather than projecting) out the degrees of freedom far away from the chemical potential. The exact low energy theory contains degrees of freedom that cannot be obtained from projective schemes. In particular, a charge 2e bosonic field which is not made out of elemental excitations emerges at low energies. Such a field accounts for dynamical spectral weight transfer across the Mott gap. At half-filling, we show that two such excitations emerge which play a crucial role in preserving the Luttinger surface along which the single-particle Green function vanishes. We also apply this method to the Anderson-U impurity and show that in addition to the Kondo interaction, bosonic degrees of freedom appear as well. We show that many of the normal state properties of the cuprates can result from this new charge
The genus zeta function of hereditary orders in central simple algebras over global fields
Denert, M.
1990-01-01
Louis Solomon introduced the notion of a zeta function {ζ_θ }(s) of an order θ in a finite-dimensional central simple K-algebra A, with K a number field or its completion {K_P} (P a non-Archimedean prime in K). In several papers, C. J. Bushnell and I. Reiner have developed the theory of zeta functions and they gave explicit formulae in some special cases. One important property of these zeta functions is the Euler product, which implies that in order to calculate {ζ_θ }(s) , it is sufficient to consider the zeta function of local orders {θ _P} . However, since these local orders {θ _P} are in general not principal ideal domains, their zeta function is a finite sum of so-called 'partial zeta functions'. The most complicated term is the 'genus zeta function', {Z_{{θ _P}}}(s) , which is related to the free {θ _P} -ideals. I. Reiner and C. J. Bushnell calculated the genus zeta function for hereditary orders in quaternion algebras (i.e., [A:K] = 4 ). The authors mention the general case but they remark that the calculations are cumbersome. In this paper we derive an explicit method to calculate the genus zeta function {Z_{{θ _P}}}(s) of any local hereditary order {θ _P} in a central simple algebra over a local field. We obtain {Z_{{θ _P}}}(s) as a finite sum of explicit terms which can be calculated with a computer. We make some remarks on the programming of the formula and give a short list of examples. The genus zeta function of the minimal hereditary orders (corresponding to the partition (1, 1, ... , 1) of n) seems to have a surprising property. In all examples, the nominator of this zeta function is a generating function for the q-Eulerian polynomials. We conclude with some remarks on a conjectured identity.
Causal independence and the energy-level density of states in local quantum field theory
International Nuclear Information System (INIS)
Within the general framework of local quantum field theory a physically motivated condition on the energy-level density of well-localized states is proposed and discussed. It is shown that any model satisfying this condition obeys a strong form of the principle of causal (statistical) independence, which manifests itself in a specific algebraic structure of the local algebras ('split property'). It is also shown that the proposed condition holds in a free field theory. (orig.)
Local spectral theory of endomorphisms of the disk algebra
Directory of Open Access Journals (Sweden)
Trivedi Shailesh
2016-03-01
Full Text Available Let A( denote the disk algebra. Every endomorphism of A( is induced by some ϕ ∈ A( with ‖ϕ‖ ≤ 1. In this paper, it is shown that if ϕ is not an automorphism of and ϕ has a fixed point in the open unit disk then the endomorphism induced by ϕ is decomposable if and only if the fixed set of ϕ is singleton. Further, we determine the local spectra of the endomorphism induced by ϕ in the cases when the fixed set of ϕ either includes unit circle or is a singleton.
Integrable structures in quantum field theory
Negro, Stefano
2016-08-01
This review was born as notes for a lecture given at the Young Researchers Integrability School (YRIS) school on integrability in Durham, in the summer of 2015. It deals with a beautiful method, developed in the mid-nineties by Bazhanov, Lukyanov and Zamolodchikov and, as such, called BLZ. This method can be interpreted as a field theory version of the quantum inverse scattering, also known as the algebraic Bethe ansatz. Starting with the case of conformal field theories (CFTs) we show how to build the field theory analogues of commuting transfer T matrices and Baxter Q-operators of integrable lattice models. These objects contain the complete information of the integrable structure of the theory, viz. the integrals of motion, and can be used, as we will show, to derive the thermodynamic Bethe ansatz and nonlinear integral equations. This same method can be easily extended to the description of integrable structures of certain particular massive deformations of CFTs; these, in turn, can be described as quantum group reductions of the quantum sine-Gordon model and it is an easy step to include this last theory in the framework of BLZ approach. Finally we show an interesting and surprising connection of the BLZ structures with classical objects emerging from the study of classical integrable models via the inverse scattering transform method. This connection goes under the name of ODE/IM correspondence and we will present it for the specific case of quantum sine-Gordon model only.
Deformations of Quantum Field Theories on Curved Spacetimes
Morales, Eric Morfa
2012-01-01
The construction and analysis of deformations of quantum field theories by warped convolutions is extended to a class of globally hyperbolic spacetimes. First, we show that any four-dimensional spacetime which admits two commuting and spacelike Killing vector fields carries a family of wedge regions with causal properties analogous to the Minkowski space wedges. Deformations of quantum field theories on these spacetimes are carried out within the operator-algebraic framework - the emerging models share many structural properties with deformations of field theories on flat spacetime. In particular, deformed quantum fields are localized in the wedges of the considered spacetime. As a concrete example, the deformation of the free Dirac field is studied. Second, quantum field theories on de Sitter spacetime with global U(1) gauge symmetry are deformed using the joint action of the internal symmetry group and a one-parameter group of boosts. The resulting theories turn out to be wedge-local and non-isomorphic to t...
An assessment of Evans' unified field theory II
Hehl, Friedrich W.; Obukhov, Yuri N.
2007-01-01
Evans developed a classical unified field theory of gravitation and electromagnetism on the background of a spacetime obeying a Riemann-Cartan geometry. In an accompanying paper I, we analyzed this theory and summarized it in nine equations. We now propose a variational principle for Evans' theory and show that it yields two field equations. The second field equation is algebraic in the torsion and we can resolve it with respect to the torsion. It turns out that for all physical cases the tor...
Khoury, Justin
2013-01-01
Chameleons are light scalar fields with remarkable properties. Through the interplay of self-interactions and coupling to matter, chameleon particles have a mass that depends on the ambient matter density. The manifestation of the fifth force mediated by chameleons therefore depends sensitively on their environment, which makes for a rich phenomenology. In this article, we review two recent results on chameleon phenomenology. The first result a pair of no-go theorems limiting the cosmological impact of chameleons and their generalizations: i) the range of the chameleon force at cosmological density today can be at most ~Mpc; ii) the conformal factor relating Einstein- and Jordan-frame scale factors is essentially constant over the last Hubble time. These theorems imply that chameleons have negligible effect on the linear growth of structure, and cannot account for the observed cosmic acceleration except as some form of dark energy. The second result pertains to the quantum stability of chameleon theories. We ...
A precise result on the arithmetic of non-principal orders in algebraic number fields
Philipp, Andreas
2011-01-01
Let $R$ be an order in an algebraic number field. If $R$ is a principal order, then many explicit results on its arithmetic are available. Among others, $R$ is half-factorial if and only if the class group of $R$ has at most two elements. Much less is known for non-principal orders. Using a new semigroup theoretical approach, we study half-factoriality and further arithmetical properties for non-principal orders in algebraic number fields.
Springer, T A
1998-01-01
"[The first] ten chapters...are an efficient, accessible, and self-contained introduction to affine algebraic groups over an algebraically closed field. The author includes exercises and the book is certainly usable by graduate students as a text or for self-study...the author [has a] student-friendly style… [The following] seven chapters... would also be a good introduction to rationality issues for algebraic groups. A number of results from the literature…appear for the first time in a text." –Mathematical Reviews (Review of the Second Edition) "This book is a completely new version of the first edition. The aim of the old book was to present the theory of linear algebraic groups over an algebraically closed field. Reading that book, many people entered the research field of linear algebraic groups. The present book has a wider scope. Its aim is to treat the theory of linear algebraic groups over arbitrary fields. Again, the author keeps the treatment of prerequisites self-contained. The material of t...
Arnold, Vladimir I; Khesin, Boris; Marsden, Jerrold E; Varchenko, AN; Vassiliev, Victor A; Viro, Oleg Yanovich; Zakalyukin, Vladimir
2013-01-01
Vladimir Arnold was one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work. At the same time he is one of the most prolific and outstanding mathematical authors. This second volume of his ""Collected Works"" focuses on hydrodynamics, bifurcation theory, and algebraic geometry.
On the Farrell-Jones Conjecture for higher algebraic K-theory
Bartels, A; Reich, H
2003-01-01
We prove the Farrell-Jones Isomorphism Conjecture about the algebraic K-theory of a group ring RG in the case where the group G is the fundamental group of a closed Riemannian manifold with strictly negative sectional curvature. The coefficient ring R is an arbitrary associative ring with unit and the result applies to all dimensions.
Modern Quantum Field Theory II - Proceeeings of the International Colloquium
Das, S. R.; Mandal, G.; Mukhi, S.; Wadia, S. R.
1995-08-01
The Table of Contents for the book is as follows: * Foreword * 1. Black Holes and Quantum Gravity * Quantum Black Holes and the Problem of Time * Black Hole Entropy and the Semiclassical Approximation * Entropy and Information Loss in Two Dimensions * Strings on a Cone and Black Hole Entropy (Abstract) * Boundary Dynamics, Black Holes and Spacetime Fluctuations in Dilation Gravity (Abstract) * Pair Creation of Black Holes (Abstract) * A Brief View of 2-Dim. String Theory and Black Holes (Abstract) * 2. String Theory * Non-Abelian Duality in WZW Models * Operators and Correlation Functions in c ≤ 1 String Theory * New Symmetries in String Theory * A Look at the Discretized Superstring Using Random Matrices * The Nested BRST Structure of Wn-Symmetries * Landau-Ginzburg Model for a Critical Topological String (Abstract) * On the Geometry of Wn Gravity (Abstract) * O(d, d) Tranformations, Marginal Deformations and the Coset Construction in WZNW Models (Abstract) * Nonperturbative Effects and Multicritical Behaviour of c = 1 Matrix Model (Abstract) * Singular Limits and String Solutions (Abstract) * BV Algebra on the Moduli Spaces of Riemann Surfaces and String Field Theory (Abstract) * 3. Condensed Matter and Statistical Mechanics * Stochastic Dynamics in a Deposition-Evaporation Model on a Line * Models with Inverse-Square Interactions: Conjectured Dynamical Correlation Functions of the Calogero-Sutherland Model at Rational Couplings * Turbulence and Generic Scale Invariance * Singular Perturbation Approach to Phase Ordering Dynamics * Kinetics of Diffusion-Controlled and Ballistically-Controlled Reactions * Field Theory of a Frustrated Heisenberg Spin Chain * FQHE Physics in Relativistic Field Theories * Importance of Initial Conditions in Determining the Dynamical Class of Cellular Automata (Abstract) * Do Hard-Core Bosons Exhibit Quantum Hall Effect? (Abstract) * Hysteresis in Ferromagnets * 4. Fundamental Aspects of Quantum Mechanics and Quantum Field Theory
Flat Holography: Aspects of the dual field theory
Bagchi, Arjun; Kakkar, Ashish; Mehra, Aditya
2016-01-01
Assuming the existence of a field theory in D dimensions dual to (D+1)-dimensional flat space, governed by the asymptotic symmetries of flat space, we make some preliminary remarks about the properties of this field theory. We review briefly some successes of the 3d bulk - 2d boundary case and then focus on the 4d bulk - 3d boundary example, where the symmetry in question is the infinite dimensional BMS4 algebra. We look at the constraints imposed by this symmetry on a 3d field theory by constructing highest weight representations of this algebra. We construct two and three point functions of BMS primary fields and surprisingly find that symmetries constrain these correlators to be identical to those of a 2d relativistic conformal field theory. We then go one dimension higher and construct prototypical examples of 4d field theories which are putative duals of 5d Minkowski spacetimes. These field theories are ultra-relativistic limits of electrodynamics and Yang-Mills theories which exhibit invariance under th...
Transformations among large c conformal field theories
Energy Technology Data Exchange (ETDEWEB)
Jankiewicz, Marcin [Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235 (United States)]. E-mail m.jankiewicz@vanderbilt.edu; Kephart, Thomas W. [Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235 (United States)]. E-mail thomas.w.kephart@vanderbilt.edu
2006-06-12
We show that there is a set of transformations that relates all of the 24 dimensional even self-dual (Niemeier) lattices, and also leads to non-lattice objects some of which can perhaps be interpreted as a basis for the construction of holomorphic conformal field theory. In the second part of this paper, we extend our observations to higher-dimensional conformal field theories build on extremal partition functions, where we generate c=24k theories. We argue that there exists generalizations of the c=24 models based on Niemeier lattices and of the non-Niemeier spin-1 theories. The extremal cases have spectra decomposable into the irreducible representations of the Fischer-Griess Monster. This additional symmetry leads us to conjecture that these extremal theories, as well as the higher-dimensional analogs of the group lattice bases Niemeiers, will eventually yield to a full construction of their associated CFTs. We observe interesting periodicities in the coefficients of extremal partition functions and characters of the extremal vertex operator algebras.
Transformations among large c conformal field theories
International Nuclear Information System (INIS)
We show that there is a set of transformations that relates all of the 24 dimensional even self-dual (Niemeier) lattices, and also leads to non-lattice objects some of which can perhaps be interpreted as a basis for the construction of holomorphic conformal field theory. In the second part of this paper, we extend our observations to higher-dimensional conformal field theories build on extremal partition functions, where we generate c=24k theories. We argue that there exists generalizations of the c=24 models based on Niemeier lattices and of the non-Niemeier spin-1 theories. The extremal cases have spectra decomposable into the irreducible representations of the Fischer-Griess Monster. This additional symmetry leads us to conjecture that these extremal theories, as well as the higher-dimensional analogs of the group lattice bases Niemeiers, will eventually yield to a full construction of their associated CFTs. We observe interesting periodicities in the coefficients of extremal partition functions and characters of the extremal vertex operator algebras
Transformations among large c conformal field theories
Jankiewicz, Marcin; Kephart, Thomas W.
2006-06-01
We show that there is a set of transformations that relates all of the 24 dimensional even self-dual (Niemeier) lattices, and also leads to non-lattice objects some of which can perhaps be interpreted as a basis for the construction of holomorphic conformal field theory. In the second part of this paper, we extend our observations to higher-dimensional conformal field theories build on extremal partition functions, where we generate c=24k theories. We argue that there exists generalizations of the c=24 models based on Niemeier lattices and of the non-Niemeier spin-1 theories. The extremal cases have spectra decomposable into the irreducible representations of the Fischer-Griess Monster. This additional symmetry leads us to conjecture that these extremal theories, as well as the higher-dimensional analogs of the group lattice bases Niemeiers, will eventually yield to a full construction of their associated CFTs. We observe interesting periodicities in the coefficients of extremal partition functions and characters of the extremal vertex operator algebras.
[Topics in field theory and string theory
International Nuclear Information System (INIS)
In the past year, I have continued to investigate the relations between conformal field theories and lattice statistical mechanical models. I have also tried to extend some of these results to higher dimensions and to find applications in string theories and other contexts
An action for F-theory: {SL}(2){{{R}}}^{+} exceptional field theory
Berman, David S.; Blair, Chris D. A.; Malek, Emanuel; Rudolph, Felix J.
2016-10-01
We construct the 12-dimensional exceptional field theory (EFT) associated to the group {SL}(2)× {{{R}}}+. Demanding the closure of the algebra of local symmetries leads to a constraint, known as the section condition, that must be imposed on all fields. This constraint has two inequivalent solutions, one giving rise to 11-dimensional supergravity and the other leading to F-theory. Thus {SL}(2)× {{{R}}}+ EFT contains both F-theory and M-theory in a single 12-dimensional formalism.
Perez, Uzziel; Sugon, Quirino M; McNamara, Daniel J; Yoshikawa, Akimasa
2015-01-01
We studied the orbit of an electron revolving around an infinitely massive nucleus of a large classical Hydrogen atom subject to an AC electric field oscillating perpendicular to the electron's circular orbit. Using perturbation theory in geometric algebra, we show that the equation of motion of the electron perpendicular to the unperturbed orbital plane satisfies a forced simple harmonic oscillator equation found in Lorentz dispersion law in Optics. We show that even though we did not introduce a damping term, the initial orbital position and velocity of the electron results to a solution whose absorbed energies are finite at the dominant resonant frequency $\\omega=\\omega_0$; the electron slowly increases its amplitude of oscillation until it becomes ionized. We computed the average power absorbed by the electron both at the perturbing frequency and at the electron's orbital frequency. We graphed the trace of the angular momentum vector at different frequencies. We showed that at different perturbing frequen...
Weighted Graph Theory Representation of Quantum Information Inspired by Lie Algebras
Belhaj, Abdelilah; Machkouri, Larbi; Sedra, Moulay Brahim; Ziti, Soumia
2016-01-01
Borrowing ideas from the relation between simply laced Lie algebras and Dynkin diagrams, a weighted graph theory representation of quantum information is addressed. In this way, the density matrix of a quantum state can be interpreted as a signless Laplacian matrix of an associated graph. Using similarities with root systems of simply laced Lie algebras, one-qubit theory is analyzed in some details and is found to be linked to a non-oriented weighted graph having two vertices. Moreover, this one-qubit theory is generalized to n-qubits. In this representation, quantum gates correspond to graph weight operations preserving the probability condition. A speculation from string theory, via D-brane quivers, is also given.
Gato-Rivera, Beatriz
2008-01-01
In 1998 the Adapted Ordering Method was developed for the study of the representation theory of the superconformal algebras in two dimensions. It allows: to determine the maximal dimension for a given type of space of singular vectors, to identify all singular vectors by only a few coefficients, to spot subsingular vectors and to set the basis for constructing embedding diagrams. In this talk I introduce the present version of the Adapted Ordering Method, published in J. Phys. A: Math. Theor. 41 (2008) 045201, which can be applied to general Lie algebras and superalgebras and their generalizations, provided they can be triangulated.
Noncommutative Geometry in M-Theory and Conformal Field Theory
Energy Technology Data Exchange (ETDEWEB)
Morariu, Bogdan
1999-05-01
In the first part of the thesis I will investigate in the Matrix theory framework, the subgroup of dualities of the Discrete Light Cone Quantization of M-theory compactified on tori, which corresponds to T-duality in the auxiliary Type II string theory. After a review of matrix theory compactification leading to noncommutative supersymmetric Yang-Mills gauge theory, I will present solutions for the fundamental and adjoint sections on a two-dimensional twisted quantum torus and generalize to three-dimensional twisted quantum tori. After showing how M-theory T-duality is realized in supersymmetric Yang-Mills gauge theories on dual noncommutative tori I will relate this to the mathematical concept of Morita equivalence of C*-algebras. As a further generalization, I consider arbitrary Ramond-Ramond backgrounds. I will also discuss the spectrum of the toroidally compactified Matrix theory corresponding to quantized electric fluxes on two and three tori. In the second part of the thesis I will present an application to conformal field theory involving quantum groups, another important example of a noncommutative space. First, I will give an introduction to Poisson-Lie groups and arrive at quantum groups using the Feynman path integral. I will quantize the symplectic leaves of the Poisson-Lie group SU(2)*. In this way we obtain the unitary representations of U{sub q}(SU(2)). I discuss the X-structure of SU(2)* and give a detailed description of its leaves using various parametrizations. Then, I will introduce a new reality structure on the Heisenberg double of Fun{sub q} (SL(N,C)) for q phase, which can be interpreted as the quantum phase space of a particle on the q-deformed mass-hyperboloid. I also present evidence that the above real form describes zero modes of certain non-compact WZNW-models.
International Nuclear Information System (INIS)
In this paper, we begin the study of zero-dimensional field theories with fields taking values in a semistrict Lie 2-algebra. These theories contain the IKKT matrix model and various M-brane related models as special cases. They feature solutions that can be interpreted as quantized 2-plectic manifolds. In particular, we find solutions corresponding to quantizations of ℝ3, S3 and a five-dimensional Hpp-wave. Moreover, by expanding a certain class of Lie 2-algebra models around the solution corresponding to quantized ℝ3, we obtain higher BF-theory on this quantized space
The foundational origin of integrability in quantum field theory
International Nuclear Information System (INIS)
There are two foundational model-independent concepts of integrability in QFT. One is 'dynamical' and generalizes the solvability in closed analytic form of the dynamical aspects as known from the Kepler two-body problem and its quantum mechanical counterpart. The other, referred to as 'kinematical' integrability, has no classical nor even quantum mechanical counterpart; it describes the relation between so called eld algebra and its local observable subalgebras and their discrete inequivalent representation classes (the DHR theory of superselection sectors). In the standard case of QFTs with mass gaps it contains the information about the representation of the (necessary compact) internal symmetry group and statistics in form of a tracial state on a 'dual group'. In Lagrangian or functional quantization one deals with the eld algebra and the division into observable /eld algebras does presently not play a role in constructive approaches to QFT. 'Kinematical' integrability is however of particular interest in conformal theories where the observable algebra fulfils the Huygens principle (light like propagation) and lives on the compactified Minkowski spacetime whereas the eld algebra, whose spacetime symmetry group is the universal covering of the conformal group lives on the universal covering of the compactified Minkowski spacetime. Since the (anomalous) dimensions of fields show up in the spectrum of the unitary representative of the center of this group , the kinematical structure contained in the relation fields/Huygens observables valuable information which in the usual terminology would be called 'dynamical'. The dynamical integrability is defined in terms of properties of 'wedge localization' and uses the fact that modular localization theory allows to 'emulate' interaction-free wedge-localized operators in a objective manner with the wedge localized interacting algebra. Emulation can be viewed as a generalization of the functorial relation between localized
Quantum Field Theory of Fluids
Gripaios, Ben; Sutherland, Dave
2015-01-01
The quantum theory of fields is largely based on studying perturbations around non-interacting, or free, field theories, which correspond to a collection of quantum-mechanical harmonic oscillators. The quantum theory of an ordinary fluid is `freer', in the sense that the non-interacting theory also contains an infinite collection of quantum-mechanical free particles, corresponding to vortex modes. By computing a variety of correlation functions at tree- and loop-level, we give evidence that a...
Bitopological spaces theory, relations with generalized algebraic structures and applications
Dvalishvili, Badri
2005-01-01
This monograph is the first and an initial introduction to the theory of bitopological spaces and its applications. In particular, different families of subsets of bitopological spaces are introduced and various relations between two topologies are analyzed on one and the same set; the theory of dimension of bitopological spaces and the theory of Baire bitopological spaces are constructed, and various classes of mappings of bitopological spaces are studied. The previously known results as well the results obtained in this monograph are applied in analysis, potential theory, general topology, a
Properties of Quaternion Algebra over the Real Number Field and Zp
Institute of Scientific and Technical Information of China (English)
QIN Ying-bing
2010-01-01
The ring of quaternion over R, denoted by R[i,j,k], is a quaternion algebra. In this paper, the roots of quadratic equation with one variable in quaternion field are investigated and it is shown that it has infinitely many roots. Then the properties of quaternion algebra over Zp are discussed, and the order of its unit group is determined. Lastly, another ring isomorphism of M2(Zp) and the quaternion algebra over Zp when p satisfies some particular conditions are presented.
Field redefinition invariance in quantum field theory
Apfeldorf, K M; Apfeldorf, Karyn M; Ordonez, Carlos
1994-01-01
We investigate the consequences of field redefinition invariance in quantum field theory by carefully performing nonlinear transformations in the path integral. We first present a ``paradox'' whereby a 1+1 freemassless scalar theory on a Minkowskian cylinder is reduced to an effectively quantum mechanical theory. We perform field redefinitions both before and after reduction to suggest that one should not ignore operator ordering issues in quantum field theory. We next employ a discretized version of the path integral for a free massless scalar quantum field in d dimensions to show that beyond the usual jacobian term, an infinite series of divergent ``extra'' terms arises in the action whenever a nonlinear field redefinition is made. The explicit forms for the first couple of these terms are derived. We evaluate Feynman diagrams to illustrate the importance of retaining the extra terms, and conjecture that these extra terms are the exact counterterms necessary to render physical quantities invariant under fie...
Proceedings of the 5. Jorge Andre Swieca Summer School Field Theory and Particle Physics
International Nuclear Information System (INIS)
Lectures on quantum field theories and particle physics are presented. The part of quantum field theories contains: constrained dynamics; Schroedinger representation in field theory; application of this representation to quantum fields in a Robertson-Walker space-time; Berry connection; problem of construction and classification of conformal field theories; lattice models; two-dimensional S matrices and conformal field theory for unifying perspective of Yang-Baxter algebras; parasupersymmetric quantum mechanics; introduction to string field theory; three dimensional gravity and two-dimensional parafermionic model. The part of particle physics contains: collider physics; strong interactions and use of strings in strong interactions. (M.C.K.)
Consistent systems of correlators in non-semisimple conformal field theory
Fuchs, Jürgen
2016-01-01
Based on the modular functor associated with a - not necessarily semisimple - finite non-degenerate ribbon category $\\mathcal D$, we present a definition of a consistent system of bulk field correlators of a conformal field theory which comprises invariance under mapping class group actions and compatibility with the sewing of surfaces. We show that when restricting to surfaces of genus zero such systems are in bijection with commutative symmetric Frobenius algebras in $\\mathcal D$, while for surfaces of any genus they are in bijection with modular Frobenius algebras in $\\mathcal D$. This extends structures familiar from rational conformal field theories to rigid logarithmic conformal field theories.
Advances In Classical Field Theory
Yahalom, Asher
2011-01-01
Classical field theory is employed by physicists to describe a wide variety of physical phenomena. These include electromagnetism, fluid dynamics, gravitation and quantum mechanics. The central entity of field theory is the field which is usually a multi component function of space and time. Those multi component functions are usually grouped together as vector fields as in the case in electromagnetic theory and fluid dynamics, in other cases they are grouped as tensors as in theories of gravitation and yet in other cases they are grouped as complex functions as in the case of quantum mechanic
Scalar Field Theory on κ-MINKOWSKI Space-Time and Translation and Lorentz Invariance
Meljanac, S.; Samsarov, A.
We investigate the properties of κ-Minkowski space-time by using representations of the corresponding deformed algebra in terms of undeformed Heisenberg-Weyl algebra. The deformed algebra consists of κ-Poincaré algebra extended with the generators of the deformed Weyl algebra. The part of deformed algebra, generated by rotation, boost and momentum generators, is described by the Hopf algebra structure. The approach used in our considerations is completely Lorentz covariant. We further use an advantage of this approach to consistently construct a star product, which has a property that under integration sign, it can be replaced by a standard pointwise multiplication, a property that was since known to hold for Moyal but not for κ-Minkowski space-time. This star product also has generalized trace and cyclic properties, and the construction alone is accomplished by considering a classical Dirac operator representation of deformed algebra and requiring it to be Hermitian. We find that the obtained star product is not translationally invariant, leading to a conclusion that the classical Dirac operator representation is the one where translation invariance cannot simultaneously be implemented along with hermiticity. However, due to the integral property satisfied by the star product, noncommutative free scalar field theory does not have a problem with translation symmetry breaking and can be shown to reduce to an ordinary free scalar field theory without nonlocal features and tachyonic modes and basically of the very same form. The issue of Lorentz invariance of the theory is also discussed.
Free-field realisations of BMS$_3$ and super-BMS$_3$ algebras
Banerjee, Nabamita; Mukhi, Sunil; Neogi, Turmoli
2015-01-01
We construct an explicit realisation of the BMS$_3$ algebra with nonzero central charges using holomorphic free fields. This can be extended by the addition of chiral matter to a realisation having arbitrary values for the two independent central charges. We show that our construction naturally extends to a coupled SU(2)-BMS$_3$ system where the SU(2) Kac-Moody symmetry is realised via the Wakimoto representation, and to the supersymmetric BMS$_3$ algebra.
K-Theory of Crossed Products of Tiling C*-Algebras by Rotation Groups
Starling, Charles
2015-02-01
Let Ω be a tiling space and let G be the maximal group of rotations which fixes Ω. Then the cohomology of Ω and Ω/ G are both invariants which give useful geometric information about the tilings in Ω. The noncommutative analog of the cohomology of Ω is the K-theory of a C*-algebra associated to Ω, and for translationally finite tilings of dimension 2 or less, the K-theory is isomorphic to the direct sum of cohomology groups. In this paper we give a prescription for calculating the noncommutative analog of the cohomology of Ω/ G, that is, the K-theory of the crossed product of the tiling C*-algebra by G. We also provide a table with some calculated K-groups for many common examples, including the Penrose and pinwheel tilings.
Kinematic analysis of parallel manipulators by algebraic screw theory
Gallardo-Alvarado, Jaime
2016-01-01
This book reviews the fundamentals of screw theory concerned with velocity analysis of rigid-bodies, confirmed with detailed and explicit proofs. The author additionally investigates acceleration, jerk, and hyper-jerk analyses of rigid-bodies following the trend of the velocity analysis. With the material provided in this book, readers can extend the theory of screws into the kinematics of optional order of rigid-bodies. Illustrative examples and exercises to reinforce learning are provided. Of particular note, the kinematics of emblematic parallel manipulators, such as the Delta robot as well as the original Gough and Stewart platforms are revisited applying, in addition to the theory of screws, new methods devoted to simplify the corresponding forward-displacement analysis, a challenging task for most parallel manipulators. Stands as the only book devoted to the acceleration, jerk and hyper-jerk (snap) analyses of rigid-body by means of screw theory; Provides new strategies to simplify the forward kinematic...
Massless conformal fields, AdS(d+1/CFTd higher spin algebras and their deformations
Directory of Open Access Journals (Sweden)
Sudarshan Fernando
2016-03-01
Full Text Available We extend our earlier work on the minimal unitary representation of SO(d,2 and its deformations for d=4,5 and 6 to arbitrary dimensions d. We show that there is a one-to-one correspondence between the minrep of SO(d,2 and its deformations and massless conformal fields in Minkowskian spacetimes in d dimensions. The minrep describes a massless conformal scalar field, and its deformations describe massless conformal fields of higher spin. The generators of Joseph ideal vanish identically as operators for the quasiconformal realization of the minrep, and its enveloping algebra yields directly the standard bosonic AdS(d+1/CFTd higher spin algebra. For deformed minreps the generators of certain deformations of Joseph ideal vanish as operators and their enveloping algebras lead to deformations of the standard bosonic higher spin algebra. In odd dimensions there is a unique deformation of the higher spin algebra corresponding to the spinor singleton. In even dimensions one finds infinitely many deformations of the higher spin algebra labelled by the eigenvalues of Casimir operator of the little group SO(d−2 for massless representations.
Schneider, Hans
1989-01-01
Linear algebra is one of the central disciplines in mathematics. A student of pure mathematics must know linear algebra if he is to continue with modern algebra or functional analysis. Much of the mathematics now taught to engineers and physicists requires it.This well-known and highly regarded text makes the subject accessible to undergraduates with little mathematical experience. Written mainly for students in physics, engineering, economics, and other fields outside mathematics, the book gives the theory of matrices and applications to systems of linear equations, as well as many related t
Galois Correspondence in Field Algebra of G-spin Model
Institute of Scientific and Technical Information of China (English)
蒋立宁; 郭懋正
2003-01-01
@@ A C*-system is a pair (B, G) consisting of a unital C*-algebra B and a continuous group homomorphism α: G → Aut(B) where G is a compact group and Aut(B) the group of automor-phisms of B. If K is a normal subgroup of G and BK = {B∈ B: k(B) = B, k ∈ K}, then BK is a G-invariant C*-subalgebra of B. On the other hand, if A is a G-invariant C*-algebra with BG A B, set G (A) = {g ∈ G: g(A) = A, A ∈ A}, G (A) is a normal subgroup of G. Clearly K G(BK) and we call K Galois closed ifK = G(BK). Similarly, A BG(A) and we call A Galois closed if A = BG(A).
Kwak, Seung Ki
The existence of momentum and winding modes of closed string on a torus leads to a natural idea that the field theoretical approach of string theory should involve winding type coordinates as well as the usual space-time coordinates. Recently developed double field theory is motivated from this idea and it implements T-duality manifestly by doubling the coordinates. In this thesis we will mainly focus on the double field theory formulation of different string theories in its low energy limit: bosonic, heterotic, type II and its massive extensions, and N = 1 supergravity theory. In chapter 2 of the thesis we study the equivalence of different formulations of double field theory. There are three different formulations of double field theory: background field E formulation, generalized metric H formulation, and frame field EAM formulation. Starting from the frame field formalism and choosing an appropriate gauge, the equivalence of the three formulations of bosonic theory are explicitly verified. In chapter 3 we construct the double field theory formulation of heterotic strings. The global symmetry enlarges to O( D, D + n) for heterotic strings and the enlarged generalized metric features this symmetry. The structural form of bosonic theory can directly be applied to the heterotic theory with the enlarged generalized metric. In chapter 4 we develop a unified framework of double field theory for type II theories. The Ramond-Ramond potentials fit into spinor representations of the duality group O( D, D) and the theory displays Spin+( D, D) symmetry with its self-duality relation. For a specific form of RR 1-form the theory reduces to the massive deformation of type IIA theory due to Romans. In chapter 5 we formulate the N = 1 supersymmetric extension of double field theory including the coupling to n abelian vector multiplets. This theory features a local O(1, 9 + n) x O(1, 9) tangent space symmetry under which the fermions transform. (Copies available exclusively from
Broken symmetries in field theory
Kok, Mark Okker de
2008-01-01
The thesis discusses the role of symmetries in Quantum Field Theory. Quantum Field Theory is the mathematical framework to describe the physics of elementary particles. A symmetry here means a transformation under which the model at hand is invariant. Three types of symmetry are distinguished: 1. In
Properties of double field theory
Penas, Victor Alejandro
2016-01-01
In this thesis we study several aspects of Double Field Theory (DFT). In general, Double Field Theory is subject to the so-called strong constraint. By using the Flux Formulation of DFT, we explore to what extent one can deal with the gauge consistency constraints of DFT without imposing the strong
Kreuzer, H.; Watanabe, K
1988-01-01
We discuss the explicit construction of diabatic states which form the basis to study the kinetics of field desorption, ionization and eventually field-induced surface chemistry. We indicate the calculation of the temperature and field dependence of energy dependent ion yields starting from a master equation.
Symmetric structure of field algebra of G-spin models determined by a normal subgroup
Energy Technology Data Exchange (ETDEWEB)
Xin, Qiaoling, E-mail: xinqiaoling0923@163.com; Jiang, Lining, E-mail: jianglining@bit.edu.cn [School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081 (China)
2014-09-15
Let G be a finite group and H a normal subgroup. D(H; G) is the crossed product of C(H) and CG which is only a subalgebra of D(G), the double algebra of G. One can construct a C*-subalgebra F{sub H} of the field algebra F of G-spin models, so that F{sub H} is a D(H; G)-module algebra, whereas F is not. Then the observable algebra A{sub (H,G)} is obtained as the D(H; G)-invariant subalgebra of F{sub H}, and there exists a unique C*-representation of D(H; G) such that D(H; G) and A{sub (H,G)} are commutants with each other.
Solutions in Exceptional Field Theory
Rudolph, Felix J
2015-01-01
Exceptional Field Theory employs an extended spacetime to make supergravity fully covariant under the U-duality groups of M-theory. This allows for the wave and monopole solutions to be combined into a single solution which obeys a twisted self-duality relation. All fundamental, solitonic and Dirichlet branes of ten- and eleven-dimensonal supergravity may be extracted from this single solution in Exceptional Field Theory.
Scale-adaptive tensor algebra for local many-body methods of electronic structure theory
Energy Technology Data Exchange (ETDEWEB)
Liakh, Dmitry I [ORNL
2014-01-01
While the formalism of multiresolution analysis (MRA), based on wavelets and adaptive integral representations of operators, is actively progressing in electronic structure theory (mostly on the independent-particle level and, recently, second-order perturbation theory), the concepts of multiresolution and adaptivity can also be utilized within the traditional formulation of correlated (many-particle) theory which is based on second quantization and the corresponding (generally nonorthogonal) tensor algebra. In this paper, we present a formalism called scale-adaptive tensor algebra (SATA) which exploits an adaptive representation of tensors of many-body operators via the local adjustment of the basis set quality. Given a series of locally supported fragment bases of a progressively lower quality, we formulate the explicit rules for tensor algebra operations dealing with adaptively resolved tensor operands. The formalism suggested is expected to enhance the applicability and reliability of local correlated many-body methods of electronic structure theory, especially those directly based on atomic orbitals (or any other localized basis functions).
Finite automata, their algebras and grammars towards a theory of formal expressions
Büchi, J Richard
1989-01-01
The author, who died in 1984, is well-known both as a person and through his research in mathematical logic and theoretical computer science. In the first part of the book he presents the new classical theory of finite automata as unary algebras which he himself invented about 30 years ago. Many results, like his work on structure lattices or his characterization of regular sets by generalized regular rules, are unknown to a wider audience. In the second part of the book he extends the theory to general (non-unary, many-sorted) algebras, term rewriting systems, tree automata, and pushdown automata. Essentially Büchi worked independent of other rersearch, following a novel and stimulating approach. He aimed for a mathematical theory of terms, but could not finish the book. Many of the results are known by now, but to work further along this line presents a challenging research program on the borderline between universal algebra, term rewriting systems, and automata theory. For the whole book and aga...
Lutfiyya, Lutfi A
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Modern Algebra includes set theory, operations, relations, basic properties of the integers, group theory, and ring theory.
Renormalization and effective field theory
Costello, Kevin
2011-01-01
This book tells mathematicians about an amazing subject invented by physicists and it tells physicists how a master mathematician must proceed in order to understand it. Physicists who know quantum field theory can learn the powerful methodology of mathematical structure, while mathematicians can position themselves to use the magical ideas of quantum field theory in "mathematics" itself. The retelling of the tale mathematically by Kevin Costello is a beautiful tour de force. --Dennis Sullivan This book is quite a remarkable contribution. It should make perturbative quantum field theory accessible to mathematicians. There is a lot of insight in the way the author uses the renormalization group and effective field theory to analyze perturbative renormalization; this may serve as a springboard to a wider use of those topics, hopefully to an eventual nonperturbative understanding. --Edward Witten Quantum field theory has had a profound influence on mathematics, and on geometry in particular. However, the notorio...
Lattice models and conformal field theories
International Nuclear Information System (INIS)
Theoretical studies concerning the connection between critical physical systems and the conformal theories are reviewed. The conformal theory associated to a critical (integrable) lattice model is derived. The obtention of the central charge, critical exponents and torus partition function, using renormalization group arguments, is shown. The quantum group structure, in the integrable lattice models, and the theory of Visaro algebra representations are discussed. The relations between off-critical integrable models and conformal theories, in finite geometries, are studied
Kleyn, Aleks
2007-01-01
The concept of F-algebra and its representation can be extended to an arbitrary bundle. We define operations of fibered F-algebra in fiber. The paper presents the representation theory of of fibered F-algebra as well as a comparison of representation of F-algebra and of representation of fibered F-algebra.
Mattson Solomon transform and algebra codes
DEFF Research Database (Denmark)
Martínez-Moro, E.; Benito, Diego Ruano
2009-01-01
In this note we review some results of the first author on the structure of codes defined as subalgebras of a commutative semisimple algebra over a finite field (see Martínez-Moro in Algebra Discrete Math. 3:99-112, 2007). Generator theory and those aspects related to the theory of Gröbner bases...
[Topics in field theory and string theory
International Nuclear Information System (INIS)
In the past year, I have continued to investigate the relations between conformal field theories and lattice statistical mechanical models, and in particular have been studying two dimensional models coupled to quantum gravity. I have continued as well to consider possible extension of these results to higher dimensions and potential applications in other contexts
Some combinatorial interpretations in perturbative quantum field theory
Yeats, Karen
2013-01-01
This paper will describe how combinatorial interpretations can help us understand the algebraic structure of two aspects of perturbative quantum field theory, namely analytic Dyson-Schwinger equations and periods of scalar Feynman graphs. The particular examples which will be looked at are, a better reduction to geometric series for Dyson-Schwinger equations, a subgraph which yields extra denominator reductions in scalar Feynman integrals, and an explanation of a trick of Brown and Schnetz to...
Non-Equilibrium Thermodynamics in Conformal Field Theory
Hollands, Stephan
2016-01-01
We present a model independent, operator algebraic approach to non-equilibrium quantum thermodynamics within the framework of two-dimensional Conformal Field Theory. Two infinite reservoirs in equilibrium at their own temperatures and chemical potentials are put in contact through a defect line, possibly by inserting a probe. As time evolves, the composite system then approaches a non-equilibrium steady state that we describe. In particular, we re-obtain recent formulas of Bernard and Doyon.
Algebra of constraints for supersymmetric Yang-Mills theory coupled to supergravity
International Nuclear Information System (INIS)
The second-order canonical vierbein formalism for the supersymmetric Yang-Mills theory coupled to supergravity is constructed. This is done by starting from the first-order canonical-covariant formalism on a group manifold previously developed. The set of first-class constraints, which verify the constraint algebra, are explicitly computed and the extended Hamiltonian which generates the time evolution of the system is written
Algebra of constraints for supersymmetric Yang-Mills theory coupled to supergravity
Energy Technology Data Exchange (ETDEWEB)
Foussats, A.; Zandron, O. (Facultad de Ciencias Exactas Ingenieria y Agrimensura, Universidad de Rosario Av., Pellegrini 250, 2000 Rosario, Argentina (AR))
1991-03-15
The second-order canonical vierbein formalism for the supersymmetric Yang-Mills theory coupled to supergravity is constructed. This is done by starting from the first-order canonical-covariant formalism on a group manifold previously developed. The set of first-class constraints, which verify the constraint algebra, are explicitly computed and the extended Hamiltonian which generates the time evolution of the system is written.
The Global Approach to Quantum Field Theory
International Nuclear Information System (INIS)
theory. This is the so-called global approach to quantum field theory where time does not play any particular role, and quantization is then naturally realized covariantly using tools such as the Peierls bracket (a covariant generalization of Poisson bracket), the Schwinger variational principle and Feynman sums over histories. However, it should be noted that the boycott of canonical methods by DeWitt is not total: when he judges they genuinely illuminate the physics of a problem, he does not hesitate to descend from the global point of view and to use them. In a few words, we have in fact described the research program initiated by DeWitt forty years ago, which has progressively evolved in order to take into account the latest development of gauge theories. While the Les Houches Lectures of 1963 were mainly concentrated on the formal structure and the quantization of Yang--Mills and gravitational fields, the present book also deals with more general gauge theories including those with open gauge algebras and structure functions, and therefore supergravity theories. More precisely, the book, more than a thousand pages in length, consists of eight parts and is completed by six appendices where certain technical aspects are singled out. An enormous variety of topics is covered, including the invariance transformations of the action functional, the Batalin-Vilkovisky formalism, Green's functions, the Peierls bracket, conservation laws, the theory of measurement, the Everett (or many worlds) interpretation of quantum mechanics, decoherence, the Schwinger variational principle and Feynm an functional integrals, the heat kernel, aspects of quantization for linear systems in stationary and non-stationary backgrounds, the S-matrix, the background field method, the effective action and the Vilkovisky-DeWitt formalism, the quantization of gauge theories without ghosts, anomalies, black holes and Hawking radiation, renormalization, and more. It should be noted that DeWitt's book
Solvability of a Lie algebra of vector fields implies their integrability by quadratures
Cariñena, J. F.; Falceto, F.; Grabowski, J.
2016-10-01
We present a substantial generalisation of a classical result by Lie on integrability by quadratures. Namely, we prove that all vector fields in a finite-dimensional transitive and solvable Lie algebra of vector fields on a manifold can be integrated by quadratures.
Subgroups of ideal class groups of real quadratic algebraic function fields
Institute of Scientific and Technical Information of China (English)
WANG; Kunpeng(王鲲鹏); ZHANG; Xianke(张贤科)
2003-01-01
Necessary and sufficient condition on real quadratic algebraic function fields K is given for theirideal class groups H(K) to contain cyclic subgroups of order n. And eight series of such real quadratic functionfields K are obtained whose ideal class groups contain cyclic subgroups of order n. In particular, the ideal classnumbers of these function fields are divisible by n.
Relating the archetypes of logarithmic conformal field theory
Energy Technology Data Exchange (ETDEWEB)
Creutzig, Thomas, E-mail: tcreutzig@mathematik.tu-darmstadt.de [Department of Physics and Astronomy, University of North Carolina, Phillips Hall, CB 3255, Chapel Hill, NC 27599-3255 (United States); Fachbereich Mathematik, Technische Universität Darmstadt, Schloßgartenstraße 7, 64289 Darmstadt (Germany); Ridout, David, E-mail: david.ridout@anu.edu.au [Department of Theoretical Physics, Research School of Physics and Engineering, Australian National University, Canberra, ACT 0200 (Australia); Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200 (Australia)
2013-07-21
Logarithmic conformal field theory is a rich and vibrant area of modern mathematical physics with well-known applications to both condensed matter theory and string theory. Our limited understanding of these theories is based upon detailed studies of various examples that one may regard as archetypal. These include the c=−2 triplet model, the Wess–Zumino–Witten model on SL(2;R) at level k=−1/2 , and its supergroup analogue on GL(1|1). Here, the latter model is studied algebraically through representation theory, fusion and modular invariance, facilitating a subsequent investigation of its cosets and extended algebras. The results show that the archetypes of logarithmic conformal field theory are in fact all very closely related, as are many other examples including, in particular, the SL(2|1) models at levels 1 and −1/2 . The conclusion is then that the archetypal examples of logarithmic conformal field theory are practically all the same, so we should not expect that their features are in any way generic. Further archetypal examples must be sought.
Introduction to quantum field theory
International Nuclear Information System (INIS)
The lectures appear to be a continuation to the introduction to elementary principles of the quantum field theory. The work is aimed at constructing the formalism of standard particle interaction model. Efforts are made to exceed the limits of the standard model in the quantum field theory context. Grand unification models including strong and electrical weak interactions, supersymmetric generalizations of the standard model and grand unification theories and, finally, supergravitation theories including gravitation interaction to the universal scheme, are considered. 3 refs.; 19 figs.; 2 tabs
Real Algebraic Tools in Stochastic Games
Abraham Neyman
2001-01-01
The present chapter brings together parts of the theory of polynomial equalities and inequalities used in the theory of stochastic games. The theory can be considered as a theory of polynomial equalities and inequalities over the field of real numbers or the field of real algebraic numbers or more generally over an arbitrary real closed field.
Noncommutative gravity and quantum field theory on noncummutative curved spacetimes
Energy Technology Data Exchange (ETDEWEB)
Schenkel, Alexander
2011-10-24
The purpose of the first part of this thesis is to understand symmetry reduction in noncommutative gravity, which then allows us to find exact solutions of the noncommutative Einstein equations. We propose an extension of the usual symmetry reduction procedure, which is frequently applied to the construction of exact solutions of Einstein's field equations, to noncommutative gravity and show that this leads to preferred choices of noncommutative deformations of a given symmetric system. We classify in the case of abelian Drinfel'd twists all consistent deformations of spatially flat Friedmann-Robertson-Walker cosmologies and of the Schwarzschild black hole. The deformed symmetry structure allows us to obtain exact solutions of the noncommutative Einstein equations in many of our models, for which the noncommutative metric field coincides with the classical one. In the second part we focus on quantum field theory on noncommutative curved spacetimes. We develop a new formalism by combining methods from the algebraic approach to quantum field theory with noncommutative differential geometry. The result is an algebra of observables for scalar quantum field theories on a large class of noncommutative curved spacetimes. A precise relation to the algebra of observables of the corresponding undeformed quantum field theory is established. We focus on explicit examples of deformed wave operators and find that there can be noncommutative corrections even on the level of free field theories, which is not the case in the simplest example of the Moyal-Weyl deformed Minkowski spacetime. The convergent deformation of simple toy-models is investigated and it is shown that these quantum field theories have many new features compared to formal deformation quantization. In addition to the expected nonlocality, we obtain that the relation between the deformed and the undeformed quantum field theory is affected in a nontrivial way, leading to an improved behavior of the
Heydeman, Matthew; Saberi, Ingmar; Stoica, Bogdan
2016-01-01
One of the many remarkable properties of conformal field theory in two dimensions is its connection to algebraic geometry. Since every compact Riemann surface is a projective algebraic curve, many constructions of interest in physics (which a priori depend on the analytic structure of the spacetime) can be formulated in purely algebraic language. This opens the door to interesting generalizations, obtained by taking another choice of field: for instance, the $p$-adics. We generalize the AdS/CFT correspondence according to this principle; the result is a formulation of holography in which the bulk geometry is discrete---the Bruhat--Tits tree for $\\mathrm{PGL}(2,\\mathbb{Q}_p)$---but the group of bulk isometries nonetheless agrees with that of boundary conformal transformations and is not broken by discretization. We suggest that this forms the natural geometric setting for tensor networks that have been proposed as models of bulk reconstruction via quantum error correcting codes; in certain cases, geodesics in ...
Clifford algebra, geometric algebra, and applications
Lundholm, Douglas
2009-01-01
These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra construction (then called geometric algebra) with geometric applications in mind, as well as in an algebraically more general form which is well suited for combinatorics, and for defining and understanding the numerous products and operations of the algebra. The various applications presented include vector space and projective geometry, orthogonal maps and spinors, normed division algebras, as well as simplicial complexes and graph theory.
Quantum field theory competitive models
Tolksdorf, Jürgen; Zeidler, Eberhard
2009-01-01
For more than 70 years, quantum field theory (QFT) can be seen as a driving force in the development of theoretical physics. Equally fascinating is the fruitful impact which QFT had in rather remote areas of mathematics. The present book features some of the different approaches, different physically viewpoints and techniques used to make the notion of quantum field theory more precise. For example, the present book contains a discussion including general considerations, stochastic methods, deformation theory and the holographic AdS/CFT correspondence. It also contains a discussion of more recent developments like the use of category theory and topos theoretic methods to describe QFT. The present volume emerged from the 3rd 'Blaubeuren Workshop: Recent Developments in Quantum Field Theory', held in July 2007 at the Max Planck Institute of Mathematics in the Sciences in Leipzig/Germany. All of the contributions are committed to the idea of this workshop series: 'To bring together outstanding experts working in...
Maxfield, Travis; Sethi, Savdeep
2015-01-01
Studying a quantum field theory involves a choice of space-time manifold and a choice of background for any global symmetries of the theory. We argue that many more choices are possible when specifying the background. In the context of branes in string theory, the additional data corresponds to a choice of supergravity tensor fluxes. We propose the existence of a landscape of field theory backgrounds, characterized by the space-time metric, global symmetry background and a choice of tensor fluxes. As evidence for this landscape, we study the supersymmetric six-dimensional (2,0) theory compactified to two dimensions. Different choices of metric and flux give rise to distinct two-dimensional theories, which can preserve differing amounts of supersymmetry.
Cubic Twistorial String Field Theory
Berkovits, Nathan; Motl, Lubos
2004-01-01
Witten has recently proposed a string theory in twistor space whose D-instanton contributions are conjectured to compute N=4 super-Yang-Mills scattering amplitudes. An alternative string theory in twistor space was then proposed whose open string tree amplitudes reproduce the D-instanton computations of maximal degree in Witten's model. In this paper, a cubic open string field theory action is constructed for this alternative string in twistor space, and is shown to be invariant under parity ...
Nekrasov and Argyres-Douglas theories in spherical Hecke algebra representation
Rim, Chaiho
2016-01-01
AGT conjecture connects Nekrasov instanton partition function of 4D quiver gauge theory with 2D Liouville conformal blocks. We re-investigate this connection using the central extension of spherical Hecke algebra in q-coordinate representation, q being the instanton expansion parameter. Based on AFLT basis together with Matsuo's interwiner we construct gauge conformal state and demonstrate its equivalence to the Liouville conformal state with careful attention to the proper scaling behavior of the state. Using the colliding limit of regular states, we obtain the formal expression of irregular conformal states corresponding to Argyres-Douglas theory which involves summation of functions over Young diagrams.
The Theory of Conceptual Fields
Vergnaud, Gerard
2009-01-01
The theory of conceptual fields is a developmental theory. It has two aims: (1) to describe and analyse the progressive complexity, on a long- and medium-term basis, of the mathematical competences that students develop inside and outside school, and (2) to establish better connections between the operational form of knowledge, which consists in…
Conformal field theories with infinitely many conservation laws
International Nuclear Information System (INIS)
Globally conformal invariant quantum field theories in a D-dimensional space-time (D even) have rational correlation functions and admit an infinite number of conserved (symmetric traceless) tensor currents. In a theory of a scalar field of dimension D-2 they were demonstrated to be generated by bilocal normal products of free massless scalar fields with an O(N), U(N), or Sp(2N) (global) gauge symmetry [B. Bakalov, N. M. Nikolov, K.-H. Rehren, and I. Todorov, “Unitary positive energy representations of scalar bilocal fields,” Commun. Math. Phys. 271, 223–246 (2007); e-print arXiv:math-ph/0604069v3; and “Infinite dimensional Lie algebras in 4D conformal quantum field theory,” J. Phys. A Math Theor. 41, 194002 (2008); e-print arXiv:0711.0627v2 [hep-th
On the renormalization of non-commutative field theories
Blaschke, Daniel N.; Garschall, Thomas; Gieres, François; Heindl, Franz; Schweda, Manfred; Wohlgenannt, Michael
2013-01-01
This paper addresses three topics concerning the quantization of non-commutative field theories (as defined in terms of the Moyal star product involving a constant tensor describing the non-commutativity of coordinates in Euclidean space). To start with, we discuss the Quantum Action Principle and provide evidence for its validity for non-commutative quantum field theories by showing that the equation of motion considered as insertion in the generating functional Z c [ j] of connected Green functions makes sense (at least at one-loop level). Second, we consider the generalization of the BPHZ renormalization scheme to non-commutative field theories and apply it to the case of a self-interacting real scalar field: Explicit computations are performed at one-loop order and the generalization to higher loops is commented upon. Finally, we discuss the renormalizability of various models for a self-interacting complex scalar field by using the approach of algebraic renormalization.
Conformal field theories with infinitely many conservation laws
Energy Technology Data Exchange (ETDEWEB)
Todorov, Ivan [Institut des Hautes Etudes Scientifiques F-91440, Bures-sur-Yvette (France)
2013-02-15
Globally conformal invariant quantum field theories in a D-dimensional space-time (D even) have rational correlation functions and admit an infinite number of conserved (symmetric traceless) tensor currents. In a theory of a scalar field of dimension D-2 they were demonstrated to be generated by bilocal normal products of free massless scalar fields with an O(N), U(N), or Sp(2N) (global) gauge symmetry [B. Bakalov, N. M. Nikolov, K.-H. Rehren, and I. Todorov, 'Unitary positive energy representations of scalar bilocal fields,' Commun. Math. Phys. 271, 223-246 (2007); e-print arXiv:math-ph/0604069v3; and 'Infinite dimensional Lie algebras in 4D conformal quantum field theory,' J. Phys. A Math Theor. 41, 194002 (2008); e-print arXiv:0711.0627v2 [hep-th
Algebras of Complete Hörmander Vector Fields, and Lie-Group Construction
Directory of Open Access Journals (Sweden)
Andrea Bonfiglioli
2014-12-01
Full Text Available The aim of this note is to characterize the Lie algebras g of the analytic vector fields in RN which coincide with the Lie algebras of the (analytic Lie groups defined on RN (with its usual differentiable structure. We show that such a characterization amounts to asking that: (i g is N-dimensional; (ii g admits a set of Lie generators which are complete vector fields; (iii g satisfies Hörmander’s rank condition. These conditions are necessary, sufficient and mutually independent. Our approach is constructive, in that for any such g we show how to construct a Lie group G = (RN, * whose Lie algebra is g. We do not make use of Lie’s Third Theorem, but we only exploit the Campbell-Baker-Hausdorff-Dynkin Theorem for ODE’s.
Representations of Conformal Nets, Universal C*-Algebras and K-Theory
Carpi, Sebastiano; Conti, Roberto; Hillier, Robin; Weiner, Mihály
2013-05-01
We study the representation theory of a conformal net {{A}} on S 1 from a K-theoretical point of view using its universal C*-algebra {C^*({A})}. We prove that if {{A}} satisfies the split property then, for every representation π of {{A}} with finite statistical dimension, {π(C^*({A}))} is weakly closed and hence a finite direct sum of type I∞ factors. We define the more manageable locally normal universal C*-algebra {C_ln^*({A})} as the quotient of {C^*({A})} by its largest ideal vanishing in all locally normal representations and we investigate its structure. In particular, if {{A}} is completely rational with n sectors, then {C_ln^*({A})} is a direct sum of n type I∞ factors. Its ideal {{K}_{A}} of compact operators has nontrivial K-theory, and we prove that the DHR endomorphisms of {C^*({A})} with finite statistical dimension act on {{K}_{A}}, giving rise to an action of the fusion semiring of DHR sectors on {K_0({K}_{A})}. Moreover, we show that this action corresponds to the regular representation of the associated fusion algebra.
Quantum processes: A Whiteheadian interpretation of quantum field theory
Bain, Jonathan
Quantum processes: A Whiteheadian interpretation of quantum field theory is an ambitious and thought-provoking exercise in physics and metaphysics, combining an erudite study of the very complex metaphysics of A.N. Whitehead with a well-informed discussion of contemporary issues in the philosophy of algebraic quantum field theory. Hättich's overall goal is to construct an interpretation of quantum field theory. He does this by translating key concepts in Whitehead's metaphysics into the language of algebraic quantum field theory. In brief, this Hättich-Whitehead (H-W, hereafter) interpretation takes "actual occasions" as the fundamental ontological entities of quantum field theory. An actual occasion is the result of two types of processes: a "transition process" in which a set of initial possibly-possessed properties for the occasion (in the form of "eternal objects") is localized to a space-time region; and a "concrescence process" in which a subset of these initial possibly-possessed properties is selected and actualized to produce the occasion. Essential to these processes is the "underlying activity", which conditions the way in which properties are initially selected and subsequently actualized. In short, under the H-W interpretation of quantum field theory, an initial set of possibly-possessed eternal objects is represented by a Boolean sublattice of the lattice of projection operators determined by a von Neumann algebra R (O) associated with a region O of Minkowski space-time, and the underlying activity is represented by a state on R (O) obtained by conditionalizing off of the vacuum state. The details associated with the H-W interpretation involve imposing constraints on these representations motivated by principles found in Whitehead's metaphysics. These details are spelled out in the three sections of the book. The first section is a summary and critique of Whitehead's metaphysics, the second section introduces the formalism of algebraic quantum field
Solvable Lie Algebras in Type IIA, Type IIB and M Theories
Andrianopoli, Laura; Ferrara, Sergio; Fré, P; Minasian, R; Trigiante, M
1997-01-01
We study some applications of solvable Lie algebras in type IIA, type IIB and M theories. RR and NS generators find a natural geometric interpretation in this framework. Special emphasis is given to the counting of the abelian nilpotent ideals (translational symmetries of the scalar manifolds) in arbitrary D dimensions. These are seen to be related, using Dynkin diagram techniques, to one-form counting in D+1 dimensions. A recipy for gauging isometries in this framework is also presented. In particular, we list the gauge groups both for compact and translational isometries. The former agree with some results already existing in gauged supergravity. The latter should be possibly related to the study of partial supersymmetry breaking, as suggested by a similar role played by solvable Lie algebras in N=2 gauged supergravity.
Directory of Open Access Journals (Sweden)
Shulin Wu
2009-01-01
Full Text Available We propose a new idea to construct an effective algorithm to compute the minimal positive solution of the nonsymmetric algebraic Riccati equations arising from transport theory. For a class of these equations, an important feature is that the minimal positive solution can be obtained by computing the minimal positive solution of a couple of fixed-point equations with vector form. Based on the fixed-point vector equations, we introduce a new algorithm, namely, two-step relaxation Newton, derived by combining two different relaxation Newton methods to compute the minimal positive solution. The monotone convergence of the solution sequence generated by this new algorithm is established. Numerical results are given to show the advantages of the new algorithm for the nonsymmetric algebraic Riccati equations in vector form.
Neural fields theory and applications
Graben, Peter; Potthast, Roland; Wright, James
2014-01-01
With this book, the editors present the first comprehensive collection in neural field studies, authored by leading scientists in the field - among them are two of the founding-fathers of neural field theory. Up to now, research results in the field have been disseminated across a number of distinct journals from mathematics, computational neuroscience, biophysics, cognitive science and others. Starting with a tutorial for novices in neural field studies, the book comprises chapters on emergent patterns, their phase transitions and evolution, on stochastic approaches, cortical development, cognition, robotics and computation, large-scale numerical simulations, the coupling of neural fields to the electroencephalogram and phase transitions in anesthesia. The intended readership are students and scientists in applied mathematics, theoretical physics, theoretical biology, and computational neuroscience. Neural field theory and its applications have a long-standing tradition in the mathematical and computational ...
Spin gauge field theory of electric and magnetic spinors
International Nuclear Information System (INIS)
In the first section, a gauge theory of an unquantized generalized electron interacting with the electromagnetic field through two vector potentials is formulated, based on invariance of the Lagrangian under an algebra of spin space transformations. The covariant derivative is essentially expressed in terms of spin space operators. It is not possible to define dual monopole spinors in a four-component theory. However, a modified eight-component generalized electron gauge theory transforms into a dual monopole theory by using a square root of the charge conjugation operator. The covariant derivatives of the two spinors are members of a continuous set, and define curvature and torsion in spin space corresponding to the two spinors. Physically important 'weak spin curvature' is closely related to the total electromagnetic field. Possible physical interpretations and extensions of the theory are discussed. (author)
Spin gauge field theory of electric and magnetic spinors
Energy Technology Data Exchange (ETDEWEB)
Chisholm, J.S.R.; Farwell, R.S. (Kent Univ., Canterbury (UK))
1981-06-05
In the first section, a gauge theory of an unquantized generalized electron interacting with the electromagnetic field through two vector potentials is formulated, based on invariance of the Lagrangian under an algebra of spin space transformations. The covariant derivative is essentially expressed in terms of spin space operators. It is not possible to define dual monopole spinors in a four-component theory. However, a modified eight-component generalized electron gauge theory transforms into a dual monopole theory by using a square root of the charge conjugation operator. The covariant derivatives of the two spinors are members of a continuous set, and define curvature and torsion in spin space corresponding to the two spinors. Physically important 'weak spin curvature' is closely related to the total electromagnetic field. Possible physical interpretations and extensions of the theory are discussed.
Solutions in Exceptional Field Theory
Energy Technology Data Exchange (ETDEWEB)
Rudolph, Felix J. [Queen Mary University of London, Centre for Research in String Theory, School of Physics, London (United Kingdom)
2016-04-15
Exceptional Field Theory employs an extended spacetime to make supergravity fully covariant under the U-duality groups of M-theory. This allows for the wave and monopole solutions to be combined into a single solution which obeys a twisted self-duality relation. All fundamental, solitonic and Dirichlet branes of ten- and eleven-dimensonal supergravity may be extracted from this single solution in Exceptional Field Theory. (copyright 2015 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
Noncommutative Gravity and Quantum Field Theory on Noncommutative Curved Spacetimes
Schenkel, Alexander
2012-01-01
The focus of this PhD thesis is on applications, new developments and extensions of the noncommutative gravity theory proposed by Julius Wess and his group. In part one we propose an extension of the usual symmetry reduction procedure to noncommutative gravity. We classify in the case of abelian Drinfel'd twists all consistent deformations of spatially flat Friedmann-Robertson-Walker cosmologies and of the Schwarzschild black hole. The deformed symmetry structure allows us to obtain exact solutions of the noncommutative Einstein equations in many of our models. In part two we develop a new formalism for quantum field theory on noncommutative curved spacetimes by combining methods from the algebraic approach to quantum field theory with noncommutative differential geometry. We also study explicit examples of deformed wave operators and find that there can be noncommutative corrections even on the level of free field theories. The convergent deformation of simple toy models is investigated and it is found that ...
Relativistic local quantum field theory for m=0 particles
International Nuclear Information System (INIS)
A method is introduced ta deal with relativistic quantum field theory for particles with m=0. Two mappings I and J, giving rise respectively to particle and anti particle states, are defined between a test space and the physical Hilbert space. The intrinsic field operator is then defined as the minimal causal linear combinations of operators belonging to the annihilation-creation algebra associated to the germ and antigerm parts of the element. Local elements are introduced as improper test elements and local field operators are constructed in the same way as the intrinsic ones. Commutation rules are given. (Author) 17 refs
Geometer energy unified field theory
Rivera, Susana; Rivera, Anacleto
GEOMETER - ENERGY UNIFIED FIELD THEORY Author: Anacleto Rivera Nivón Co-author: Susana Rivera Cabrera This work is an attempt to find the relationship between the Electromagnetic Field and the Gravitational Field. Despite it is based on the existence of Strings of Energy, it is not the same kind of strings that appears on other theories like Superstring Theory, Branas Theory, M - Theory, or any other related string theories. Here, the Strings are concentrated energy lines that vibrates, and experiences shrinking and elongations, absorbing and yielding on each contraction and expansion all that is found in the Universe: matter and antimatter, waves and energy in all manifestations. In contrast to superstring theory, which strings are on the range of the Length of Planck, these Strings can be on the cosmological size, and can contain many galaxies, or clusters, or groups of galaxies; but also they can reach as small sizes as subatomic levels. Besides, and contrary to what it is stated in some other string theories that need the existence of ten or more dimensions, the present proposal sustains in only four particular dimensions. It has been developed a mathematical support that will try to help to improve the understanding of the phenomena that take place at the Universe.
Cubic twistorial string field theory
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Berkovits, Nathan; Motl, Lubos E-mail: motl@feynman.harvard.edu
2004-04-01
Witten has recently proposed a string theory in twistor space whose D-instanton contributions are conjectured to compute N=4 super-Yang-Mills scattering amplitudes. An alternative string theory in twistor space was then proposed whose open string tree amplitudes reproduce the D-instanton computations of maximal degree in Witten's model. In this paper, a cubic open string field theory action is constructed for this alternative string in twistor space, and is shown to be invariant under parity transformations which exchange MHV and googly amplitudes. Since the string field theory action is gauge-invariant and reproduces the correct cubic super-Yang-Mills interactions, it provides strong support for the conjecture that the string theory correctly computes N-point super-Yang-Mills tree amplitudes. (author)
Cubic Twistorial String Field Theory
Berkovits, N; Berkovits, Nathan; Motl, Lubos
2004-01-01
Witten has recently proposed a string theory in twistor space whose D-instanton contributions are conjectured to compute N=4 super-Yang-Mills scattering amplitudes. An alternative string theory in twistor space was then proposed whose open string tree amplitudes reproduce the D-instanton computations of maximal degree in Witten's model. In this paper, a cubic open string field theory action is constructed for this alternative string in twistor space, and is shown to be invariant under parity transformations which exchange MHV and googly amplitudes. Since the string field theory action is gauge-invariant and reproduces the correct cubic super-Yang-Mills interactions, it provides strong support for the conjecture that the string theory correctly computes N-point super-Yang-Mills tree amplitudes.
Nonlocal and quasilocal field theories
Tomboulis, E. T.
2015-12-01
We investigate nonlocal field theories, a subject that has attracted some renewed interest in connection with nonlocal gravity models. We study, in particular, scalar theories of interacting delocalized fields, the delocalization being specified by nonlocal integral kernels. We distinguish between strictly nonlocal and quasilocal (compact support) kernels and impose conditions on them to insure UV finiteness and unitarity of amplitudes. We study the classical initial value problem for the partial integro-differential equations of motion in detail. We give rigorous proofs of the existence but accompanying loss of uniqueness of solutions due to the presence of future, as well as past, "delays," a manifestation of acausality. In the quantum theory we derive a generalization of the Bogoliubov causality condition equation for amplitudes, which explicitly exhibits the corrections due to nonlocality. One finds that, remarkably, for quasilocal kernels all acausal effects are confined within the compact support regions. We briefly discuss the extension to other types of fields and prospects of such theories.
Lectures on Conformal Field Theory
Qualls, Joshua D
2015-01-01
These lectures notes are based on courses given at National Taiwan University, National Chiao-Tung University, and National Tsing Hua University in the spring term of 2015. Although the course was offered primarily for graduate students, these lecture notes have been prepared for a more general audience. They are intended as an introduction to conformal field theories in various dimensions, with applications related to topics of particular interest: topics include the conformal bootstrap program, boundary conformal field theory, and applications related to the AdS/CFT correspondence. We assume the reader to be familiar with quantum mechanics at the graduate level and to have some basic knowledge of quantum field theory. Familiarity with string theory is not a prerequisite for this lectures, although it can only help.
[Studies in quantum field theory
International Nuclear Information System (INIS)
During the period 4/1/89--3/31/90 the theoretical physics group supported by Department of Energy Contract No. AC02-78ER04915.A015 and consisting of Professors Bender and Shrauner, Associate Professor Papanicolaou, Assistant Professor Ogilvie, and Senior Research Associate Visser has made progress in many areas of theoretical and mathematical physics. Professors Bender and Shrauner, Associate Professor Papanicolaou, Assistant Professor Ogilvie, and Research Associate Visser are currently conducting research in many areas of high energy theoretical and mathematical physics. These areas include: strong-coupling approximation; classical solutions of non-Abelian gauge theories; mean-field approximation in quantum field theory; path integral and coherent state representations in quantum field theory; lattice gauge calculations; the nature of perturbation theory in large order; quark condensation in QCD; chiral symmetry breaking; the 1/N expansion in quantum field theory; effective potential and action in quantum field theories, including OCD; studies of the early universe and inflation, and quantum gravity
Sine-square deformation and Mobius quantization of two-dimensional conformal field theory
Okunishi, Kouichi
2016-01-01
Motivated by sine-square deformation (SSD) for quantum critical systems in 1+1-dimension, we discuss a Mobius quantization approach to the two-dimensional conformal field theory (CFT), which bridges the conventional radial quantization and the dipolar quantization recently proposed by Ishibashi and Tada. We then find that the continuous Virasoro algebra of the dipolar quantization can be interpreted as a continuum limit of the Virasoro algebra for scaled generators in the SSD limit of the Mobius quantization approach.
Background Independent String Field Theory
Bars, Itzhak
2014-01-01
We develop a new background independent Moyal star formalism in bosonic open string field theory. The new star product is formulated in a half-phase-space, and because phase space is independent of any background fields, the interactions are background independent. In this basis there is a large amount of symmetry, including a supersymmetry OSp(d|2) that acts on matter and ghost degrees of freedom, and simplifies computations. The BRST operator that defines the quadratic kinetic term of string field theory may be regarded as the solution of the equation of motion A*A=0 of a purely cubic background independent string field theory. We find an infinite number of non-perturbative solutions to this equation, and are able to associate them to the BRST operator of conformal field theories on the worldsheet. Thus, the background emerges from a spontaneous-type breaking of a purely cubic highly symmetric theory. The form of the BRST field breaks the symmetry in a tractable way such that the symmetry continues to be us...
S-duality and the prepotential of N=2* theories (I): the ADE algebras
Billo', M; Fucito, F; Lerda, A; Morales, J F
2015-01-01
The prepotential of N=2* supersymmetric theories with unitary gauge groups in an Omega-background satisfies a modular anomaly equation that can be recursively solved order by order in an expansion for small mass. By requiring that S-duality acts on the prepotential as a Fourier transform we generalise this result to N=2* theories with gauge algebras of the D and E type and show that their prepotentials can be written in terms of quasi-modular forms of SL(2,Z). The results are checked against microscopic multi-instanton calculus based on localization for the A and D series and reproduce the known 1-instanton prepotential of the pure N=2 theories for any gauge group of ADE type. Our results can also be used to obtain the multi-instanton terms in the exceptional theories for which the microscopic instanton calculus and the ADHM construction are not available.
S-duality and the prepotential in N={2}^{star } theories (I): the ADE algebras
Billó, M.; Frau, M.; Fucito, F.; Lerda, A.; Morales, J. F.
2015-11-01
The prepotential of N={2}^{star } supersymmetric theories with unitary gauge groups in an Ω background satisfies a modular anomaly equation that can be recursively solved order by order in an expansion for small mass. By requiring that S-duality acts on the prepotential as a Fourier transform we generalise this result to N={2}^{star } theories with gauge algebras of the D and E type and show that their prepotentials can be written in terms of quasi-modular forms of SL(2, {Z}) . The results are checked against microscopic multi-instanton calculus based on localization for the A and D series and reproduce the known 1-instanton prepotential of the pure N=2 theories for any gauge group of ADE type. Our results can also be used to obtain the multi-instanton terms in the exceptional theories for which the microscopic instanton calculus and the ADHM construction are not available.
Grabowski, Jan
2015-01-01
In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We transfer a definition of Gekhtman, Shapiro and Vainshtein to the algebraic setting, yielding the notion of a multi-graded cluster algebra. We then study gradings for finite type cluster algebras without coefficients, giving a full classification. Translating ...
Twisted Logarithmic Modules of Vertex Algebras
Bakalov, Bojko
2016-07-01
Motivated by logarithmic conformal field theory and Gromov-Witten theory, we introduce a notion of a twisted module of a vertex algebra under an arbitrary (not necessarily semisimple) automorphism. Its main feature is that the twisted fields involve the logarithm of the formal variable. We develop the theory of such twisted modules and, in particular, derive a Borcherds identity and commutator formula for them. We investigate in detail the examples of affine and Heisenberg vertex algebras.
Studies of gauge field theories in terms of local gauge-invariant quantities
International Nuclear Information System (INIS)
In the framework of the functional-integral approach to quantum gauge field theories in the present thesis a quantization procedure in terms of gauge-invariant fields is proposed and realized on the example of two- and four-dimensional Abelian models (Thirring model and QED) as well as the one-flavour QCD. For this the algebra of from the gauge-dependent field configuration of the basing quantum field theory formed gauge-invariant Grassmann-algebra valued differential forms, which carries the structure of a Z2-graded differential algebra, is studied in more detail. Thereafter follows the implementation of a suitable chosen set of gauge-invariant fields as well as certain algebraic relations into the functional integral, by which the original gauge-dependent field configuration can be integrated out. This procedure called ''reduction of the functional integral'' leads finally to an effective bosonized (quantum) theory of interacting gauge-invariant and by this physical fields. The presented procedure can be considered as general bosonization scheme for quantum field theories in arbitrary space-time dimensions. The physical evaluation of the obtained effective theories is demonstrated on the example of the calculation of the chiral anomaly as well as certain vacuum expectation values in the framework of the studied Abelian models. As it is thereby shown one is confronted with a series of novel phenomena and problems, which allow at suitable treatment deeper insights in non-perturbative questions
Cameron, Peter J
2007-01-01
This Second Edition of a classic algebra text includes updated and comprehensive introductory chapters,. new material on axiom of Choice, p-groups and local rings, discussion of theory and applications, and over 300 exercises. It is an ideal introductory text for all Year 1 and 2 undergraduate students in mathematics. - ;Developed to meet the needs of modern students, this Second Edition of the classic algebra text by Peter Cameron covers all the abstract algebra an undergraduate student is likely to need. Starting with an introductory overview of numbers, sets and functions, matrices, polynomials, and modular arithmetic, the text then introduces the most important algebraic structures: groups, rings and fields, and their properties. This is followed by coverage of vector spaces and modules with. applications to abelian groups and canonical forms before returning to the construction of the number systems, including the existence of transcendental numbers. The final chapters take the reader further into the th...
The Global Approach to Quantum Field Theory
Energy Technology Data Exchange (ETDEWEB)
Folacci, Antoine; Jensen, Bruce [Faculte des Sciences, Universite de Corse (France); Department of Mathematics, University of Southampton (United Kingdom)
2003-12-12
formalism of quantum field theory. This is the so-called global approach to quantum field theory where time does not play any particular role, and quantization is then naturally realized covariantly using tools such as the Peierls bracket (a covariant generalization of Poisson bracket), the Schwinger variational principle and Feynman sums over histories. However, it should be noted that the boycott of canonical methods by DeWitt is not total: when he judges they genuinely illuminate the physics of a problem, he does not hesitate to descend from the global point of view and to use them. In a few words, we have in fact described the research program initiated by DeWitt forty years ago, which has progressively evolved in order to take into account the latest development of gauge theories. While the Les Houches Lectures of 1963 were mainly concentrated on the formal structure and the quantization of Yang--Mills and gravitational fields, the present book also deals with more general gauge theories including those with open gauge algebras and structure functions, and therefore supergravity theories. More precisely, the book, more than a thousand pages in length, consists of eight parts and is completed by six appendices where certain technical aspects are singled out. An enormous variety of topics is covered, including the invariance transformations of the action functional, the Batalin-Vilkovisky formalism, Green's functions, the Peierls bracket, conservation laws, the theory of measurement, the Everett (or many worlds) interpretation of quantum mechanics, decoherence, the Schwinger variational principle and Feynm an functional integrals, the heat kernel, aspects of quantization for linear systems in stationary and non-stationary backgrounds, the S-matrix, the background field method, the effective action and the Vilkovisky-DeWitt formalism, the quantization of gauge theories without ghosts, anomalies, black holes and Hawking radiation, renormalization, and more. It should
The Nonlinear Field Space Theory
Jakub Mielczarek; Tomasz Trześniewski
2016-01-01
In recent years the idea that not only the configuration space of particles, i.e. spacetime, but also the corresponding momentum space may have nontrivial geometry has attracted significant attention, especially in the context of quantum gravity. The aim of this letter is to extend this concept to the domain of field theories, by introducing field spaces (i.e. phase spaces of field values) that are not affine spaces. After discussing the motivation and general aspects of our approach we prese...
State sum construction of two-dimensional topological quantum field theories on spin surfaces
Novak, Sebastian
2014-01-01
We provide a combinatorial model for spin surfaces. Given a triangulation of an oriented surface, a spin structure is encoded by assigning to each triangle a preferred edge, and to each edge an orientation and a sign, subject to certain admissibility conditions. The behaviour of this data under Pachner moves is then used to define a state sum topological field theory on spin surfaces. The algebraic data is a Delta-separable Frobenius algebra whose Nakayama automorphism is an involution. We find that a simple extra condition on the algebra guarantees that the amplitude is zero unless the combinatorial data satisfies the admissibility condition required for the reconstruction of the spin structure.
A novel Lie algebra of the genetic code over the Galois field of four DNA bases.
Sánchez, Robersy; Grau, Ricardo; Morgado, Eberto
2006-07-01
Starting from the four DNA bases order in the Boolean lattice, a novel Lie Algebra of the genetic code is proposed. Here, the main partitions of the genetic code table were obtained as equivalent classes of quotient spaces of the genetic code vector space over the Galois field of the four DNA bases. The new algebraic structure shows strong connections among algebraic relationships, codon assignments and physicochemical properties of amino acids. Moreover, a distance defined between codons expresses a physicochemical meaning. It was also noticed that the distance between wild type and mutant codons tends to be small in mutational variants of four genes: human phenylalanine hydroxylase, human beta-globin, HIV-1 protease and HIV-1 reverse transcriptase. These results strongly suggest that deterministic rules in genetic code origin must be involved. PMID:16780898
Boicescu, V; Georgescu, G; Rudeanu, S
1991-01-01
The Lukasiewicz-Moisil algebras were created by Moisil as an algebraic counterpart for the many-valued logics of Lukasiewicz. The theory of LM-algebras has developed to a considerable extent both as an algebraic theory of intrinsic interest and in view of its applications to logic and switching theory.This book gives an overview of the theory, comprising both classical results and recent contributions, including those of the authors. N-valued and &THgr;-valued algebras are presented, as well as &THgr;-algebras with negation.Mathematicians interested in lattice theory or symbolic logic, and computer scientists, will find in this monograph stimulating material for further research.
Field-theory methods in coagulation theory
International Nuclear Information System (INIS)
Coagulating systems are systems of chaotically moving particles that collide and coalesce, producing daughter particles of mass equal to the sum of the masses involved in the respective collision event. The present article puts forth basic ideas underlying the application of methods of quantum-field theory to the theory of coagulating systems. Instead of the generally accepted treatment based on the use of a standard kinetic equation that describes the time evolution of concentrations of particles consisting of a preset number of identical objects (monomers in the following), one introduces the probability W(Q, t) to find the system in some state Q at an instant t for a specific rate of transitions between various states. Each state Q is characterized by a set of occupation numbers Q = (n1, n2, ..., ng, ...), where ng is the total number of particles containing precisely g monomers. Thereupon, one introduces the generating functional Ψ for the probability W(Q, t). The time evolution of Ψ is described by an equation that is similar to the Schrödinger equation for a one-dimensional Bose field. This equation is solved exactly for transition rates proportional to the product of the masses of colliding particles. It is shown that, within a finite time interval, which is independent of the total mass of the entire system, a giant particle of mass about the mass of the entire system may appear in this system. The particle in question is unobservable in the thermodynamic limit, and this explains the well-known paradox of mass-concentration nonconservation in classical kinetic theory. The theory described in the present article is successfully applied in studying the time evolution of random graphs.
Relativistic dynamics of charges in external fields: the Pauli algebra approach
International Nuclear Information System (INIS)
The Pauli algebra is used to study the relativistic motion of charged particles in external electromagnetic fields. Plane-wave solutions of Maxwell's equations, in particular standing waves with electric and magnetic fields which are everywhere parallel, are easily represented and analysed in a gauge-independent treatment. We derive the 4-momentum of electromagnetic fields and show how the integrals for fields of a charged particle can be evaluated with integration limits specified in the rest frame of the moving charge. The general solution of the Lorentz force equation is shown to be a Lorentz transformation with time-dependent rotation and boost parameters which, in the case of parallel fields, are trivially related to the magnetic and electric fields, respectively. With the Pauli algebra, an earlier derivation in the Dirac algebra of the 4-velocity in constant uniform fields is greatly simplified and extended to arbitrary initial motion, and solutions are found to motion in a Coulomb field and in the presence of arbitrarily polarised plane waves. (author)
The Theory of Prime Ideals of Leavitt Path Algebras over Arbitrary Graphs
Rangaswamy, Kulumani M
2011-01-01
Given an arbitrary graph E and a field K, the prime ideals as well as the primitive ideals of the Leavitt path algebra L_K(E) are completely described in terms of their generators. The stratification of the prime spectrum of L_K(E) is indicated with information on its individual stratum. Necessary and sufficient conditions are given on the graph E under which every prime ideal of L_K(E) is primitive. Leavitt path algebras of Krull dimension zero are characterized and those with various prescribed Krull dimension are constructed. The minimal prime ideals of L_K(E) are are described in terms of the graphical properties of E and using this, complete descriptions of the height one as well as the co-height one prime ideals of L_K(E) are given.
Arithmetic gravity and Yang-Mills theory: An approach to adelic physics via algebraic spaces
Schmidt, Rene
2008-01-01
This work is a dissertation thesis written at the WWU Muenster (Germany), supervised by Prof. Dr. Raimar Wulkenhaar. We present an approach to adelic physics based on the language of algebraic spaces. Relative algebraic spaces X over a base S are considered as fundamental objects which describe space-time. This yields a formulation of general relativity which is covariant with respect to changes of the chosen domain of numbers S. With regard to adelic physics the choice of S as an excellent Dedekind scheme is of interest (because this way also the finite prime spots, i.e. the p-adic degrees of freedom are taken into account). In this arithmetic case, it turns out that X is a Neron model. This enables us to make concrete statements concerning the structure of the space-time described by X. Furthermore, some solutions of the arithmetic Einstein equations are presented. In a next step, Yang-Mills gauge fields are incorporated.
Electromagnetic field theories for engineering
Salam, Md Abdus
2014-01-01
A four year Electrical and Electronic engineering curriculum normally contains two modules of electromagnetic field theories during the first two years. However, some curricula do not have enough slots to accommodate the two modules. This book, Electromagnetic Field Theories, is designed for Electrical and Electronic engineering undergraduate students to provide fundamental knowledge of electromagnetic fields and waves in a structured manner. A comprehensive fundamental knowledge of electric and magnetic fields is required to understand the working principles of generators, motors and transformers. This knowledge is also necessary to analyze transmission lines, substations, insulator flashover mechanism, transient phenomena, etc. Recently, academics and researches are working for sending electrical power to a remote area by designing a suitable antenna. In this case, the knowledge of electromagnetic fields is considered as important tool.
Lavrauw, Michel; Sheekey, John
2014-01-01
We classify the orbits of elements of the tensor product spaces ${\\mathbb{F}}^2\\otimes {\\mathbb{F}}^3 \\otimes {\\mathbb{F}}^3$ for all finite; real; and algebraically closed fields under the action of two natural groups. The result can also be interpreted as the classification of the orbits in the $17$-dimensional projective space of the Segre variety product of a projective line and two projective planes. This extends the classification of the orbits in the $7$-dimensional projective space of...
Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras
Zhang, Tianjie; Gao, Xing; Guo, Li
2016-10-01
The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular, the Hopf algebra of rooted trees serves as the "baby model" of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free Rota-Baxter algebra with the structure of a cocycle Hopf algebra.
Conformal field theories from deformations of theories with $W_n$ symmetry
Babaro, Juan Pabo; Ranjbar, Arash
2016-01-01
Following a method proposed by Ribault in reference [arXiv:0803.2099], we construct a set of non-rational conformal field theories that consist of deformations of Toda field theory for ${sl}(n)$. While preserving conformal invariance, the deformation breaks the $W_n$ symmetry. However, the theories may still enjoy a remnant infinite-dimensional affine symmetry. The case $n=3$ is used to illustrate this phenomenon, together with further deformations that yield enhanced Kac-Moody symmetry algebras. For generic $n$ we compute $N$-point correlation functions on the Riemann sphere and show that these can be expressed in terms of $sl(n)$ Toda field theory $((N-2)n+2)$-point correlation functions.
Energy Technology Data Exchange (ETDEWEB)
Fucito, F.; Tanzini, A. [Rome Univ. 2 (Italy). Dipt. di Fisica; Vilar, L.C.Q.; Ventura, O.S.; Sasaki, C.A.G. [Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, RJ (Brazil); Sorella, S.P. [Universidade do Estado (UERJ), Rio de Janeiro, RJ (Brazil). Inst. de Fisica
1997-07-01
The aim of these notes is to provide a simple and pedagogical (as much as possible) introduction to what is nowadays commonly called Algebraic Renormalization. As the same itself let it understand, the Algebraic Renormalization gives a systematic set up in order to analyse the quantum extension of a given set of classical symmetries. The framework is purely algebraic, yielding a complete characterization of all possible anomalies and invariant counterterms without making use of any explicit computation of the Feynman diagrams. This goal is achieved by collecting, with the introduction of suitable ghost fields, all the symmetries into a unique operation summarized by a generalized Slavnov-Taylor (or master equation) identity which is the starting point for the quantum analysis. The Slavnov-Taylor identity allows to define a nilpotent operator whose cohomology classes in the space of the integrated local polynomials in the fields and their derivatives with dimensions bounded by power counting give all nontrivial anomalies and counterterms. I other words, the proof of the renormalizability is reduced to the computation of some cohomology classes. (author) 28 refs., 2 figs.
Gauge Transformation of Double Field Theory for Open String
Ma, Chen-Te
2014-01-01
I combine the symmetry structure of both the normal and transverse coordinates to construct the Dirac-Born-Infeld (DBI) theory. Normal coordinates are dual to Neumann boundary conditions and transverse coordinates dual to Dirichlet boundary conditions. The gauge transformation of the generalized metric is also governed by the generalized Lie derivative as the massless closed string field theory. The massless closed string field theory gives $C$-bracket, but the DBI theory gives the $F$-bracket from the closed algebra. The $F$-bracket can be a $\\alpha^{\\prime}$ deformation of the $C$-bracket. From the symmetry, we can deduce the suitable action with non-zero flux in open string. From the equations of motion of the scalar dilaton, it offers the generalized scalar curvature. Finally, I show the classical equivalence between the double sigma model and normal sigma model for open string.
Field theory without Feynman diagrams: One-loop effective actions
International Nuclear Information System (INIS)
In this paper the connection between standard perturbation theory techniques and the new Bern-Kosower calculational rules for gauge theory is clarified. For one-loop effective actions of scalars, Dirac spinors, and vector bosons in a background gauge field, Bern-Kosower type rules are derived without the use of either string theory or Feynman diagrams. The effective action is written as a one-dimensional path integral, which can be calculated to any order in the gauge coupling; evaluation leads to Feynman parameter integrals directly, bypassing the usual algebra required from Feynman diagrams, and leading to compact and organized expressions. This formalism is valid off-shell, is explicitly gauge invariant, and can be extended to a number of other field theories. (orig.)
A Gross-Zagier formula for quaternion algebras over totally real fields
Goren, Eyal Z
2011-01-01
We prove a higher dimensional generalization of Gross and Zagier's theorem on the factorization of differences of singular moduli. Their result is proved by giving a counting formula for the number of isomorphisms between elliptic curves with complex multiplication by two different imaginary quadratic fields $K$ and $K^\\prime$, when the curves are reduced modulo a supersingular prime and its powers. Equivalently, the Gross-Zagier formula counts optimal embeddings of the ring of integers of an imaginary quadratic field into particular maximal orders in $B_{p, \\infty}$, the definite quaternion algebra over $\\QQ$ ramified only at $p$ and infinity. Our work gives an analogous counting formula for the number of simultaneous embeddings of the rings of integers of primitive CM fields into superspecial orders in definite quaternion algebras over totally real fields of strict class number 1. Our results can also be viewed as a counting formula for the number of isomorphisms modulo $\\frak{p} | p$ between abelian variet...
Refringence, field theory and normal modes
International Nuclear Information System (INIS)
In a previous paper [Barcelo C et al 2001 Class. Quantum Grav. 18 3595-610 (Preprint gr-qc/0104001)] we have shown that the occurrence of curved spacetime 'effective Lorentzian geometries' is a generic result of linearizing an arbitrary classical field theory around some nontrivial background configuration. This observation explains the ubiquitous nature of the 'analogue models' for general relativity that have recently been developed based on condensed matter physics. In the simple (single scalar field) situation analysed in our previous paper, there is a single unique effective metric; more complicated situations can lead to bi-metric and multi-metric theories. In the present paper we will investigate the conditions required to keep the situation under control and compatible with experiment - either by enforcing a unique effective metric (as would be required to be strictly compatible with the Einstein equivalence principle), or at the worst by arranging things so that there are multiple metrics that are all 'close' to each other (in order to be compatible with the Eoetvoes experiment). The algebraically most general situation leads to a physical model whose mathematical description requires an extension of the usual notion of Finsler geometry to a Lorentzian-signature pseudo-Finsler geometry; while this is possibly of some interest in its own right, this particular case does not seem to be immediately relevant for either particle physics or gravitation. The key result is that wide classes of theories lend themselves to an effective metric description. This observation provides further evidence that the notion of 'analogue gravity' is rather generic
Cooperstein, Bruce
2010-01-01
Vector SpacesFieldsThe Space FnVector Spaces over an Arbitrary Field Subspaces of Vector SpacesSpan and IndependenceBases and Finite Dimensional Vector SpacesBases and Infinite Dimensional Vector SpacesCoordinate VectorsLinear TransformationsIntroduction to Linear TransformationsThe Range and Kernel of a Linear TransformationThe Correspondence and Isomorphism TheoremsMatrix of a Linear TransformationThe Algebra of L(V, W) and Mmn(F)Invertible Transformations and MatricesPolynomialsThe Algebra of PolynomialsRoots of PolynomialsTheory of a Single Linear OperatorInvariant Subspaces of an Operator
Realization of $W_{1+\\infty}$ and Virasoro Algebras in Supersymmetric Theories on Four Manifolds
Johansen, Andrei
1994-01-01
We demonstrate that a supersymmetric theory twisted on a K\\"ahler four manifold $M=\\Sigma_1 \\times \\Sigma_2 ,$ where $\\Sigma_{1,2}$ are 2D Riemann surfaces, possesses a "left-moving" conformal stress tensor on $\\Sigma_1$ ($\\Sigma_2$) in the BRST cohomology. The central charge of the Virasoro algebra has a purely geometric origin and is proportional to the Euler characteristic of the $\\Sigma_2$ ($\\Sigma_1$) surface. This structure is shown to be invariant under renormalization group. We also g...
An ideal topology type convergent theorem on scale effect algebras
Institute of Scientific and Technical Information of China (English)
WU JunDe; ZHOU XuanChang; Minhyung CHO
2007-01-01
The famous Antosik-Mikusinski convergent theorem on the Abel topological groups has very extensive applications in measure theory, summation theory and other analysis fields. In this paper, we establish the theorem on a class of effect algebras equipped with the ideal topology. This paper shows also that the ideal topology of effect algebras is a useful topology in studying the quantum logic theory.
Finite and Infinite W Algebras and their Applications
Tjin, T
1993-01-01
In this paper we present a systematic study of $W$ algebras from the Hamiltonian reduction point of view. The Drinfeld-Sokolov (DS) reduction scheme is generalized to arbitrary $sl_2$ embeddings thus showing that a large class of W algebras can be viewed as reductions of affine Lie algebras. The hierarchies of integrable evolution equations associated to these classical W algebras are constructed as well as the generalized Toda field theories which have them as Noether symmetry algebras. The problem of quantising the DS reductions is solved for arbitrary $sl_2$ embeddings and it is shown that any W algebra can be embedded into an affine Lie algebra. This also provides us with an algorithmic method to write down free field realizations for arbitrary W algebras. Just like affine Lie algebras W algebras have finite underlying structures called `finite W algebras'. We study the classical and quantum theory of these algebras, which play an important role in the theory of ordinary W algebras, in detail as well as s...
A Lagrangian effective field theory
Vlah, Zvonimir; White, Martin; Aviles, Alejandro
2015-01-01
We have continued the development of Lagrangian, cosmological perturbation theory for the low-order correlators of the matter density field. We provide a new route to understanding how the effective field theory (EFT) of large-scale structure can be formulated in the Lagrandian framework and a new resummation scheme, comparing our results to earlier work and to a series of high-resolution N-body simulations in both Fourier and configuration space. The `new' terms arising from EFT serve to tam...
Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras
Sole, A.; Kac, V. G.; Valeri, D.
2014-01-01
We put the Adler-Gelfand-Dickey approach to classical W-algebras in the framework of Poisson vertex algebras. We show how to recover the bi-Poisson structure of the KP hierarchy, together with its generalizations and reduction to the N-th KdV hierarchy, using the formal distribution calculus and the lambda-bracket formalism. We apply the Lenard-Magri scheme to prove integrability of the corresponding hierarchies. We also give a simple proof of a theorem of Kupershmidt and Wilson in this frame...
Electroweak spin gauge theories and the frame field
International Nuclear Information System (INIS)
The paper presents electroweak spin gauge theories and the frame field. The contents are divided into four sections. Section I lays down the general principles of spin gauge theories. Section II shows how the matrix elements of the Glashow-Salem-Weinberg theory of electroweak interactions of the electron and its neutrino are the consequence of a particular spin gauge invariance. Section III introduces a new set of concepts concerning mass. These start with the idea that a fermion mass is the result of a coupling of the fermion field to the 'frame field'. Section four uses a postulate that the extended covariant derivative for first generation quarks is obtained from that for the leptons by simply changing the Clifford algebra representation, and introducing extra mass terms to fit the masses of the up and down quarks. A discussion of these ideas is carried out. (UK)
Twisted boundary states and representation of generalized fusion algebra
Ishikawa, Hiroshi; Tani, Taro
2005-01-01
The mutual consistency of boundary conditions twisted by an automorphism group G of the chiral algebra is studied for general modular invariants of rational conformal field theories. We show that a consistent set of twisted boundary states associated with any modular invariant realizes a non-negative integer matrix representation (NIM-rep) of the generalized fusion algebra, an extension of the fusion algebra by representations of the twisted chiral algebra associated with the automorphism gro...
Locality in the gauge-covariant field theory of strings
Energy Technology Data Exchange (ETDEWEB)
Kaku, Michio
1985-11-07
Recently, we wrote down the gauge-covariant field theory of the free bosonic, super, and heterotic strings. These second quantized actions were derived from path integrals in the same way as Feynman derived the Schroedinger equation. These actions possess all the local gauge invariance of the super Virasoro algebra. These actions, however, are non-local. It has been conjectured that these actions can be made local by adding auxiliary fields. In this paper, we prove this conjecture to all orders, making our action explicitly local. (orig.).
Virasoro central charges for Nichols algebras
Semikhatov, A M
2011-01-01
Virasoro central charge associated with Nichols algebras is invariant under the Weyl groupoid action and takes very suggestive values for some items in Heckenberger's list of rank-2 Nichols algebras. In particular, this might be taken as an indication of the existence of reasonable logarithmic extensions of W3==WA2, WB2, and WG2 models of conformal field theory. In the W3 case, the construction of an octuplet extended algebra is outlined.
Bosonic colored group field theory
Energy Technology Data Exchange (ETDEWEB)
Ben Geloun, Joseph [Universite Paris XI, Laboratoire de Physique Theorique, Orsay Cedex (France); University of Abomey-Calavi, Cotonou (BJ). International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair); Universite Cheikh Anta Diop, Departement de Mathematiques et Informatique, Faculte des Sciences et Techniques, Dakar (Senegal); Magnen, Jacques [Ecole Polytechnique, Centre de Physique Theorique, Palaiseau Cedex (France); Rivasseau, Vincent [Universite Paris XI, Laboratoire de Physique Theorique, Orsay Cedex (France)
2010-12-15
Bosonic colored group field theory is considered. Focusing first on dimension four, namely the colored Ooguri group field model, the main properties of Feynman graphs are studied. This leads to a theorem on optimal perturbative bounds of Feynman amplitudes in the ''ultraspin'' (large spin) limit. The results are generalized in any dimension. Finally, integrating out two colors we write a new representation, which could be useful for the constructive analysis of this type of models. (orig.)
Unitarity of Superstring Field Theory
Sen, Ashoke
2016-01-01
We complete the proof of unitarity of (compactified) heterotic and type II string field theories by showing that in the cut diagrams only physical states appear in the sum over intermediate states. This analysis takes into account the effect of mass and wave-function renormalization, and the possibility that the true vacuum may be related to the perturbative vacuum by small shifts in the string fields.
A first course in abstract algebra rings, groups, and fields
Anderson, Marlow
2014-01-01
Numbers, Polynomials, and Factoring The Natural Numbers The Integers Modular Arithmetic Polynomials with Rational CoefficientsFactorization of PolynomialsSection I in a NutshellRings, Domains, and Fields Rings Subrings and Unity Integral Domains and Fields Ideals Polynomials over a Field Section II in a NutshellRing Homomorphisms and Ideals Ring HomomorphismsThe Kernel Rings of Cosets The Isomorphism Theorem for Rings Maximal and Prime Ideals The Chinese Remainder Theorem Section III in a NutshellGroups Symmetries of Geometric Figures PermutationsAbstract Groups Subgroups Cyclic Groups Section
Quantum Field Theory in Graphene
Fialkovsky, I. V.; Vassilevich, D. V.
2011-01-01
This is a short non-technical introduction to applications of the Quantum Field Theory methods to graphene. We derive the Dirac model from the tight binding model and describe calculations of the polarization operator (conductivity). Later on, we use this quantity to describe the Quantum Hall Effect, light absorption by graphene, the Faraday effect, and the Casimir interaction.
Boundary Logarithmic Conformal Field Theory
Kogan, I I; Kogan, Ian I.; Wheater, John F.
2000-01-01
We discuss the effect of boundaries in boundary logarithmic conformal field theory and show, with reference to both $c=-2$ and $c=0$ models, how they produce new features even in bulk correlation functions which are not present in the corresponding models without boundaries. We show how Cardy's relation between boundary states and bulk quantities is modified.
SOME RESULTS ON INFINITE DIMENSIONAL NOVIKOV ALGEBRAS
Institute of Scientific and Technical Information of China (English)
赵玉凤; 孟道骥
2003-01-01
This paper gives some sufficient conditions for determining the simplicity of infinite di-mensional Novikov algebras of characteristic 0, and also constructs a class of simple Novikovalgebras by extending the base field. At last, the deformation theory of Novikov algebras isintroduced.
Lectures on extended affine Lie algebras
Neher, Erhard
2010-01-01
We give an introduction to the structure theory of extended affine Lie algebras, which provide a common framework for finite-dimensional semisimple, affine and toroidal Lie algebras. The notes are based on a lecture series given during the Fields Institute summer school at the University of Ottawa in June 2009.
Boundedly controlled topology foundations of algebraic topology and simple homotopy theory
Anderson, Douglas R
1988-01-01
Several recent investigations have focused attention on spaces and manifolds which are non-compact but where the problems studied have some kind of "control near infinity". This monograph introduces the category of spaces that are "boundedly controlled" over the (usually non-compact) metric space Z. It sets out to develop the algebraic and geometric tools needed to formulate and to prove boundedly controlled analogues of many of the standard results of algebraic topology and simple homotopy theory. One of the themes of the book is to show that in many cases the proof of a standard result can be easily adapted to prove the boundedly controlled analogue and to provide the details, often omitted in other treatments, of this adaptation. For this reason, the book does not require of the reader an extensive background. In the last chapter it is shown that special cases of the boundedly controlled Whitehead group are strongly related to lower K-theoretic groups, and the boundedly controlled theory is compared to Sie...
Higher AGT Correspondences, W-algebras, and Higher Quantum Geometric Langlands Duality from M-Theory
Tan, Meng-Chwan
2016-01-01
We further explore the implications of our framework in [arXiv:1301.1977, arXiv:1309.4775], and physically derive, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent, (i) a 5d AGT correspondence for any compact Lie group, (ii) a 5d and 6d AGT correspondence on ALE space of type ADE, and (iii) identities between the ordinary, q-deformed and elliptic affine W-algebras associated with the 4d, 5d and 6d AGT correspondence, respectively, which also define a quantum geometric Langlands duality and its higher analogs formulated by Feigin-Frenkel-Reshetikhin in [3,4]. As an offshoot, we are led to the sought-after connection between the gauge-theoretic realization of the geometric Langlands correspondence by Kapustin-Witten [5,6] and its algebraic CFT formulation by Beilinson-Drinfeld [7], where one can also understand Wilson and 't Hooft-Hecke line operators in 4d gauge theory as monodromy loop operators in 2d CFT, for example. In turn, this will allow ...
The algebras of large N matrix mechanics
Energy Technology Data Exchange (ETDEWEB)
Halpern, M.B.; Schwartz, C.
1999-09-16
Extending early work, we formulate the large N matrix mechanics of general bosonic, fermionic and supersymmetric matrix models, including Matrix theory: The Hamiltonian framework of large N matrix mechanics provides a natural setting in which to study the algebras of the large N limit, including (reduced) Lie algebras, (reduced) supersymmetry algebras and free algebras. We find in particular a broad array of new free algebras which we call symmetric Cuntz algebras, interacting symmetric Cuntz algebras, symmetric Bose/Fermi/Cuntz algebras and symmetric Cuntz superalgebras, and we discuss the role of these algebras in solving the large N theory. Most important, the interacting Cuntz algebras are associated to a set of new (hidden!) local quantities which are generically conserved only at large N. A number of other new large N phenomena are also observed, including the intrinsic nonlocality of the (reduced) trace class operators of the theory and a closely related large N field identification phenomenon which is associated to another set (this time nonlocal) of new conserved quantities at large N.
Embedding quantum and random optics in a larger field theory
Morgan, Peter
2008-01-01
Introducing creation and annihilation operators for negative frequency components extends the algebra of smeared local observables of quantum optics to include an associated classical random field optics.
Theory of Antisymmetric Tensor Fields
Dvoeglazov, V V
2003-01-01
It has long been claimed that the antisymmetric tensor field of the second rank is pure longitudinal after quantization. In my opinion, such a situation is quite unacceptable. I repeat the well-known procedure of the derivation of the set of Proca equations. It is shown that it can be written in various forms. Furthermore, on the basis of the Lagrangian formalism I calculate dynamical invariants (including the Pauli-Lubanski vector of relativistic spin for this field). Even at the classical level the Pauli-Lubanski vector can be equal to zero after applications of well-known constraints. The importance of the normalization is pointed out for the problem of the description of quantized fields of maximal spin 1. The correct quantization procedure permits us to propose a solution of this puzzle in the modern field theory. Finally, the discussion of the connection of the Ogievetskii-Polubarinov-Kalb-Ramond field and the electrodynamic gauge is presented.
Young, Matthew B
2016-01-01
We introduce a new class of representations of the cohomological Hall algebras of Kontsevich and Soibelman which we call cohomological Hall modules, or CoHM for short. These representations are constructed from self-dual representations of a quiver with contravariant involution $\\sigma$ and provide a mathematical model for the space of BPS states in orientifold string theory. We use the CoHM to define a generalization of cohomological Donaldson-Thomas theory of quivers which allows the quiver representations to have orthogonal and symplectic structure groups. The associated invariants are called orientifold Donaldson-Thomas invariants. We prove the integrality conjecture for orientifold Donaldson-Thomas invariants of $\\sigma$-symmetric quivers. We also formulate precise conjectures regarding the geometric meaning of these invariants and the freeness of the CoHM of a $\\sigma$-symmetric quiver. We prove the freeness conjecture for disjoint union quivers, loop quivers and the affine Dynkin quiver of type $\\widet...
Elements of algebraic coding systems
Cardoso da Rocha, Jr, Valdemar
2014-01-01
Elements of Algebraic Coding Systems is an introductory textto algebraic coding theory. In the first chapter, you'll gain insideknowledge of coding fundamentals, which is essential for a deeperunderstanding of state-of-the-art coding systems.This book is a quick reference for those who are unfamiliar withthis topic, as well as for use with specific applications such as cryptographyand communication. Linear error-correcting block codesthrough elementary principles span eleven chapters of the text.Cyclic codes, some finite field algebra, Goppa codes, algebraic decodingalgorithms, and applications in public-key cryptography andsecret-key cryptography are discussed, including problems and solutionsat the end of each chapter. Three appendices cover the Gilbertbound and some related derivations, a derivation of the Mac-Williams' identities based on the probability of undetected error,and two important tools for algebraic decoding-namely, the finitefield Fourier transform and the Euclidean algorithm for polynomials.
Bohmian Mechanics and Quantum Field Theory
Duerr, Detlef; Goldstein, Sheldon; Tumulka, Roderich; Zanghi, Nino
2003-01-01
We discuss a recently proposed extension of Bohmian mechanics to quantum field theory. For more or less any regularized quantum field theory there is a corresponding theory of particle motion, which in particular ascribes trajectories to the electrons or whatever sort of particles the quantum field theory is about. Corresponding to the nonconservation of the particle number operator in the quantum field theory, the theory describes explicit creation and annihilation events: the world lines fo...
Indian Academy of Sciences (India)
Tomás L Gómez
2001-02-01
This is an expository article on the theory of algebraic stacks. After introducing the general theory, we concentrate in the example of the moduli stack of vector bundles, giving a detailed comparison with the moduli scheme obtained via geometric invariant theory.
Baal, Pierre Van
2014-01-01
""… a pleasant novelty that manages the impossible: a full course in field theory from a derivation of the Dirac equation to the standard electroweak theory in less than 200 pages. Moreover, the final chapter consists of a careful selection of assorted problems, which are original and either anticipate or detail some of the topics discussed in the bulk of the chapters. Instead of building a treatise out of a collection of lecture notes, the author took the complementary approach and constructed a course out of a number of well-known and classic treatises. The result is fresh and useful. … the
Einstein's theory of unified fields
Tonnelat, Marie Antoinette
2014-01-01
First published in1966, here is presented a comprehensive overview of one of the most elusive scientific speculations by the pre-eminent genius of the 20th century. The theory is viewed by some scientists with deep suspicion, by others with optimism, but all agree that it represents an extreme challenge. As the author herself affirms, this work is not intended to be a complete treatise or 'didactic exposition' of the theory of unified fields, but rather a tool for further study, both by students and professional physicists. Dealing with all the major areas of research whic
Spin from defects in two-dimensional quantum field theory
Novak, Sebastian
2015-01-01
We build two-dimensional quantum field theories on spin surfaces starting from theories on oriented surfaces with networks of topological defect lines and junctions. The construction uses a combinatorial description of the spin structure in terms of a triangulation equipped with extra data. The amplitude for the spin surfaces is defined to be the amplitude for the underlying oriented surface together with a defect network dual to the triangulation. Independence of the triangulation and of the other choices follows if the line defect and junctions are obtained from a Delta-separable Frobenius algebra with involutive Nakayama automorphism in the monoidal category of topological defects. For rational conformal field theory we can give a more explicit description of the defect category, and we work out two examples related to free fermions in detail: the Ising model and the so(n) WZW model at level 1.
Duality and modular invariance in rational conformal field theories
International Nuclear Information System (INIS)
We investigate the polynomial equations which should be satisfied by the duality data for a rational conformal field theory. We show that by these duality data we can construct some vector spaces which are isomorphic to the spaces of conformal blocks. One can construct explicitly the inner product for the former if one deals with a unitary theory. These vector spaces endowed with an inner product are the algebraic reminiscences of the Hilbert spaces in a Chern-Simons theory. As by-products, we show that the polynomial equations involving the modular transformations for the one-point blocks on the torus are not independent. And along the way, we discuss the reconstruction of the quantum group in a rational conformal theory. Finally, we discuss the solution of structure constants for a physical theory. Making some assumption, we obtain a neat solution. And this solution in turn implies that the quantum groups of the left sector and of the right sector must be the same, although the chiral algebras need not to be the same. Some examples are given. (orig.)
Protected gates for topological quantum field theories
Energy Technology Data Exchange (ETDEWEB)
Beverland, Michael E.; Pastawski, Fernando; Preskill, John [Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125 (United States); Buerschaper, Oliver [Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin (Germany); Koenig, Robert [Institute for Advanced Study and Zentrum Mathematik, Technische Universität München, 85748 Garching (Germany); Sijher, Sumit [Institute for Quantum Computing and Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1 (Canada)
2016-02-15
We study restrictions on locality-preserving unitary logical gates for topological quantum codes in two spatial dimensions. A locality-preserving operation is one which maps local operators to local operators — for example, a constant-depth quantum circuit of geometrically local gates, or evolution for a constant time governed by a geometrically local bounded-strength Hamiltonian. Locality-preserving logical gates of topological codes are intrinsically fault tolerant because spatially localized errors remain localized, and hence sufficiently dilute errors remain correctable. By invoking general properties of two-dimensional topological field theories, we find that the locality-preserving logical gates are severely limited for codes which admit non-abelian anyons, in particular, there are no locality-preserving logical gates on the torus or the sphere with M punctures if the braiding of anyons is computationally universal. Furthermore, for Ising anyons on the M-punctured sphere, locality-preserving gates must be elements of the logical Pauli group. We derive these results by relating logical gates of a topological code to automorphisms of the Verlinde algebra of the corresponding anyon model, and by requiring the logical gates to be compatible with basis changes in the logical Hilbert space arising from local F-moves and the mapping class group.
Conformal Field Theory, Automorphic Forms and Related Topics
Weissauer, Rainer; CFT 2011
2014-01-01
This book, part of the series Contributions in Mathematical and Computational Sciences, reviews recent developments in the theory of vertex operator algebras (VOAs) and their applications to mathematics and physics. The mathematical theory of VOAs originated from the famous monstrous moonshine conjectures of J.H. Conway and S.P. Norton, which predicted a deep relationship between the characters of the largest simple finite sporadic group, the Monster, and the theory of modular forms inspired by the observations of J. MacKay and J. Thompson. The contributions are based on lectures delivered at the 2011 conference on Conformal Field Theory, Automorphic Forms and Related Topics, organized by the editors as part of a special program offered at Heidelberg University that summer under the sponsorship of the MAThematics Center Heidelberg (MATCH).
Quantum Fields on Noncommutative Spacetimes: Theory and Phenomenology
Directory of Open Access Journals (Sweden)
Aiyalam P. Balachandran
2010-06-01
Full Text Available In the present work we review the twisted field construction of quantum field theory on noncommutative spacetimes based on twisted Poincaré invariance. We present the latest development in the field, in particular the notion of equivalence of such quantum field theories on a noncommutative spacetime, in this regard we work out explicitly the inequivalence between twisted quantum field theories on Moyal and Wick-Voros planes; the duality between deformations of the multiplication map on the algebra of functions on spacetime F(R^4 and coproduct deformations of the Poincaré-Hopf algebra HP acting on F(R^4; the appearance of a nonassociative product on F(R^4 when gauge fields are also included in the picture. The last part of the manuscript is dedicated to the phenomenology of noncommutative quantum field theories in the particular approach adopted in this review. CPT violating processes, modification of two-point temperature correlation function in CMB spectrum analysis and Pauli-forbidden transition in Be^4 are all effects which show up in such a noncommutative setting. We review how they appear and in particular the constraint we can infer from comparison between theoretical computations and experimental bounds on such effects. The best bound we can get, coming from Borexino experiment, is >10^{24} TeV for the energy scale of noncommutativity, which corresponds to a length scale <10^{-43} m. This bound comes from a different model of spacetime deformation more adapted to applications in atomic physics. It is thus model dependent even though similar bounds are expected for the Moyal spacetime as well as argued elsewhere.
Transversity results and computations in symplectic field theory
Energy Technology Data Exchange (ETDEWEB)
Fabert, Oliver
2008-02-21
Although the definition of symplectic field theory suggests that one has to count holomorphic curves in cylindrical manifolds R x V equipped with a cylindrical almost complex structure J, it is already well-known from Gromov-Witten theory that, due to the presence of multiply-covered curves, we in general cannot achieve transversality for all moduli spaces even for generic choices of J. In this thesis we treat the transversality problem of symplectic field theory in two important cases. In the first part of this thesis we are concerned with the rational symplectic field theory of Hamiltonian mapping tori, which is also called the Floer case. For this observe that in the general geometric setup for symplectic field theory, the contact manifolds can be replaced by mapping tori M{sub {phi}} of symplectic manifolds (M,{omega}{sub M}) with symplectomorphisms {phi}. While the cylindrical contact homology of M{sub {phi}} is given by the Floer homologies of powers of {phi}, the other algebraic invariants of symplectic field theory for M{sub {phi}} provide natural generalizations of symplectic Floer homology. For symplectically aspherical M and Hamiltonian {phi} we study the moduli spaces of rational curves and prove a transversality result, which does not need the polyfold theory by Hofer, Wysocki and Zehnder and allows us to compute the full contact homology of M{sub {phi}} {approx_equal} S{sup 1} x M. The second part of this thesis is devoted to the branched covers of trivial cylinders over closed Reeb orbits, which are the trivial examples of punctured holomorphic curves studied in rational symplectic field theory. Since all moduli spaces of trivial curves with virtual dimension one cannot be regular, we use obstruction bundles in order to find compact perturbations making the Cauchy-Riemann operator transversal to the zero section and show that the algebraic count of elements in the resulting regular moduli spaces is zero. Once the analytical foundations of symplectic
Rigorous Computation of Fundamental Units in Algebraic Number Fields
Fontein, Felix
2010-01-01
We present an algorithm that unconditionally computes a representation of the unit group of a number field of discriminant $\\Delta_K$, given a full-rank subgroup as input, in asymptotically fewer bit operations than the baby-step giant-step algorithm. If the input is assumed to represent the full unit group, for example, under the assumption of the Generalized Riemann Hypothesis, then our algorithm can unconditionally certify its correctness in expected time $O(\\Delta_K^{n/(4n + 2) + \\epsilon}) = O(\\Delta_K^{1/4 - 1/(8n+4) + \\epsilon})$ where $n$ is the unit rank.
Theory of field reversed configurations
International Nuclear Information System (INIS)
This final report surveys the results of work conducted on the theory of field reversed configurations. This project has spanned ten years, beginning in early 1980. During this period, Spectra Technology was one of the leading contributors to the advances in understanding FRC. The report is organized into technical topic areas, FRC formation, equilibrium, stability, and transport. Included as an appendix are papers published in archival journals that were generated in the course of this report. 33 refs
AdS Field Theory from Conformal Field Theory
Fitzpatrick, A Liam
2012-01-01
We provide necessary and sufficient conditions for a Conformal Field Theory to have a description in terms of a perturbative Effective Field Theory in AdS. The first two conditions are well-known: the existence of a perturbative `1/N' expansion and an approximate Fock space of states generated by a finite number of low-dimension operators. We add a third condition, that the Mellin amplitudes of the CFT correlators must be well-approximated by functions that are bounded by a polynomial at infinity in Mellin space, or in other words, that the Mellin amplitudes have an effective theory-type expansion. We explain the relationship between our conditions and unitarity, and provide an analogy with scattering amplitudes that becomes exact in the flat space limit of AdS. The analysis also yields a simple connection between conformal blocks and AdS diagrams, providing a new calculational tool very much in the spirit of the S-Matrix program. We also begin to explore the potential pathologies associated with higher spin ...