Mydlík, M; Derzsiová, K
2010-11-01
Professor Frantisek Por MD and Professor Robert Klopstock MD were contemporaries, both born in 1899, one in Zvolen, the other in Dombovar, at the time of Austro-Hungarian Monarchy. Prof. Por attended the Faculty of Medicine in Budapest from 1918 to 1920, and Prof. Klopstock studied at the same place between 1917 and 1919. From 1920 until graduation on 6th February 1926, Prof. Por continued his studies at the German Faculty of Medicine, Charles University in Prague. Prof. Klopstock had to interrupt his studies in Budapest due to pulmonary tuberculosis; he received treatment at Tatranske Matliare where he befriended Franz Kafka. Later, upon Kafka's encouragement, he changed institutions and continued his studies at the German Faculty of Medicine, Charles University in Prague, where he graduated the first great go. It is very likely that, during their studies in Budapest and Prague, both professors met repeatedly, even though their life paths later separated. Following his graduation, Prof. Por practiced as an internist in Prague, later in Slovakia, and from 1945 in Kosice. In 1961, he was awarded the title of university professor of internal medicine at the Faculty of Medicine, Pavol Jozef Safarik University in Kosice, where he practiced until his death in 1980. Prof. Klopstock continued his studies in Kiel and Berlin. After his graduation in 1933, he practiced in Berlin as a surgeon and in 1938 left for USA. In 1962, he was awarded the title of university professor of pulmonary surgery in NewYork, where he died in 1972. PMID:21250499
Johnstone, Robert E; Fleisher, Lee A
2016-06-01
Robert D. Dripps, M.D. (1911 to 1973), helped found academic anesthesiology. Newly reviewed teaching slides from the University of Pennsylvania (Philadelphia, Pennsylvania) contain six anesthesia records from 1965 to 1967 that involved Dripps. They illustrate the clinical philosophy he taught-to consider administration of each anesthetic a research study. Intense public criticism in 1967 for improper experimentation on patients during anesthesia changed his clinical and research philosophies and teaching. PMID:27028470
In Remembrance of Robert J. Arceci, M.D., Ph.D. | Office of Cancer Genomics
It is with great sadness and a profound sense of loss that OCG recognizes the untimely passing of Dr. Robert J. Arceci. Dr. Arceci was a co-Principal Investigator for the Acute Myeloid Leukemia (AML) project within the TARGET initiative, which aims to discover novel, more effective treatments for childhood cancers. Dr. Arceci was passionate about the use of cancer genomics to both inform therapeutic approaches in the clinic and expand the field of precision medicine.
Lisi, A. Garrett
2002-01-01
Classical anti-commuting spinor fields and their dynamics are derived from the geometry of the Clifford bundle over spacetime via the BRST formulation. In conjunction with Kaluza-Klein theory, this results in a geometric description of all the fields and dynamics of the standard model coupled to gravity and provides the starting point for a new approach to quantum gravity.
Idempotents of Clifford Algebras
Ablamowicz, R.; Fauser, B.; Podlaski, K.; Rembielinski, J.
2003-01-01
A classification of idempotents in Clifford algebras C(p,q) is presented. It is shown that using isomorphisms between Clifford algebras C(p,q) and appropriate matrix rings, it is possible to classify idempotents in any Clifford algebra into continuous families. These families include primitive idempotents used to generate minimal one sided ideals in Clifford algebras. Some low dimensional examples are discussed.
Khovanova, Tanya
2008-01-01
I show how to associate a Clifford algebra to a graph. I describe the structure of these Clifford graph algebras and provide many examples and pictures. I describe which graphs correspond to isomorphic Clifford algebras and also discuss other related sets of graphs. This construction can be used to build models of representations of simply-laced compact Lie groups.
Clifford Algebra with Mathematica
Aragon-Camarasa, G.; Aragon-Gonzalez, G; Aragon, J. L.; Rodriguez-Andrade, M. A.
2008-01-01
The Clifford algebra of a n-dimensional Euclidean vector space provides a general language comprising vectors, complex numbers, quaternions, Grassman algebra, Pauli and Dirac matrices. In this work, a package for Clifford algebra calculations for the computer algebra program Mathematica is introduced through a presentation of the main ideas of Clifford algebras and illustrative examples. This package can be a useful computational tool since allows the manipulation of all these mathematical ob...
Unitary spaces on Clifford algebras
Marchuk, N. G.; Shirokov, D. S.
2007-01-01
For the complex Clifford algebra Cl(p,q) of dimension n=p+q we define a Hermitian scalar product. This scalar product depends on the signature (p,q) of Clifford algebra. So, we arrive at unitary spaces on Clifford algebras. With the aid of Hermitian idempotents we suggest a new construction of, so called, normal matrix representations of Clifford algebra elements. These representations take into account the structure of unitary space on Clifford algebra.
Fernández, V. V.; Moya, A. M.; Rodrigues Jr., W. A.
2002-01-01
In this paper we introduce the concept of metric Clifford algebra $\\mathcal{C\\ell}(V,g)$ for a $n$-dimensional real vector space $V$ endowed with a metric extensor $g$ whose signature is $(p,q)$, with $p+q=n$. The metric Clifford product on $\\mathcal{C\\ell}(V,g)$ appears as a well-defined \\emph{deformation}(induced by $g$) of an euclidean Clifford product on $\\mathcal{C\\ell}(V)$. Associated with the metric extensor $g,$ there is a gauge metric extensor $h$ which codifies all the geometric inf...
On Clifford neurons and Clifford multi-layer perceptrons.
Buchholz, Sven; Sommer, Gerald
2008-09-01
We study the framework of Clifford algebra for the design of neural architectures capable of processing different geometric entities. The benefits of this model-based computation over standard real-valued networks are demonstrated. One particular example thereof is the new class of so-called Spinor Clifford neurons. The paper provides a sound theoretical basis to Clifford neural computation. For that purpose the new concepts of isomorphic neurons and isomorphic representations are introduced. A unified training rule for Clifford MLPs is also provided. The topic of activation functions for Clifford MLPs is discussed in detail for all two-dimensional Clifford algebras for the first time. PMID:18514482
Clifford algebra as quantum language
Baugh, James; Finkelstein, David Ritz; Galiautdinov, Andrei; Saller, Heinrich
2000-01-01
We suggest Clifford algebra as a useful simplifying language for present quantum dynamics. Clifford algebras arise from representations of the permutation groups as they arise from representations of the rotation groups. Aggregates using such representations for their permutations obey Clifford statistics. The vectors supporting the Clifford algebras of permutations and rotations are plexors and spinors respectively. Physical spinors may actually be plexors describing quantum ensembles, not s...
'Twisted duality' for Clifford Algebras
Robinson, P. L.
2014-01-01
Viewing the complex Clifford algebra $C(V)$ of a real inner product space $V$ as a superalgebra, we offer several proofs of the fact that if $W$ is a subspace of the complexification of $V$ then the supercommutant of the Clifford algebra $C(W)$ is precisely the Clifford algebra $C(W^{\\perp})$.
Lounesto, Pertti
1993-01-01
This volume contains a facsimile reproduction of Marcel Riesz's notes of a set of lectures he delivered at the University of Maryland, College Park, between October 1957 and January 1958, which has not been formally published to date This seminal material (arranged in four chapters), which contributed greatly to the start of modern research on Clifford algebras, is supplemented in this book by notes which Riesz dictated to E Folke Bolinder in the following year and which were intended to be a fifth chapter of the Riesz lecture notes In addition, Riesz's work on Clifford algebra is put into an historical perspective in a separate review by P Lounesto As well as providing an introduction to Clifford algebra, this volume will be of value to those interested in the history of mathematics
Bošković Aleksandar
2007-01-01
Full Text Available The paper presents some concepts of the recently deceased American anthropologist Clifford Geertz, putting them into the specific context of his rich and interesting career, influences that he had, as well as some reactions to his ideas. A particular attention is placed upon the concept of culture, as the key concept in the 20th century American anthropology.
On parallelizing the Clifford algebra product for CLIFFORD
Ablamowicz, Rafal; Fauser, Bertfried
2012-01-01
We present, as a proof of concept, a way to parallelize the Clifford product in CL_{p,q} for a diagonalized quadratic form as a new procedure `cmulWpar' in the \\Clifford package for \\Maple(R). The procedure uses a new `Threads' module available under Maple 15 (and later) and a new \\Clifford procedure `cmulW' which computes the Clifford product of any two Grassmann monomials in \\CL_{p,q} with a help of Walsh functions. We benchmark `cmulWpar' and compare it to two other procedures `cmulNUM' an...
Expository notes on Clifford algebras and spinors with a detailed discussion of Majorana, Weyl, and Dirac spinors. The paper is meant as a review of background material, needed, in particular, in now fashionable theoretical speculations on neutrino masses. It has a more mathematical flavour than the over twenty-six-year-old Introduction to Majorana masses [M84] and includes historical notes and biographical data on past participants in the story. (author)
Transgression and Clifford algebras
Rohr, Rudolf Philippe
2007-01-01
Let $W$ be a differential (not necessarily commutative) algebra which carries a free action of a polynomial algebra $SP$ with homogeneous generators $p_1, >..., p_r$. We show that for $W$ acyclic, the cohomology of the quotient $H(W/)$ is isomorphic to a Clifford algebra $\\text{Cl}(P,B)$, where the (possibly degenerate) bilinear form $B$ depends on $W$. This observation is an analogue of an old result of Borel in a non-commutative context. As an application, we study the case of $W$ given by ...
Timorin, Vladlen
2002-01-01
Consider a smooth map from a neighborhood of the origin in a real vector space to a neighborhood of the origin in a Euclidean space. Suppose that this map takes all germs of lines passing through the origin to germs of Euclidean circles, or lines, or a point. We prove that under some simple additional assumptions this map takes all lines passing though the origin to the same circles as a Hopf map coming from a representation of a Clifford algebra does. We also describe a connection between ou...
Heckenberger, I.; Schueler, A.
2000-01-01
We study the q-Clifford algebras Cl_q(N,c), called FRT-Clifford algebras, introduced by Faddeev, Reshetikhin and Takhtajan. It is shown that Cl_q(N,c) acts on the q-exterior algebra \\Lambda(O_q^N). Moreover, explicit formulas for the embedding of U_q(so_N) into Cl_q(N,c) and its relation to the vector and spin representations of U_q(so_N) are given and proved. Key Words: q-Clifford algebra, Drinfeld-Jimbo algebra, spin representation
Diffeological Clifford algebras and pseudo-bundles of Clifford modules
Pervova, Ekaterina
2015-01-01
We consider the diffeological version of the Clifford algebra of a (diffeological) finite-dimensional vector space; we start by commenting on the notion of a diffeological algebra (which is the expected analogue of the usual one) and that of a diffeological module (also an expected counterpart of the usual notion). After considering the natural diffeology of the Clifford algebra, and its expected properties, we turn to our main interest, which is constructing pseudo-bundles of diffeological C...
Clifford Algebras and magnetic monopoles
It is known that the introduction of magnetic monopolies in electromagnetism does still present formal problems from the point of view of classical field theory. The author attempts to overcome at least some of them by making recourse to the Clifford Algebra formalism. In fact, while the events of a two-dimensional Minkowski space-time M(1,1) are sufficiently well represented by ordinary Complex Numbers, when dealing with the events of the four-dimensional Minkowski space M(1,3)identical to M/sub 4/ one has of course to look for hypercomplex numbers or, more generally, for the elements of a Clifford Algebra. The author uses the Clifford Algebras in terms of ''multivectors'', and in particular by Hestenes' language, which suits space-time quite well. He recalls that the Clifford product chiγ is the sum of the internal product chi . γ and of the wedge product chiΛγ
Clifford (Geometric) Algebra Wavelet Transform
Hitzer, Eckhard
2013-01-01
While the Clifford (geometric) algebra Fourier Transform (CFT) is global, we introduce here the local Clifford (geometric) algebra (GA) wavelet concept. We show how for $n=2,3 (\\mod 4)$ continuous $Cl_n$-valued admissible wavelets can be constructed using the similitude group $SIM(n)$. We strictly aim for real geometric interpretation, and replace the imaginary unit $i \\in \\C$ therefore with a GA blade squaring to $-1$. Consequences due to non-commutativity arise. We express the admissibility...
A theory of neural computation with Clifford algebras
Buchholz, Sven
2005-01-01
The present thesis introduces Clifford Algebra as a framework for neural computation. Neural computation with Clifford algebras is model-based. This principle is established by constructing Clifford algebras from quadratic spaces. Then the subspace grading inherent to any Clifford algebra is introduced. The above features of Clifford algebras are then taken as motivation for introducing the Basic Clifford Neuron (BCN). As a second type of Clifford neuron the Spinor Clifford Neuron is presente...
Clifford wavelets, singular integrals, and Hardy spaces
Mitrea, Marius
1994-01-01
The book discusses the extensions of basic Fourier Analysis techniques to the Clifford algebra framework. Topics covered: construction of Clifford-valued wavelets, Calderon-Zygmund theory for Clifford valued singular integral operators on Lipschitz hyper-surfaces, Hardy spaces of Clifford monogenic functions on Lipschitz domains. Results are applied to potential theory and elliptic boundary value problems on non-smooth domains. The book is self-contained to a large extent and well-suited for graduate students and researchers in the areas of wavelet theory, Harmonic and Clifford Analysis. It will also interest the specialists concerned with the applications of the Clifford algebra machinery to Mathematical Physics.
Beyond Stabilizer Codes II: Clifford Codes
Klappenecker, Andreas; Roetteler, Martin
2000-01-01
Knill introduced a generalization of stabilizer codes, in this note called Clifford codes. It remained unclear whether or not Clifford codes can be superior to stabilizer codes. We show that Clifford codes are stabilizer codes provided that the abstract error group has an abelian index group. In particular, if the errors are modelled by tensor products of Pauli matrices, then the associated Clifford codes are necessarily stabilizer codes.
Method of averaging in Clifford algebras
Shirokov, D. S.
2014-01-01
In this paper we consider different operators acting on Clifford algebras. We consider Reynolds operator of Salingaros' vee group. This operator average" an action of Salingaros' vee group on Clifford algebra. We consider conjugate action on Clifford algebra. We present a relation between these operators and projection operators onto fixed subspaces of Clifford algebras. Using method of averaging we present solutions of system of commutator equations.
Richard Handler
2008-01-01
Full Text Available En la primavera de 1991, Adam Kuper, por entonces director de Current Anthropology, y por derecho propio destacado historiador de la disciplina antropológica, me pidió realizar una entrevista a Clifford Geertz. Acepté encantado y ese mismo verano viajé a Princeton, Nueva Jersey, donde transcurrí aproximadamente tres horas con Geertz en su oficina, en el Instituto de Estudios Avanzados. Geertz me dio una cordial bienvenida y habló conmigo sin tapujos, dándome (como podrá comprobar el lector un claro y completo relato de su carrera (para una completa y exacta versión, los lectores pueden consultar ahora su autobiografía en After the Fact: Two Countries, Four Decades, one Anthropolgist, Harvard University Press, 1995. De la transcripción de la entrevista realicé un manuscrito, donde limpié las repeticiones y dubitaciones de la conversación, pero manteniendo fielmente la charla, tal y como tuvo lugar. Se lo mandé a Geertz, que propuso algunas correcciones pero que por lo demás aceptó todo.
Mathematics of CLIFFORD - A Maple package for Clifford and Grassmann algebras
Ablamowicz, Rafal; Fauser, Bertfried
2002-01-01
CLIFFORD performs various computations in Grassmann and Clifford algebras. It can compute with quaternions, octonions, and matrices with entries in Cl(B) - the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for Clifford product are implemented: 'cmulNUM' - based on Chevalley's recursive formula, and 'cmulRS' - based on non-recursive Rota-Stein sausage. Grassmann and Clifford bases can be used. Properties of reversion in undotted ...
K. Biekart (Kees); D.R. Gasper (Des)
2013-01-01
textabstractProfessor Robert Chambers is a Research Associate at the Institute of Development Studies (IDS), University of Sussex (Brighton, UK), where he has been based for the last 40 years, including as Professorial Research Fellow. He became involved in the field of development management in the
Symbolic Computations in Higher Dimensional Clifford Algebras
Ablamowicz, Rafal; Fauser, Bertfried
2012-01-01
We present different methods for symbolic computer algebra computations in higher dimensional (\\ge9) Clifford algebras using the \\Clifford\\ and \\Bigebra\\ packages for \\Maple(R). This is achieved using graded tensor decompositions, periodicity theorems and matrix spinor representations over Clifford numbers. We show how to code the graded algebra isomorphisms and the main involutions, and we provide some benchmarks.
Clifford theory for group representations
Karpilovsky, G
1989-01-01
Let N be a normal subgroup of a finite group G and let F be a field. An important method for constructing irreducible FG-modules consists of the application (perhaps repeated) of three basic operations: (i) restriction to FN. (ii) extension from FN. (iii) induction from FN. This is the `Clifford Theory' developed by Clifford in 1937. In the past twenty years, the theory has enjoyed a period of vigorous development. The foundations have been strengthened and reorganized from new points of view, especially from the viewpoint of graded rings and crossed products.The purpos
Linear operators in Clifford algebras
We consider the real vector space structure of the algebra of linear endomorphisms of a finite-dimensional real Clifford algebra (2, 4, 5, 6, 7, 8). A basis of that space is constructed in terms of the operators MeI,eJ defined by x→eI.x.eJ, where the eI are the generators of the Clifford algebra and I is a multi-index (3, 7). In particular, it is shown that the family (MeI,eJ) is exactly a basis in the even case. (orig.)
Clifford algebraic symmetries in physics
This paper reviews the following appearances of Clifford algebras in theoretical physics: statistical mechanics; general relativity; quantum electrodynamics; internal symmetries; the vee product; classical electrodynamics; charged-particle motion; and the Lorentz group. It is concluded that the power of the Clifford-algebraic description resides in its ability to perform representation-free calculations which are generalizations of the traditional vector algebra and that this considerable computational asset, in combination with the intrinsic symmetry, provides a practical framework for much of theoretical physics. 5 references
Rigidity theorems of Clifford Torus
SOUSA JR. LUIZ A. M.
2001-01-01
Full Text Available Let M be an n-dimensional closed minimally immersed hypersurface in the unit sphere Sn + 1. Assume in addition that M has constant scalar curvature or constant Gauss-Kronecker curvature. In this note we announce that if M has (n - 1 principal curvatures with the same sign everywhere, then M is isometric to a Clifford Torus .
Williams, Jack
2011-01-01
The 16th-Century intellectual Robert Recorde is chiefly remembered for introducing the equals sign into algebra, yet the greater significance and broader scope of his work is often overlooked. This book presents an authoritative and in-depth analysis of the man, his achievements and his historical importance. This scholarly yet accessible work examines the latest evidence on all aspects of Recorde's life, throwing new light on a character deserving of greater recognition. Topics and features: presents a concise chronology of Recorde's life; examines his published works; describes Recorde's pro
On Computational Complexity of Clifford Algebra
Budinich, Marco
2009-01-01
After a brief discussion of the computational complexity of Clifford algebras, we present a new basis for even Clifford algebra Cl(2m) that simplifies greatly the actual calculations and, without resorting to the conventional matrix isomorphism formulation, obtains the same complexity. In the last part we apply these results to the Clifford algebra formulation of the NP-complete problem of the maximum clique of a graph introduced in a previous paper.
Complex structure of a real Clifford algebra
Hanson, Jason
2011-01-01
The classification of real Clifford algebras in terms of matrix algebras is well--known. Here we consider the real Clifford algebra ${\\mathcal Cl}(r,s)$ not as a matrix algebra, but as a Clifford module over itself. We show that ${\\mathcal Cl}(r,s)$ possesses a basis independent complex structure only when the square of the volume element $\\omega$ is -1, in which case it is uniquely given up to sign by right multiplication with $\\omega$.
Clifford Fibrations and Possible Kinematics
Alan S. McRae
2009-07-01
Full Text Available Following Herranz and Santander [Herranz F.J., Santander M., Mem. Real Acad. Cienc. Exact. Fis. Natur. Madrid 32 (1998, 59-84, physics/9702030] we will construct homogeneous spaces based on possible kinematical algebras and groups [Bacry H., Levy-Leblond J.-M., J. Math. Phys. 9 (1967, 1605-1614] and their contractions for 2-dimensional spacetimes. Our construction is different in that it is based on a generalized Clifford fibration: Following Penrose [Penrose R., Alfred A. Knopf, Inc., New York, 2005] we will call our fibration a Clifford fibration and not a Hopf fibration, as our fibration is a geometrical construction. The simple algebraic properties of the fibration describe the geometrical properties of the kinematical algebras and groups as well as the spacetimes that are derived from them. We develop an algebraic framework that handles all possible kinematic algebras save one, the static algebra.
Introduction to Clifford's Geometric Algebra
Hitzer, Eckhard
2013-01-01
Geometric algebra was initiated by W.K. Clifford over 130 years ago. It unifies all branches of physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing, tracking, geographic information systems and neural computing. This tutorial explains the basics of geometric algebra, with concrete examples of the plane, of 3D space, of spacetime, and the popular conformal model. Geometric algebras are ideal to represen...
Lederman, R J
1999-01-01
Robert Schumann, one of the giants of early romantic music, was born in Saxony in 1810 and died in an asylum shortly after his 46th birthday. Early in life, he demonstrated extraordinary skills in both music and journalism; he remained active in both areas until his final illness. His marriage to the remarkable pianist, Clara Wieck, provided him with both much-needed emotional support and a highly effective champion of his music throughout her lengthy career. Schumann's plans to be a concert pianist were thwarted at least partially by an injury to his right hand, the nature of which has been the subject of much speculation. After considering what few facts are available, the author concludes that this may have represented focal dystonia. His compositional output waxed and waned dramatically over his professional life, reflecting to some degree his emotional state. It is considered most likely that he suffered from a major affective disorder, bipolar type. This ultimately led to a suicide attempt in February 1854, and to his eventual death in July 1856. Despite wide-spread and reasonable suspicion that he may have died from neurosyphilis, severe malnutrition from self-starvation seems more likely. PMID:10718523
Braided Clifford algebras as braided quantum groups
Durdevic, M
1995-01-01
The paper deals with braided Clifford algebras, understood as Chevalley-Kahler deformations of braided exterior algebras. It is shown that Clifford algebras based on involutive braids can be naturally endowed with a braided quantum group structure. Basic group entities are constructed explicitly.
Representations of Clifford Algebras and its Applications
Okubo, Susumu
1994-01-01
A real representation theory of real Clifford algebra has been studied in further detail, especially in connection with Fierz identities. As its application, we have constructed real octonion algebras as well as related octonionic triple system in terms of 8-component spinors associated with the Clifford algebras $C(0,7)$ and $C(4,3)$.
An introduction to Clifford algebras and spinors
Vaz, Jayme
2016-01-01
This text explores how Clifford algebras and spinors have been sparking a collaboration and bridging a gap between Physics and Mathematics. This collaboration has been the consequence of a growing awareness of the importance of algebraic and geometric properties in many physical phenomena, and of the discovery of common ground through various touch points: relating Clifford algebras and the arising geometry to so-called spinors, and to their three definitions (both from the mathematical and physical viewpoint). The main point of contact are the representations of Clifford algebras and the periodicity theorems. Clifford algebras also constitute a highly intuitive formalism, having an intimate relationship to quantum field theory. The text strives to seamlessly combine these various viewpoints and is devoted to a wider audience of both physicists and mathematicians. Among the existing approaches to Clifford algebras and spinors this book is unique in that it provides a didactical presentation of the topic and ...
Algebraic spinors on Clifford manifolds
A Clifford manifold of n dimensions is defined by the fundamental relation {eμ(x), eν(x)}=2gμν(x)1 between the n frame field components {eμ(x)} and the metric matrix {gμν(x)}. At any point x, the tangent space, orthonormal frames and the spin group are defined in terms of the frame field. Different types of field are classified in terms of their properties under the general linear coordinate transformation group on the manifold, and under spin group transformations. Connections for different types of field are determined by their covariance properties under these two groups. The bivector spin connection is then uniquely determined by the 'uniformity assumption' for Clifford algebraic grades. A key result is established, that the frame field is necessarily covariantly constant on a Clifford manifold, with both vector and spin connections. 'Spin elements' are formed by contracting the frame field with Riemannian vector fields, and possess a 'two-sided' commutator covariant derivative. A set of Riemannian fields orthonormal with respect to the manifold defines an orthonormal set of spin elements in the tangent space, from which idempotents can be constructed. If S is an asymptotically flat (n-1)-dimensional submanifold on which a constant idempotent is defined in terms of a constant spin frame, parallel transport along geodesics from each point of S defines a unique position-dependent extension of the idempotent in a patch P of the manifold. In an earlier model which describes the electroweak interactions of leptons, with a simplification of the Glashow Lagrangian, the 'right-hand' part of the two-sided spin connection gives rise to new gravitational terms. The nature of these new terms is discussed. (author)
Clifford Hopf gebra for two-dimensional space
Fauser, Bertfried; Oziewicz, Zbigniew
2000-01-01
A Clifford algebra Cl(V,\\eta\\in V^*\\otimes V^*) jointly with a Clifford cogebra Cl(V,\\xi\\in V\\otimes V) is said to be a Clifford biconvolution Cl(\\eta,\\xi). We show that a Clifford biconvolution for dim_R Cl=4 does possess an antipode iff det(id-\\xi\\circ\\eta)\
On computational complexity of Clifford algebra
Budinich, Marco
2009-05-01
After a brief discussion of the computational complexity of Clifford algebras, we present a new basis for even Clifford algebra Cl(2m) that simplifies greatly the actual calculations and, without resorting to the conventional matrix isomorphism formulation, obtains the same complexity. In the last part we apply these results to the Clifford algebra formulation of the NP-complete problem of the maximum clique of a graph introduced by Budinich and Budinich ["A spinorial formulation of the maximum clique problem of a graph," J. Math. Phys. 47, 043502 (2006)].
The Closed Subsemigroups of a Clifford Semigroup
Fu Yin-yin; Zhao Xian-zhong
2014-01-01
In this paper we study the closed subsemigroups of a Clifford semigroup. It is shown that{∪}Gα | Y′ ∈ P (Y ) is the set of all closed subsemigroups ofα∈Y′a Clifford semigroup S = [Y;Gα;ϕα,β], where Y′ denotes the subsemilattice of Y generated by Y′. In particular, G is the only closed subsemigroup of itself for a group G and each one of subsemilattices of a semilattice is closed. Also, it is shown that the semiring P (S ) is isomorphic to the semiring P (Y ) for a Clifford semigroup S=[Y;Gα;ϕα,β].
Invariants of the local Clifford group
We study the algebra of complex polynomials which remain invariant under the action of the local Clifford group under conjugation. Within this algebra, we consider the linear spaces of homogeneous polynomials degree by degree and construct bases for these vector spaces for each degree, thereby obtaining a generating set of polynomial invariants. Our approach is based on the description of Clifford operators in terms of linear operations over GF(2). Such a study of polynomial invariants of the local Clifford group is mainly of importance in quantum coding theory, in particular in the classification of binary quantum codes. Some applications in entanglement theory and quantum computing are briefly discussed as well
2006-01-01
10. novembril toimub TLÜ-s Ameerika kultuurantropoloogi Clifford Geertzi mälestusõhtu, kus esinevad rektor Rein Raud, TLÜ Eesti Humanitaarinstituudi antropoloogia keskuse dotsent Lorenzo Cañás Bottos ja kultuuriteooria lektor Marek Tamm
Obstructions to Clifford System Extensions of Algebras
Antonio M Cegarra; Antonio R Garzón
2001-05-01
In this paper we do phrase the obstruction for realization of a generalized group character, and then we give a classification of Clifford systems in terms of suitable low-dimensional cohomology groups.
Clifford algebras and physical and engineering sciences
Furui, Sadataka
2013-10-01
Clifford algebra in physical and engineering science are studied. Roles of triality symmetry of Cartan's spinor in axial anomaly of particle physics and quaternion and octonion in the memristic circuits are discussed.
A diagrammatic categorification of a Clifford algebra
Tian, Yin
2013-01-01
We give a graphical calculus for a categorification of a Clifford algebra and its Fock space representation via differential graded categories. The categorical action is motivated by the gluing action between the contact categories of infinite strips.
The Clifford algebra of a finite morphism
Krashen, Daniel; Lieblich, Max
2015-01-01
We develop a general theory of Clifford algebras for finite morphisms of schemes and describe applications to the theory of Ulrich bundles and connections to period-index problems for curves of genus 1.
Classical particle with spin and Clifford algebra
Equations of motion of classical particle with spin in electromagnetic field are derived in terms of the Clifford algebra of the Minkowsky space. The use of the Clifford algebra simplifies the derivation of these equations as well as their form and process of their solving. The equations also get an evident geometric interpretation. The perturbation theory for these equations is formulated which allows to analyze the motion and the polarization of particles in various electromagnetic fields
Clifford algebra, geometric algebra, and applications
Lundholm, Douglas; Svensson, Lars
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 v...
The Hyperbolic Clifford Algebra of Multivecfors
Rodrigues Jr., W. A.; de Souza, Q. A. G.
2007-01-01
In this paper we give a thoughtful exposition of the hyperbolic Clifford algebra of multivecfors which is naturally associated with a hyperbolic space, whose elements are called vecfors. Geometrical interpretation of vecfors and multivecfors are given. Poincare automorphism (Hodge dual operator) is introduced and several useful formulas derived. The role of a particular ideal in the hyperbolic Clifford algebra whose elements are representatives of spinors and resume the algebraic properties o...
Inequalities for spinor norms in Clifford algebras
In hypercomplex analysis one considers mappings from the euclidean space Rn to its Clifford algebra Rn, where an inequality |uv| ≤ Kn |u||v| holds for u, v in Rn. In this paper the smallest possible value of the constant Kn is determined. As a byproduct the authors present a more detailed description of the faithful matrix representations of Clifford algebras, which might also be useful for other purposes. (author). 8 refs
Octonionic representations of Clifford Algebras and triality
The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the nonassociativity and noncommutativity of this division algebra are answered. Octonionic representations for Clifford algebras lead to a notion of octonionic spinors and are used to give octoninic representations of the respective orthogonal groups. Finally, the triality automorphisms are shown to exhibit a manifest Σ3x SO(8) structure in this framework
On Clifford representation of Hopf algebras and Fierz identities
Rodríguez-Romo, S
1996-01-01
We present a short review of the action and coaction of Hopf algebras on Clifford algebras as an introduction to physically meaningful examples. Some q-deformed Clifford algebras are studied from this context and conclusions are derived.
Modulo 2 periodicity of complex Clifford algebras and electromagnetic field
Varlamov, Vadim V.
1997-01-01
Electromagnetic field is considered in the framework of Clifford algebra $\\C_2$ over a field of complex numbers. It is shown here that a modulo 2 periodicity of complex Clifford algebras may be connected with electromagnetic field.
Quaternion types of Clifford algebra elements, basis-free approach
Shirokov, D S
2011-01-01
We consider Clifford algebras over the field of real or complex numbers as a quotient algebra without fixed basis. We present classification of Clifford algebra elements based on the notion of quaternion type. This classification allows us to reveal and prove a number of new properties of Clifford algebras. We rely on the operations of conjugation to introduce the notion of quaternion type. Also we find relations between the concepts of quaternion type and rank of Clifford algebra element.
Concepts of trace, determinant and inverse of Clifford algebra elements
Shirokov, Dmitry
2011-01-01
In our paper we consider the notion of determinant of Clifford algebra elements. We present some new formulas for determinant of Clifford algebra elements for the cases of dimension 4 and 5. Also we consider the notion of trace of Clifford algebra elements. We use the generalization of the Pauli's theorem for 2 sets of elements that satisfy the main anticommutation conditions of Clifford algebra.
A method of quaternion typification of Clifford algebra elements
Shirokov, Dmitry
2008-01-01
We present a new classification of Clifford algebra elements. Our classification is based on the notion of quaternion type. Using this classification we develop a method of analysis of commutators and anticommutators of Clifford algebra elements. This method allows us to find out and prove a number of new properties of Clifford algebra elements.
Inverse and Determinant in 0 to 5 Dimensional Clifford Algebra
Dadbeh, Peruzan
2011-01-01
This paper presents equations for the inverse of a Clifford number in Clifford algebras of up to five dimensions. In presenting these, there are also presented formulas for the determinant and adjugate of a general Clifford number of up to five dimensions, matching the determinant and adjugate of the matrix representations of the algebra. These equations are independent of the metric used.
Quregisters, symmetry groups and Clifford algebras
Cervantes, Dalia; Morales-Luna, Guillermo
2015-01-01
The Clifford algebra over the three-dimensional real linear space includes its linear structure and its exterior algebra, the subspaces spanned by multivectors of the same degree determine a gradation of the Clifford algebra. Through these geometric notions, natural one-to-one and two-to-one homomorphisms from $\\mbox{SO}(3)$ into $\\mbox{SU}(2)$ are built conventionally, and the set of qubits, is identified with a subgroup of $\\mbox{SU}(2)$. These constructions are suitable to be extended to c...
Agama dalam Tentukur Antropologi Simbolik Clifford Geertz
Yusri Mohamad Ramli
2012-06-01
Full Text Available Clifford Geertz can be regarded as one of the most influential figures in religious studies, particularly in the field of anthropology. His unique symbolic anthropology approach had attracted researchers because of his emphasis on deductive reasoning in explaining the meaning of religion and in viewing cultural values that exist in religion. Research based on the content analysis of his works found that Clifford Geertz thought very strongly influenced by Ibn Khaldun as both of them emphasize on the practical reality of religious phenomena in the society. These symbols are then making a cultural system of what we call religion.
Algebra de clifford del espacio tiempo
Spinel G., Ma. Carolina
2012-01-01
En un artículo previo, presentamos la estructura y relaciones básicas del algebra de Clifford Gn generada por el producto geométrico de los vectores de un espacio vectorial Vn sobre el cuerpo de los reales en la versión moderna de Hestenes. Este artículo se dedica a los aspectos fundamentales algebra de Clifford del espacio-tiempo plano (A.E.T.) muestra algunos hechos interesantes relacionados con teoría de Dirac, que ponen de manifiesto la importancia sencillez de la aplicación de algebras d...
Adinkras for Clifford Algebras, and Worldline Supermultiplets
Doran, C F; Gates, S J; Hübsch, T; Iga, K M; Landweber, G D; Miller, R L
2008-01-01
Adinkras are a graphical depiction of representations of the N-extended supersymmetry algebra in one dimension, on the worldline. These diagrams represent the component fields in a supermultiplet as vertices, and the action of the supersymmetry generators as edges. In a previous work, we showed that the chromotopology (topology with colors) of an Adinkra must come from a doubly even binary linear code. Herein, we relate Adinkras to Clifford algebras, and use this to construct, for every such code, a supermultiplet corresponding to that code. In this way, we correlate the well-known classification of representations of Clifford algebras to the classification of Adinkra chromotopologies.
Stabilizer quantum codes over the Clifford algebra
The key problem for constructing a stabilizer quantum code is how to create a set of generators for the stabilizer of the stabilizer quantum code, i.e. check matrix. In this paper, we suggest an approach based on the Clifford algebra to create the check matrix for the stabilizer quantum codes. In the proposed approach, the recursive relation of the matrix transform over the Clifford algebra is employed to generate the check matrix. With the proposed approach, a quantum code with any length can be constructed easily. Especially some new codes, which are impossible via previous approaches, are constructed
Quantized Fields à la Clifford and Unification
Pavšič, Matej
It is shown that the generators of Clifford algebras behave as creation and annihilation operators for fermions and bosons. They can create extended objects, such as strings and branes, and can induce curved metric of our space-time. At a fixed point, we consider the Clifford algebra Cl(8) of the 8D phase space, and show that one quarter of the basis elements of Cl(8) can represent all known particles of the first generation of the Standard model, whereas the other three quarters are invisible to us and can thus correspond to dark matter.
Unifying Clifford algebra formalism for relativistic fields
It is shown that a Clifford algebra formalism provides a unifying description of spin-0, -1/2, and -1 fields. Since the operators and operands are both expressed in terms of the same Clifford algebra, the formalism obtains some results which are considerably different from those of the standard of formalisms for these fields. In particular, the conservation laws are obtained uniquely and unambiguously from the equations of motion in this formalism and do not suffer from the ambiguities and inconsistencies of the standard methods
A note on Clifford-Klein forms
Jastrzȩbski, P.; Tralle, A.
2014-09-01
We consider the problem of finding Clifford-Klein forms in a class of homogeneous spaces determined by inclusions of real Lie algebras of a special type which we call strongly regular. This class of inclusions is described in terms of their Satake diagrams. For example, the complexifications of such inclusions contain the class of subalgebras generated by automorphisms of finite order. We show that the condition of strong regularity implies the restriction on the real rank of subalgebras. This in part explains why the known examples of Clifford-Klein forms are rare. We make detailed calculations of some known examples from the point of view of the Satake diagrams.
Graded Skew Clifford Algebras that are Twists of Graded Clifford Algebras
Nafari, Manizheh; Vancliff, Michaela
2012-01-01
We prove that if $A$ is a regular graded skew Clifford algebra and is a twist of a regular graded Clifford algebra $B$ by an automorphism, then the subalgebra of $A$ generated by a certain normalizing sequence of homogeneous degree-two elements is a twist of a polynomial ring by an automorphism, and is a skew polynomial ring. We also present an example that demonstrates that this can fail when $A$ is not a twist of $B$.
Thomas Clifford Allbutt and Comparative Pathology
Leung, Danny C. K.
2008-01-01
This paper reconceptualizes Thomas Clifford Allbutt's contributions to the making of scientific medicine in late nineteenth-century England. Existing literature on Allbutt usually describes his achievements, such as his design of the pocket thermometer and his advocacy of the use of the ophthalmoscope in general medicine, as independent events;…
Gravitoelectromagnetism in a complex Clifford algebra
Ulrych, S.
2006-01-01
A linear vector model of gravitation is introduced in the context of quantum physics as a generalization of electromagnetism. The gravitoelectromagnetic gauge symmetry corresponds to a hyperbolic unitary extension of the usual complex phase symmetry of electromagnetism. The reversed sign for the gravitational coupling is obtained by means of the pseudoscalar of the underlying complex Clifford algebra.
The Teodorescu Operator in Clifford Analysis
F.BRACKX; H.De SCHEPPER; M.E.LUNA-ELIZARRAR(A)S; M.SHAPIRO
2012-01-01
Euclidean Clifford analysis is a higher dimensional function theory centred around monogenic functions,i.e.,null solutions to a first order vector valued rotation invariant differential operator (θ) called the Dirac operator.More recently,Hermitian Clifford analysis has emerged as a new branch,offering yet a refinement of the Euclidean case; it focuses on the simultaneous null solutions,called Hermitian monogenic functions,to two Hermitian Dirac operators (θ)z_ and (θ)z_(+) which are invariant under the action of the unitary group.In Euclidean Clifford analysis,the Teodorescu operator is the right inverse of the Dirac operator (θ).In this paper,Teodorescu operators for the Hermitian Dirac operators (θ)z_ and (θ)z(+) are constructed.Moreover,the structure of the Euclidean and Hermitian Teodorescu operators is revealed by analyzing the more subtle behaviour of their components.Finally,the obtained inversion relations are still refined for the differential operators issuing from the Euclidean and Hermitian Dirac operators by splitting the Clifford algebra product into its dot and wedge parts.Their relationship with several complex variables theory is discussed.
Clifford Fourier transform on vector fields.
Ebling, Julia; Scheuermann, Gerik
2005-01-01
Image processing and computer vision have robust methods for feature extraction and the computation of derivatives of scalar fields. Furthermore, interpolation and the effects of applying a filter can be analyzed in detail and can be advantages when applying these methods to vector fields to obtain a solid theoretical basis for feature extraction. We recently introduced the Clifford convolution, which is an extension of the classical convolution on scalar fields and provides a unified notation for the convolution of scalar and vector fields. It has attractive geometric properties that allow pattern matching on vector fields. In image processing, the convolution and the Fourier transform operators are closely related by the convolution theorem and, in this paper, we extend the Fourier transform to include general elements of Clifford Algebra, called multivectors, including scalars and vectors. The resulting convolution and derivative theorems are extensions of those for convolution and the Fourier transform on scalar fields. The Clifford Fourier transform allows a frequency analysis of vector fields and the behavior of vector-valued filters. In frequency space, vectors are transformed into general multivectors of the Clifford Algebra. Many basic vector-valued patterns, such as source, sink, saddle points, and potential vortices, can be described by a few multivectors in frequency space. PMID:16138556
Clifford algebra in finite quantum field theories
We consider the most general power counting renormalizable and gauge invariant Lagrangean density L invariant with respect to some non-Abelian, compact, and semisimple gauge group G. The particle content of this quantum field theory consists of gauge vector bosons, real scalar bosons, fermions, and ghost fields. We assume that the ultimate grand unified theory needs no cutoff. This yields so-called finiteness conditions, resulting from the demand for finite physical quantities calculated by the bare Lagrangean. In lower loop order, necessary conditions for finiteness are thus vanishing beta functions for dimensionless couplings. The complexity of the finiteness conditions for a general quantum field theory makes the discussion of non-supersymmetric theories rather cumbersome. Recently, the F = 1 class of finite quantum field theories has been proposed embracing all supersymmetric theories. A special type of F = 1 theories proposed turns out to have Yukawa couplings which are equivalent to generators of a Clifford algebra representation. These algebraic structures are remarkable all the more than in the context of a well-known conjecture which states that finiteness is maybe related to global symmetries (such as supersymmetry) of the Lagrangean density. We can prove that supersymmetric theories can never be of this Clifford-type. It turns out that these Clifford algebra representations found recently are a consequence of certain invariances of the finiteness conditions resulting from a vanishing of the renormalization group β-function for the Yukawa couplings. We are able to exclude almost all such Clifford-like theories. (author)
Gravitoelectromagnetism in a complex Clifford algebra
A linear vector model of gravitation is introduced in the context of quantum physics as a generalization of electromagnetism. The gravitoelectromagnetic gauge symmetry corresponds to a hyperbolic unitary extension of the usual complex phase symmetry of electromagnetism. The reversed sign for the gravitational coupling is obtained by means of the pseudoscalar of the underlying complex Clifford algebra
Physical Holonomy, Thomas Precession, and Clifford Algebra
After a general discussion of the physical significance of holonomy group transformations, a relation between the transports of Fermi-Walker and Levi-Civita in Special Relativity is pointed out. A well-known example -the Thomas-Wigner angle - is rederived in a completely frame-independent manner using Clifford algebra. 14 refs. (Author)
Random symmetric matrices on Clifford algebras
Bakry, Dominique; Zani, Marguerite
2013-01-01
We consider Brownian motions and other processes (Ornstein-Uhlenbeck processes, spherical Brownian motions) on various sets of symmetric matrices constructed from algebra structures, and look at their associated spectral measure processes. This leads to the identification of the multiplicity of the eigenvalues, together with the identification of the spectral measures. For Clifford algebras, we thus recover Bott's periodicity.
A Clifford Algebra Description of Polarization Optics
Yevick, David; Soliman, George
2014-03-01
The polarization changes induced by optical components are represented as Clifford algebra transformations. This yields a unified formalism for polarized and partially polarized light and for the frequency dependence of polarization in the presence of polarization mode dispersion and polarization dependent loss. Work supported by NSERC.
Solari, Hernán G.
2013-01-01
To present the personality of Bob Gilmore is a formidable task, as his scientific contributions include group theory, laser physics, non-linear dynamics, catastrophe theory, thermodynamics, dynamical systems, quantum theory and more. But even if we succeed in describing his contributions, much of Gilmore's being would be lost. Bob as advisor, Bob as father, Bob as teacher, Bob as scientific communicator reveal as much of Bob Gilmore as his scientific papers and his books. Very much as in the Group Theory so close to him, there is a Robert Gilmore in abstract as well as representations of Robert Gilmore. We will make an attempt to find the "principle of the rule", the abstract level of Robert Gilmore as well as Robert Gilmore, himself, as a representation of the duality science-humanism.
Disproof of Bell's Theorem by Clifford Algebra Valued Local Variables
Christian, Joy
2007-01-01
It is shown that Bell's theorem fails for the Clifford algebra valued local realistic variables. This is made evident by exactly reproducing quantum mechanical expectation value for the EPR-Bohm type spin correlations observable by means of a local, deterministic, Clifford algebra valued variable, without necessitating either remote contextuality or backward causation. Since Clifford product of multivector variables is non-commutative in general, the spin correlations derived within our local...
Sparse Representations of Clifford and Tensor algebras in Maxima
Prodanov, Dimiter; Toth, Viktor T.
2016-01-01
Clifford algebras have broad applications in science and engineering. The use of Clifford algebras can be further promoted in these fields by availability of computational tools that automate tedious routine calculations. We offer an extensive demonstration of the applications of Clifford algebras in electromagnetism using the geometric algebra G3 = Cl(3,0) as a computational model in the Maxima computer algebra system. We compare the geometric algebra-based approach with conventional symboli...
Clifford Algebra-Valued Wavelet Transform on Multivector Fields
Bahri, Mawardi; Adji, Sriwulan; Zhao, Jiman
2010-01-01
This paper presents a construction of the n = 2 (mod 4) Clifford algebra Cln,0-valued admissible wavelet transform using the admissible similitude group SIM(n), a subgroup of the affine group of Rn. We express the admissibility condition in terms of the Cln,0 Clifford Fourier transform (CFT). We show that its fundamental properties such as inner product, norm relation, and inversion formula can be established whenever the Clifford admissible wavelet satisfies a particular ad...
The Extended Relativity Theory in Clifford Spaces
Castro, C
2004-01-01
A brief review of some of the most important features of the Extended Relativity theory in Clifford-spaces ($C$-spaces) is presented whose " point" coordinates are noncommuting Clifford-valued quantities and which incorporate the lines, areas, volumes,.... degrees of freedom associated with the collective particle, string, membrane,... dynamics of $p$-loops (closed p-branes) living in target $D$-dimensional spacetime backgrounds. $C$-space Relativity naturally incorporates the ideas of an invariant length (Planck scale), maximal acceleration, noncommuting coordinates, supersymmetry, holography, higher derivative gravity with torsion and variable dimensions/signatures that allows to study the dynamics of all (closed) p-branes, for all values of $ p $, on a unified footing. It resolves the ordering ambiguities in QFT and the problem of time in Cosmology. A discussion of the maximal-acceleration Relativity principle in phase-spaces follows along with the study of the invariance group of symmetry transformations ...
The Extended Relativity Theory in Clifford Spaces
Castro, C
2004-01-01
A brief review of some of the most important features of the Extended Relativity theory in Clifford-spaces ( $C$-spaces) is presented whose " point" coordinates are noncommuting Clifford-valued quantities and which incoporate the lines, areas, volumes, .... degrees of freedom associated with the collective particle, string, membrane, ... dynamics of the $p$-loop histories (closed p-branes) living in target $D$-dimensional spacetime backgrounds. $C$-space Relativity naturally incoporates the ideas of an invariant length (Planck scale), maximal acceleration, noncommuting coordinates, supersymmetry, holography, superluminal propagation, higher derivative gravity with torsion and variable dimensions/signatures that allows to study the dynamics of all (closed ) p-branes, for all values of $ p $, in a unified footing. It resolves the ordering ambiguities in QFT and the problem of time in Cosmology. A discussion of the maximal-acceleration Relativity principle in phase-spaces follows along with the study of the inva...
A Clifford analysis approach to superspace
A new framework for studying superspace is given, based on methods from Clifford analysis. This leads to the introduction of both orthogonal and symplectic Clifford algebra generators, allowing for an easy and canonical introduction of a super-Dirac operator, a super-Laplace operator and the like. This framework is then used to define a super-Hodge coderivative, which, together with the exterior derivative, factorizes the Laplace operator. Finally both the cohomology of the exterior derivative and the homology of the Hodge operator on the level of polynomial-valued super-differential forms are studied. This leads to some interesting graphical representations and provides a better insight in the definition of the Berezin-integral
Quantum Clifford algebras from spinor representations
A general theory of quantum Clifford algebras is presented, based on a quantum generalization of the Cartan theory of spinors. We concentrate on the case when it is possible to apply the quantum-group formalism of bicovariant bimodules. The general theory is then singularized to the quantum SL(n,C) group case, to generate explicit forms for the whole class of braidings required. The corresponding spinor representations are introduced and investigated. Starting from our Clifford algebras we introduce the quantum-Euclidean underlying spaces compatible with different choices of *-structures from where the analogues of Dirac and Laplace operators are built. Using the formalism developed, quantum Spin(n) groups are defined. copyright 1996 American Institute of Physics
Hecke Algebras, SVD, and Other Computational Examples with {\\sc CLIFFORD}
Ablamowicz, Rafal
1999-01-01
{\\sc CLIFFORD} is a Maple package for computations in Clifford algebras $\\cl (B)$ of an arbitrary symbolic or numeric bilinear form B. In particular, B may have a non-trivial antisymmetric part. It is well known that the symmetric part g of B determines a unique (up to an isomorphism) Clifford structure on $\\cl(B)$ while the antisymmetric part of B changes the multilinear structure of $\\cl(B).$ As an example, we verify Helmstetter's formula which relates Clifford product in $\\cl(g)$ to the Cl...
Twin bent functions and Clifford algebras
Leopardi, Paul C.
2015-01-01
This paper examines a pair of bent functions on $\\mathbb{Z}_2^{2m}$ and their relationship to a necessary condition for the existence of an automorphism of an edge-coloured graph whose colours are defined by the properties of a canonical basis for the real representation of the Clifford algebra $\\mathbb{R}_{m,m}.$ Some other necessary conditions are also briefly examined.
Angles between subspaces computed in Clifford Algebra
Hitzer, Eckhard
2013-01-01
We first review the definition of the angle between subspaces and how it is computed using matrix algebra. Then we introduce the Grassmann and Clifford algebra description of subspaces. The geometric product of two subspaces yields the full relative angular information in an explicit manner. We explain and interpret the result of the geometric product of subspaces gaining thus full practical access to the relative orientation information.
75 FR 49979 - Tony T. Bui, M.D.; Revocation of Registration
2010-08-16
....'' Robert A. Leslie, M.D., 68 FR 15227, 15230 (2003). I ``may rely on any one or a combination of factors... girlfriend admitting to a soft- drink adulteration that never occurred.'' Id. at 32. Moreover, even after...
Fischli, Fredi; Olsen, Niels
2012-01-01
Die Monografie Robert & Trix Haussmann eröffnet die Publikationsreihe STUDIOLO / Edition Patrick Frey, eine Kollaboration des Verlags mit dem Ausstellungsraum STUDIOLO. Die Kuratoren Fredi Fischli und Niels Olsen betreiben in einem Atelierhaus in Zürich ein vielfältiges Programm gegenwärtiger Kunstproduktion. Die Ausstellung The Log-O-Rhythmic Slide Rule im Frühjahr 2012 widmete sich dem Werk von Trix und Robert Haussmann und ist Ausgangspunkt für die folgende Publikation, die mit Bildern, Es...
CONVOLUTION THEOREMS FOR CLIFFORD FOURIER TRANSFORM AND PROPERTIES
Mawardi Bahri
2014-10-01
Full Text Available The non-commutativity of the Clifford multiplication gives different aspects from the classical Fourier analysis.We establish main properties of convolution theorems for the Clifford Fourier transform. Some properties of these generalized convolutionsare extensions of the corresponding convolution theorems of the classical Fourier transform.
An investigation of symmetry operations with Clifford algebra
After presenting Clifford algebra and quaternions, the symmetry operations with Clifford algebra and quaternions are defined. This symmetry operations are applied to a Platonic solid, which is called as dodecahedron. Also, the vertices of a dodecahedron presented in the Cartesian coordinates are calculated (Authors)
Clifford algebras and the classical dynamical Yang-Baxter equation
Alekseev, Anton; Meinrenken, E.
2003-01-01
We describe a relationship of the classical dynamical Yang-Baxter equation with the following elementary problem for Clifford algebras: Given a vector space $V$ with quadratic form $Q_V$, how is the exponential of an element in $\\wedge^2(V)$ under exterior algebra multiplication related to its exponential under Clifford multiplication?
Explicit isomorphisms of real Clifford algebras
N. Değırmencı
2006-06-01
Full Text Available It is well known that the Clifford algebra Clp,q associated to a nondegenerate quadratic form on Ã¢Â„ÂnÃ¢Â€Â‰(n=p+q is isomorphic to a matrix algebra K(m or direct sum K(mÃ¢ÂŠÂ•K(m of matrix algebras, where K=Ã¢Â„Â,Ã¢Â„Â‚,Ã¢Â„Â. On the other hand, there are no explicit expressions for these isomorphisms in literature. In this work, we give a method for the explicit construction of these isomorphisms.
N=2-extended supersymmetries and Clifford algebras
By searching for the largest numbers of one-parameter Lie algebras for one-dimensional supersymmetric harmonic oscillators, we study the impact of fermionic variables associated with fundamental Clifford algebras such as cl2 and cl4. Amongst the sets of associated generators we point out the largest closed superstructures identified as invariance or spectrum generating superalgebras. The additional supersymmetries which do not close under the generalized Lie product lead to new constants of motion. Direct connections with other recent contributions are also singled out. (orig.)
On Clifford's theorem for singular curves
Franciosi, Marco
2011-01-01
Let C be a 2-connected Gorenstein curve either reduced or contained in a smooth algebraic surface and let S be a subcanonical cluster (i.e. a 0-dim scheme such that the space H^0(C, I_S K_C) contains a generically invertible section). Under some general assumptions on S or C we show that h^0(C, I_S K_C) <= p_a(C) - deg (S)/2 and if equality holds then either S is trivial, or C is honestly hyperelliptic or 3-disconnected. As a corollary we give a generalization of Clifford's theorem for reduced curves.
78 FR 59060 - Gabriel Sanchez, M.D.; Decision and Order
2013-09-25
... any compression of the nerves, or the spinal column, or the nerve root,'' and that it was ``difficult... sufficiently reliable to be accepted and relied upon in this .'' See Cynthia M. Cadet, M.D., 76 FR 19450, 19458... factors are . . . considered in the disjunctive.'' Robert A. Leslie, M.D., 68 FR 15227, 15230 (2003)....
Multifractal vector fields and stochastic Clifford algebra
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.
Multifractal vector fields and stochastic Clifford algebra
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.
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
On the relation of Manin's quantum plane and quantum Clifford algebras
In a recent work we have shown that quantum Clifford algebras - i.e. Clifford algebras of an arbitrary bilinear form - are closely related to the deformed structures as q-spin groups, Hecke algebras, q-Young operators and deformed tensor products. The question to relate Manin's approach to quantum Clifford algebras is addressed here. Explicit computations using the CLIFFORD Maple package are exhibited. The meaning of non-commutative geometry is reexamined and interpreted in Clifford algebraic terms. (author)
Dervan, Peter B.
1988-01-01
I met the great John D. Roberts at Caltech for the first time in 1972. He seemed larger than life: a tall, handsome, rugged man with a shock of curly grey hair, broad warm grin, huge embracing handshake, and twinkle in his eye. This was the man who has inspired a generation of organic chemists by his intellectual leadership, his innovation in developing new techniques to explore mechanistic organic chemistry, his contributions to undergraduate and graduate education and the advancement of s...
Clifford algebra approach to the coincidence problem for planar lattices.
Rodríguez, M A; Aragón, J L; Verde-Star, L
2005-03-01
The problem of coincidences of planar lattices is analyzed using Clifford algebra. It is shown that an arbitrary coincidence isometry can be decomposed as a product of coincidence reflections and this allows planar coincidence lattices to be characterized algebraically. The cases of square, rectangular and rhombic lattices are worked out in detail. One of the aims of this work is to show the potential usefulness of Clifford algebra in crystallography. The power of Clifford algebra for expressing geometric ideas is exploited here and the procedure presented can be generalized to higher dimensions. PMID:15724067
Valence-band of cubic semiconductors: Clifford algebra approach II
Dargys, A, E-mail: dargys@pfi.l [Semiconductor Physics Institute, A. Gostauto 11, LT-01108 Vilnius (Lithuania)
2010-07-15
Application of Clifford algebra in the analysis of valence-band spin properties in semiconductors is considered. In the first part (Dargys A 2009 Phys. Scr. 80 065701), for this purpose the isomorphism between multivectors and their matrix representations was used to transform the problem to Clifford algebra. Here equivalence rules are established between the spinors of Hilbert space and basis elements of the five-dimensional Clifford algebra Cl{sub 4,1}. Then, the rules are applied to the total angular momentum components and the two-band hole Hamiltonian. The resulting biquaternionic Schroedinger equation for hole spin is solved as an example.
Valence-band of cubic semiconductors: Clifford algebra approach II
Application of Clifford algebra in the analysis of valence-band spin properties in semiconductors is considered. In the first part (Dargys A 2009 Phys. Scr. 80 065701), for this purpose the isomorphism between multivectors and their matrix representations was used to transform the problem to Clifford algebra. Here equivalence rules are established between the spinors of Hilbert space and basis elements of the five-dimensional Clifford algebra Cl4,1. Then, the rules are applied to the total angular momentum components and the two-band hole Hamiltonian. The resulting biquaternionic Schroedinger equation for hole spin is solved as an example.
On bundles of rank 3 computing Clifford indices
Lange, H
2012-01-01
Let $C$ be a smooth irreducible projective algebraic curve defined over the complex numbers. The notion of the Clifford index of $C$ was extended a few years ago to semistable bundles of any rank. Recent work has been focussed mainly on the rank-2 Clifford index, although interesting results have also been obtained for the case of rank 3. In this paper we extend this work, obtaining improved lower bounds for the rank-3 Clifford index. This allows the first computations of the rank-3 index in non-trivial cases and examples for which the rank-3 index is greater than the rank-2 index.
Kultuurantropoloog Clifford Geertzi mälestusõhtu
2006-01-01
Tallinna Ülikoolis peetakse homme Ameerika kultuurantropoloogi Clifford Geertzi mälestusõhtut, esinevad rektor Rein Raud, Eesti Humanitaarinstituudi dotsent Lorenzo Cañás Bottos ja kultuuriteooria lektor Marek Tamm
The γ5-problem and anomalies - a Clifford algebra approach
It is shown that a strong correspondence between noncyclicity and anomalies exists. This allows, by fundamental properties of Clifford algebras, to build a simple and consistent scheme for treating γ5 without using (d-4)-dimensional objects. (orig.)
Symplectic, orthogonal and linear Lie groups in Clifford algebra
Shirokov, D. S.
2014-01-01
In this paper we prove isomorphisms between 5 Lie groups (of arbitrary dimension and fixed signatures) in Clifford algebra and classical matrix Lie groups - symplectic, orthogonal and linear groups. Also we obtain isomorphisms of corresponding Lie algebras.
A Clifford algebra associated to generalized Fibonacci quaternions
Flaut, Cristina
2014-01-01
In this paper we find a Clifford algebra associated to generalized Fibonacci quaternions. In this way, we provide a nice algorithm to obtain a division quaternion algebra starting from a quaternion non-division algebra and vice-versa.
Cayley-Dickson and Clifford Algebras as Twisted Group Algebras
Bales, John W.
2011-01-01
The effect of some properties of twisted groups on the associated algebras, particularly Cayley-Dickson and Clifford algebras. It is conjectured that the Hilbert space of square-summable sequences is a Cayley-Dickson algebra.
Clifford algebra and the projective model of Hyperbolic spaces
Sokolov, Andrey
2016-01-01
I apply the algebraic framework developed in [1] to study geometry of hyperbolic spaces in 1, 2, and 3 dimensions. The background material on projectivised Clifford algebras and their application to Cayley-Klein geometries is described in [2].
Composite bundles in Clifford algebras. Gravitation theory. Part I
Sardanashvily, G
2016-01-01
Based on a fact that complex Clifford algebras of even dimension are isomorphic to the matrix ones, we consider bundles in Clifford algebras whose structure group is a general linear group acting on a Clifford algebra by left multiplications, but not a group of its automorphisms. It is essential that such a Clifford algebra bundle contains spinor subbundles, and that it can be associated to a tangent bundle over a smooth manifold. This is just the case of gravitation theory. However, different these bundles need not be isomorphic. To characterize all of them, we follow the technique of composite bundles. In gravitation theory, this technique enables us to describe different types of spinor fields in the presence of general linear connections and under general covariant transformations.
On generalized Clifford algebras and their physical applications
Jagannathan, R.
2010-01-01
Generalized Clifford algebras (GCAs) and their physical applications were extensively studied for about a decade from 1967 by Alladi Ramakrishnan and his collaborators under the name of L-matrix theory. Some aspects of GCAs and their physical applications are outlined here. The topics dealt with include: GCAs and projective representations of finite abelian groups, Alladi Ramakrishnan's sigma operation approach to the representation theory of Clifford algebra and GCAs, Dirac's positive energy...
Scalable randomized benchmarking of non-Clifford gates
Cross, Andrew W.; Magesan, Easwar; Bishop, Lev S.; Smolin, John A.; Gambetta, Jay M.
2015-01-01
Randomized benchmarking is a widely used experimental technique to characterize the average error of quantum operations. Benchmarking procedures that scale to enable characterization of $n$-qubit circuits rely on efficient procedures for manipulating those circuits and, as such, have been limited to subgroups of the Clifford group. However, universal quantum computers require additional, non-Clifford gates to approximate arbitrary unitary transformations. We define a scalable randomized bench...
On continuity of homomorphisms between topological Clifford semigroups
I. Pastukhova
2014-01-01
Generalizing an old result of Bowman we prove that a homomorphism $f:X\\to Y$ between topological Clifford semigroups is continuous if the idempotent band $E_X=\\{x\\in X:xx=x\\}$ of $X$ is a $V$-semilattice;the topological Clifford semigroup $Y$ is ditopological;the restriction $f|E_X$ is continuous;for each subgroup $H\\subset X$ the restriction $f|H$ is continuous.
Clifford Algebras in Symplectic Geometry and Quantum Mechanics
Binz, Ernst; de Gosson, Maurice A.; Hiley, Basil J.
2011-01-01
The necessary appearance of Clifford algebras in the quantum description of fermions has prompted us to re-examine the fundamental role played by the quaternion Clifford algebra, C(0,2). This algebra is essentially the geometric algebra describing the rotational properties of space. Hidden within this algebra are symplectic structures with Heisenberg algebras at their core. This algebra also enables us to define a Poisson algebra of all homogeneous quadratic polynomials on a two-dimensional s...
Real representations of finite Clifford algebras. I. Classification
A classification of real matrix irreducible representations of finite-dimensional real Clifford algebras has been made. In contrast to the case of complex representation, three distinct types of representations can be obtained which we call normal, almost complex, and quaternionic. The dimension of the latter two cases is twice as large as that of the normal representation. A criteria for a given Clifford algebra to possess a particular type of the representations is also given with some applications
Representation of Crystallographic Subperiodic Groups in Clifford's Geometric Algebra
Hitzer, Eckhard; Ichikawa, Daisuke
2013-01-01
This paper explains how, following the representation of 3D crystallographic space groups in Clifford's geometric algebra, it is further possible to similarly represent the 162 so called subperiodic groups of crystallography in Clifford's geometric algebra. A new compact geometric algebra group representation symbol is constructed, which allows to read off the complete set of geometric algebra generators. For clarity moreover the chosen generators are stated explicitly. The group symbols are ...
Clifford algebras and the quantization of the free Dirac field
In this paper we study the Clifford algebra of the Minkowski space and prove that any of its irreducible representations carries a canonical representation of a cover group of the Lorentz group, a canonical sesquilinear hermitian form, a canonical conjugation and a canonical antilinear operator called the charge conjugation. We also consider the problem of the quantization of the free Dirac field, in connection with the infinite dimensional Clifford algebra associated to the space of classical fields. (Author)
The Stabilized Poincare-Heisenberg algebra: a Clifford algebra viewpoint
Gresnigt, N. G.; Renaud, P. F.; Butler, P. H.
2006-01-01
The stabilized Poincare-Heisenberg algebra (SPHA) is the Lie algebra of quantum relativistic kinematics generated by fifteen generators. It is obtained from imposing stability conditions after attempting to combine the Lie algebras of quantum mechanics and relativity which by themselves are stable, however not when combined. In this paper we show how the sixteen dimensional Clifford algebra CL(1,3) can be used to generate the SPHA. The Clifford algebra path to the SPHA avoids the traditional ...
Dirac cohomology for the degenerate affine Hecke Clifford algebra
Chan, Kei Yuen
2013-01-01
We define an analogue of the Dirac operator for the degenerate affine Hecke-Clifford algebra. A main result is to relate the central characters of the degenerate affine Hecke-Clifford algebra with the central characters of the Sergeev algebra via Dirac cohomology. The action of the Dirac operator on certain modules is also computed. Results in this paper could be viewed as a projective version of the Dirac cohomology of the degenerate affine Hecke algebra.
Sparavigna, Amelia Carolina
2013-01-01
Here I am proposing a translation and discussion of the De Colore, one of the short scientific treatises written by Robert Grosseteste. In this very short treatise of the mid-1220s, Grosseteste continued the discussion on light and colours he started in the De Iride. He describes two manners of counting colours: one gives an infinity of tones, the other counts seven colours. In both cases, colours are created by the purity or impurity of the transparent medium when light is passing through it. This medieval framework survived until Newton's experiments with prisms.
Lauren van Haaften-Schick
2011-06-01
Full Text Available This January, while preparing a new course, Robert Seydel was struck and killed by an unexpected heart attack. He was a critically under-appreciated artist and one of the most beloved and admired professors at Hampshire College.At the time of his passing, Seydel was on the brink of a major artistic and career milestone. His Book of Ruth was being prepared for publication by Siglio Press. His publisher describes the book as: “an alchemical assemblage that composes the life of his alter ego, Ruth Greisman— spinster, Sunday painter, and friend to Joseph Cornell and Marcel Duchamp. Through collages, drawings, and journal entries from Ruth’s imagined life, Seydel invokes her interior world in novelistic rhythms.”This convergence of his professional triumph with the tragedy of his death makes now a particularly appropriate time to think about Robert Seydel and his work. This feature contains a selection of excerpts from Book of Ruth (courtesy of Siglio Press alongside a pair of texts remembering him and giving critical and biographical insights into his art and his person. These texts, from a former student and a colleague respectively, were originally prepared for Seydel's memorial at Hampshire College and have since been revised for publication in continent.
Robert Cailliau honoured by Belgium
2004-01-01
On 15 November, Robert Cailliau received the distinction of Commandeur de l'Ordre de Léopold from Belgium, his home country, for his pioneering work in developing the World Wide Web. Robert Cailliau worked closely with Tim Berners-Lee, the inventor of the Web.
k-deformed Poincare algebras and quantum Clifford-Hopf algebras
da Rocha, Roldao; Bernardini, Alex E.; Vaz Jr, Jayme
2008-01-01
The Minkowski spacetime quantum Clifford algebra structure associated with the conformal group and the Clifford-Hopf alternative k-deformed quantum Poincare algebra is investigated in the Atiyah-Bott-Shapiro mod 8 theorem context. The resulting algebra is equivalent to the deformed anti-de Sitter algebra U_q(so(3,2)), when the associated Clifford-Hopf algebra is taken into account, together with the associated quantum Clifford algebra and a (not braided) deformation of the periodicity Atiyah-...
Shirokov, Dmitry
2009-01-01
In this paper we further develop the method of quaternion typification of Clifford algebra elements suggested by the author in the previous papers. On the basis of new classification of Clifford algebra elements it is possible to find out and prove a number of new properties of Clifford algebra. In particular, we find subalgebras and Lie subalgebras of Clifford algebra and subalgebras of the Lie algebra of the pseudo-unitary Lie group.
Clifford modules and invariants of quadratic forms
Karoubi, Max
2010-01-01
Let A be a commutative ring with 1/2 in A. In this paper, we define new characteristic classes for finitely generated projective A-modules V provided with a non degenerate quadratic form. These classes belong to the usual K-theory of A. They generalize in some sense the classical "cannibalistic" Bott classes in topological K-theory, when A is the ring of continuous functions on a compact space X. To define these classes, we replace the topological Thom isomorphism by a Morita equivalence between A-modules and C(V)-modules, where C(V) denotes the Clifford algebra of V, assuming that the class of C(V) in the graded Brauer group of A is trivial. We then essentially use ideas going back to Atiyah, Bott and Shapiro together with an alternative definition of the Adams operations due to Atiyah. When C(V) is not trivial in the graded Brauer group, the characteristic classes take their values in an algebraic analog of twisted K-theory. Finally, we also make use of a letter written by J.-P. Serre to the author, in orde...
Spin Singularities: Clifford Kaleidoscopes and Particle Masses
Cohen, Marcus S
2009-01-01
Are particles singularities- vortex lines, tubes, or sheets in some global ocean of dark energy? We visit the zoo of Lagrangian singularities, or caustics in a spin(4,C) phase flow over compactifed Minkowsky space, and find that their varieties and energies parallel the families and masses of the elementary particles. Singularities are classified by tensor products of J Coxeter groups s generated by reflections. The multiplicity, s, is the number reflections needed to close a cycle of null zigzags: nonlinear resonances of J chiral pairs of lightlike matter spinors with (4-J) Clifford mirrors: dyads in the remaining unperturbed vacuum pairs. Using singular perturbations to "peel" phase-space singularities by orders in the vacuum intensity, we find that singular varieties with quantized mass, charge, and spin parallel the families of leptons (J=1), mesons (J=2), and hadrons (J=3). Taking the symplectic 4 form - the volume element in the 8- spinor phase space- as a natural Lagrangian, these singularities turn ou...
Quantum Clifford algebra from classical differential geometry
We show the emergence of Clifford algebras of nonsymmetric bilinear forms as cotangent algebras of Kaluza-Klein (KK) spaces pertaining to teleparallel space-times. These spaces are canonically determined by the horizontal differential invariants of Finsler bundles of the type, B'(M)→S(M), where B'(M) is the set of all the tangent frames to a differentiable manifold M, and where S(M) is the sphere bundle. If M is space-time itself, M4, the 'geometric phase space', S(M4), has dimension seven. This reformulation of the horizontal invariants as pertaining to a KK space removes the mismatch between the dimensionality of the tangent frames to M4 and the dimensionality of S(M4). In the KK space, a symmetric tangent metric induces a cotangent metric which is not symmetric in general. An interior covariant derivative in the sense of Kaehler is defined. It involves the antisymmetric part of the cotangent metric, which thus enters electrodynamics and the Dirac equation
The naked spinor a rewrite of Clifford algebra
Morris, Dennis
2015-01-01
This book is about spinors. The whole mathematical theory of spinors is within Clifford algebra, and so this book is about Clifford algebra. Spinor theory is really the theory of empty space, and so this book is about empty space. The whole of Clifford algebra is rewritten in a much simpler form, and so the whole of spinor theory is rewritten in a much simpler form. Not only does this book make Clifford algebra simple and obvious, but it lifts the fog and mirrors from this area of mathematics to make it clear and obvious. In doing so, the true nature of spinors is revealed to the reader, and, with that, the true nature of empty space. To understand this book you will need an elementary knowledge of linear algebra (matrices) an elementary knowledge of finite groups and an elementary knowledge of the complex numbers. From no more than that, you will gain a very deep understanding of Clifford algebra, spinors, and empty space. The book is well written with all the mathematical steps laid before the reader in a w...
On Generalized Clifford Algebras and Spin Lattice Systems
The incessantly growing area of applications of Clifford algebras and naturalness of their use in formulating problems for direct calculation entitles one to call them Clifford numbers. The generalized ''universal'' Clifford numbers are here introduced via k-ubic form Qk replacing quadratic one in familiar construction of an appropriate ideal of tensor algebra. One of the epimorphic images of universal algebras k - Cn = T(V)/I(Qk) is the algebra Cln(k) with n generators and these are the algebras to be used here. Because generalized Clifford algebras Cln(k) possess inherent Zk x Zk x Λ XZk grading - this makes them an efficient apparatus to deal with spin lattice systems. This efficiency is illustrated here by derivation of two major observations. Namely - partition functions for vector and planar Potts models and other model with Zn invariant Hamiltonian are polynomials in generalized hyperbolic functions of the n-th order. Secondly, the problem of algorithmic calculation of the partition function for any vector Potts model as treated here is reduced to the calculation of Tr(γi1..γis ), where γ's are the generators of the generalized Clifford algebra. Finally the expression for Tr(γi1..γis), for arbitrary collection of such y matrices is derived. (author)
Method of generalized Reynolds operators and Pauli's theorem in Clifford algebras
Shirokov, D. S.
2014-01-01
We consider real and complex Clifford algebras of arbitrary even and odd dimensions and prove generalizations of Pauli's theorem for two sets of Clifford algebra elements that satisfy the main anticommutative conditions. In our proof we use some special operators - generalized Reynolds operators. This method allows us to obtain an algorithm to compute elements that connect two different sets of Clifford algebra elements.
The Clifford-Fourier transform $\\mathcal{F}_O$ and monogenic extensions
Lopez, Arnoldo Bezanilla; Sanchez, Omar Leon
2014-01-01
Several versions of the Fourier transform have been formulated in the framework of Clifford algebra. We present a (Clifford-Fourier) transform, constructed using the geometric properties of Clifford algebra. We show the corresponding results of operational calculus. We obtain a technique to construct monogenic extensions of a certain type of continuous functions, and versions of the Paley-Wiener theorems are formulated.
Hole spin precession in semiconductors: Clifford algebra approach
Recently the Clifford (geometrical) algebra was addressed to describe dynamical properties of electron spin in semiconductors (Dargys 2009 Phys. Scr. 79 055702). In this paper, the Clifford algebra is used to investigate heavy and light hole spins in valence bands of cubic semiconductors. Owing to strong spin-orbit interaction in the valence band, the precession trajectories of hole spin polarization are ellipses, or even lines, rather than circles as usually found in the case of electrons. The paper shows how one can investigate the valence band spectrum and free-hole spin precession trajectories within a framework of the Clifford algebra Cl4,1. General formulae that describe free heavy- and light-mass hole spin precession are presented.
Clifford algebraic approach to superfields and some consequences
Frames provided by Clifford algebras C/sub n/ are considered for the purpose of expanding a field multiplet (containing, possibly, both bosons and fermions). After giving a brief--mainly geometrical--description of Clifford algebras, the main tools of the present scheme are introduced: a scalar product in C/sub n/, a conjugation operation, and a ''Lorentz covariant derivative.'' It is described how these Clifford algebraic tools can be applied in order to obtain free massless Lagrangian expressions for a number of field theoretical models. It is also shown how gauge fields can arise within this scheme. It appears possible that the suggested formalism can lead naturally to spinor field operators as ''gauge fields.'' A specific example which can lead to a two-component ''gauge spinor'' is discussed. Possible lines of investigation which could solidify this potentially rich approach are suggested. (1 figure, 1 table)
Clifford Algebra Implying Three Fermion Generations Revisited
The author's idea of algebraic compositeness of fundamental particles, allowing to understand the existence in Nature of three fermion generations, is revisited. It is based on two postulates. Primo, for all fundamental particles of matter the Dirac square-root procedure √p2 → Γ(N)·p works, leading to a sequence N=1, 2, 3, ... of Dirac-type equations, where four Dirac-type matrices Γ(N)μ are embedded into a Clifford algebra via a Jacobi definition introducing four ''centre-of-mass'' and (N - 1) x four ''relative'' Dirac-type matrices. These define one ''centre-of-mass'' and N - 1 ''relative'' Dirac bispinor indices. Secundo, the ''centre-of-mass'' Dirac bispinor index is coupled to the Standard Model gauge fields, while N - 1 ''relative'' Dirac bispinor indices are all free indistinguishable physical objects obeying Fermi statistics along with the Pauli principle which requires the full antisymmetry with respect to ''relative'' Dirac indices. This allows only for three Dirac-type equations with N = 1, 3, 5 in the case of N odd, and two with N = 2, 4 in the case of N even. The first of these results implies unavoidably the existence of three and only three generations of fundamental fermions, namely leptons and quarks, as labelled by the Standard Model signature. At the end, a comment is added on the possible shape of Dirac 3 x 3 mass matrices for four sorts of spin-1/2 fundamental fermions appearing in three generations. For charged leptons a prediction is mτ = 1776.80 MeV, when the input of experimental me and mμ is used. (author)
Extending Fourier transformations to Hamilton's quaternions and Clifford's geometric algebras
Hitzer, Eckhard
2013-10-01
We show how Fourier transformations can be extended to Hamilton's algebra of quaternions. This was initially motivated by applications in nuclear magnetic resonance and electric engineering. Followed by an ever wider range of applications in color image and signal processing. Hamilton's algebra of quaternions is only one example of the larger class of Clifford's geometric algebras, complete algebras encoding a vector space and all its subspace elements. We introduce how Fourier transformations are extended to Clifford algebras and applied in electromagnetism, and in the processing of images, color images, vector field and climate data.
Clifford, Dirac, and Majorana algebras, and their representations
We show that the Dirac algebra is an algebra in five dimensions. It has traditionally been confused with the two distinct algebras in four dimensions, which we have identified as the Majorana algebra and the Clifford algebra in Minkowski space-time. A careful discussion of the subtle inter-relationship between these three algebras is achieved by employing a basis of differential forms. In addition, we provide for the first time a 4 x 4 complex matrix representation of the Clifford algebra in Minkowski spacetime, and compare it to the matrix representations of the Dirac and Majorana algebras. A remark on Eddington's E-numbers is included
On the relationship between twistors and Clifford algebras
Basis p-forms of a complexified Minkowski spacetime can be used to realize a Clifford algebra isomorphic to the Dirac algebra of γ matrices. Twistor space is then constructed as a spin of this abstract algebra through a Witt decomposition of the Minkowski space. We derive explicit formulas relating the basis p-forms to index one twistors. Using an isomorphism between the Clifford algebra and a space of index two twistors, we expand a suitably defined antisymmetric index two twistor basis on p-forms of ranks zero, one, and four. Together with the inverse formula they provide a complete passage between twistors and p-forms. (orig.)
The investigation of platonic solids symmetry operations with clifford algebra
The geometric algebra produces the new fields of view in the modern mathematical physics, definition of bodies and rearranging for equations of mathematics and physics. The new mathematical approaches play an important role in the progress of physics. After presenting Clifford algebra and quarantine's, the symmetry operations with Clifford algebra and quarantine's are defined. This symmetry operations are applied to a Platonic solids, which are called as tetrahedron, cube, octahedron, icosahedron and dodecahedron. Also, the vertices of Platonic solids presented in the Cartesian coordinates are calculated
Weak Values: Approach through the Clifford and Moyal Algebras
In this paper we calculate various transition probability amplitudes, TPAs, known as 'weak values' for the Schrödinger and Pauli particles. It is shown that these values are related to the Bohm momentum, the Bohm energy and the quantum potential in each case. The results for the Schrödinger particle are obtained in three ways, the standard approach, the Clifford algebra approach of Hiley and Callaghan, and the Moyal approach. To obtain the results for the Pauli particle, we combine the Clifford and Moyal algebras into one structure. The consequences of these results are discussed.
Certain Clifford-like algebra and quantum vertex algebras
Li, Haisheng; Tan, Shaobin; Wang, Qing
2015-01-01
In this paper, we study in the context of quantum vertex algebras a certain Clifford-like algebra introduced by Jing and Nie. We establish bases of PBW type and classify its $\\mathbb N$-graded irreducible modules by using a notion of Verma module. On the other hand, we introduce a new algebra, a twin of the original algebra. Using this new algebra we construct a quantum vertex algebra and we associate $\\mathbb N$-graded modules for Jing-Nie's Clifford-like algebra with $\\phi$-coordinated modu...
1999-01-01
Ameerika maakunstniku Robert Smithsoni (1938-1973) retrospektiivnäitus 12. sept.-ni Stockholmi Moodsa Kunsti Muuseumis. Kunstnikule maailmakuulsuse toonud tööst, 1970. a. Utah' osariigi Suurde Soolajärve ehitatud spiraalmuulist "Spiral Jetty".
Genetics Home Reference: Roberts syndrome
... mechanism underlying Roberts syndrome involves loss of ESCO2 acetyltransferase activity. Hum Mol Genet. 2008 Jul 15;17( ... Zou H. Two human orthologues of Eco1/Ctf7 acetyltransferases are both required for proper sister-chromatid cohesion. ...
Robert Lepiksoni maailm / Eha Komissarov
Komissarov, Eha, 1947-
2000-01-01
Näitus "Minu maailm. Robert Lepikson fotograafina ja kollektsionäärina" galeriis "Vaal". Väljas on loodusfotod, fotod autodest, kunstikogust Eduard Steinbergi (1937) ja Vladimir Nemuhhini (1925) maalid, paar Salvador Dali värvilist litograafiat
Clifford Hopf-gebra and Bi-universal Hopf-gebra
Oziewicz, Z
1997-01-01
We consider a pair of independent scalar products, one scalar product on vectors, and another independent scalar product on dual space of co-vectors. The Clifford co-product of multivectors is calculated from the dual Clifford algebra. With respect to this co-product unit is not group-like and vectors are not primitive. The Clifford product and the Clifford co-product fits to the bi-gebra with respect to the family of the (pre)-braids. The Clifford bi-gebra is in a braided category iff at least one of these scalar products vanish.
Clifford Hopf-gebra and Bi-universal Hopf-gebra
Oziewicz, Zbigniew
1997-01-01
We consider a pair of independent scalar products, one scalar product on vectors, and another independent scalar product on dual space of co-vectors. The Clifford co-product of multivectors is calculated from the dual Clifford algebra. With respect to this co-product unit is not group-like and vectors are not primitive. The Clifford product and the Clifford co-product fits to the bi-gebra with respect to the family of the (pre)-braids. The Clifford bi-gebra is in a braided category iff at lea...
Gonzalo Cataño
2003-01-01
Full Text Available El 23 de febrero de 2003 murió en la ciudad de Nueva York el sociólogo norteamericano Robert King Merton. Teniendo presente que no es fácil resumir en unas pocas páginas la obra de quien se dedicó durante 66 años del siglo XX a una intensa actividad teórica y docente en Sociología, el artículo sintetiza algunos de los rasgos mas sobresalientes de su extensa obra.Su carrera académica se desarrolló a partir del legado de las figuras de la tradición sociológica europea y norteamericana que habían afirmado de manera definitiva los conceptos, los métodos y los marcos de referencia del estudio de la sociedad. Siguiendo las huellas de los clásicos, sus intereses teóricos lo llevaron por los más diversos campos del análisis social con una mente abierta, lo cual lo llevó a incursionar en casi todas las especialidades de la Sociología. Los trabajos de Merton sobre la anomia, la estructura burocrática y las relaciones de la ciencia con el orden social dieron lugar al desarrollo de campos específicos del análisis social. En América Latina como en el resto del mundo su influencia en el desarrollo de la Sociología se ha hecho sentir a pesar de haber sido relegada con la irrupción del marxismo a finales de los años sesenta y comienzos de los setenta. Su ejemplo se ha convertido en modelo de rol para generaciones enteras de analistas sociales, en el patrón a seguir por quienes eligen la Sociologías como una ocupación vocacionalmente orientada.
Gonzalo Cataño
2006-01-01
Full Text Available El 23 de febrero de 2003 murió en la ciudad de Nueva York el sociólogo norteamericano Robert King Merton. Teniendo presente que no es fácil resumir en unas pocas páginas la obra de quien se dedicó durante 66 años del siglo XX a una intensa actividad teórica y docente en Sociología, el artículo sintetiza algunos de los rasgos mas sobresalientes de su extensa obra. Su carrera académica se desarrolló a partir del legado de las figuras de la tradición sociológica europea y norteamericana que habían afirmado de manera definitiva los conceptos, los métodos y los marcos de referencia del estudio de la sociedad. Siguiendo las huellas de los clásicos, sus intereses teóricos lo llevaron por los más diversos campos del análisis social con una mente abierta, lo cual lo llevó a incursionar en casi todas las especialidades de la Sociología. Los trabajos de Merton sobre la anomia, la estructura burocrática y las relaciones de la ciencia con el orden social dieron lugar al desarrollo de campos específicos del análisis social. En América Latina como en el resto del mundo su influencia en el desarrollo de la Sociología se ha hecho sentir a pesar de haber sido relegada con la irrupción del marxismo a finales de los años sesenta y comienzos de los setenta. Su ejemplo se ha convertido en modelo de rol para generaciones enteras de analistas sociales, en el patrón a seguir por quienes eligen la Sociologías como una ocupación vocacionalmente orientada.
... Awards Enhancing Diversity Find People About NINDS NINDS Muscular Dystrophy Information Page Clinical Trials Finding the Optimum Regimen ... en Español Additional resources from MedlinePlus What is Muscular Dystrophy? The muscular dystrophies (MD) are a group of ...
Square Roots of -1 in Real Clifford Algebras
Hitzer, Eckhard; Ablamowicz, Rafal
2012-01-01
It is well known that Clifford (geometric) algebra offers a geometric interpretation for square roots of -1 in the form of blades that square to minus 1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Systematic research has been done [1] on the biquaternion roots of -1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra $Cl(3,0)$ of $\\mathbb{R}^3$. Further research on general algebras $Cl(p,q)$ has explicitly derived the geometric roots of -1 for $p+q \\leq 4$ [2]. The current research abandons this dimension limit and uses the Clifford algebra to matrix algebra isomorphisms in order to algebraically characterize the continuous manifolds of square roots of -1 found in the different types of Clifford algebras, depending on the type of associated ring ($\\mathbb{R}$, $\\mathbb{H}$, $\\mathbb{R}^2$, $\\mathbb{H}^2$, or $\\mathbb{C}$). At the end of the paper explicit computer generated tables of representative sq...
'Twisted duality' in the ${\\rm C^*}$ Clifford algebra
Robinson, P. L.
2014-01-01
Let $V$ be a real inner product space and $C[V]$ its ${\\rm C}^*$ Clifford algebra. We prove that if $Z$ is a subspace of $V$ then $C[Z^{\\perp}]$ coincides with the supercommutant of $C[Z]$ in $C[V]$.
Algebraic spinor spaces in the Clifford algebras of Minkowski spaces
Algebraic spinor spaces in the Clifford algebras of two- and four-dimensional Minkowski spaces are considered. Their description in terms of primitive idempotens and their classification with respect to the action of the Lorentz group are given. (author). 6 refs
The $E_8$ geometry from a Clifford perspective
Dechant, Pierre-Philippe
2016-01-01
This paper considers the geometry of $E_8$ from a Clifford point of view in three complementary ways. Firstly, in earlier work, I had shown how to construct the four-dimensional exceptional root systems from the 3D root systems using Clifford techniques, by constructing them in the 4D even subalgebra of the 3D Clifford algebra; for instance the icosahedral root system $H_3$ gives rise to the largest (and therefore exceptional) non-crystallographic root system $H_4$. Arnold's trinities and the McKay correspondence then hint that there might be an indirect connection between the icosahedron and $E_8$. Secondly, in a related construction, I have now made this connection explicit for the first time: in the 8D Clifford algebra of 3D space the $120$ elements of the icosahedral group $H_3$ are doubly covered by $240$ 8-component objects, which endowed with a `reduced inner product' are exactly the $E_8$ root system. It was previously known that $E_8$ splits into $H_4$-invariant subspaces, and we discuss the folding ...
Fractional Dirac operators and deformed field theory on Clifford algebra
Fractional Dirac equations are constructed and fractional Dirac operators on Clifford algebra in four dimensional are introduced within the framework of the fractional calculus of variations recently introduced by the author. Many interesting consequences are revealed and discussed in some details.
Categorification of Clifford algebra via geometric induction and restriction
Gruson, Caroline; Serganova, Vera
2016-01-01
We use geometric parabolic induction functors and the adjoint functors for the supergroups Osp(2m+1,2n) (where m and n vary) to categorify the action of the infinite-dimensional Clifford algebra on the Fock space of semi-infinite forms.
Matrix solutions of wave equations and Clifford algebras
We are extending the formation of matrix solutions un for linear and nonlinear wave equations by construction of unitary anti-Hermitian-anti-commuting matrices up to the eighth order. We use Clifford algebras C(0,n) with periodicity in modulo 8 to construct coupled matrix solutions. We also propose to use the matrix solutions for describing the intrinsic rotations of particles. (author)
On the representation of generalized Dirac (Clifford) algebras
Some results of Brauer and Weyl and of Jordan and Wigner on irreducible representations of generalized Dirac (Clifford) algebras have been proved, adopting a new and simple approach which (i) makes the whole subject straight-forward for physicists and (ii) simplifies the demonstration of the fundamental theorem of Pauli. (Auth.)
Modulo 8 periodicity of real Clifford algebras and particle physics
After a review of the properties of real Clifford algebras, we discuss the isomorphism existing between these algebras and matrix algebras over the real, complex or quaternion field. This is done for all dimensions and all possible signatures of the metric. The modulo 8 periodicity theorem is discussed and extended. A comment is made about the appearance of 'hidden' symmetries in supergravity theories. (orig.)
Gorzawski, Arkadiusz; Papotti, Giulia; Salvachua Ferrando, Belen Maria; Wenninger, Jorg; Pieloni, Tatiana; CERN. Geneva. ATS Department
2015-01-01
Colliding the beams during the squeeze to profit from Landau damping due to head--on beam-beam and b* leveling are two operational modes that may have to be used in a not so distant future at the LHC. This MD aimed at improving the process control during the squeeze with much improved handling of reference for the orbit feedback system and at evaluating instruments and techniques to maintain the beam in collisions with active feedback on a good observable.
75 FR 26993 - Alvin Darby, M.D.; Denial of Application
2010-05-13
... dependency on crack cocaine and `primo,' a mixture of cocaine and marijuana smoked together.'' GX 3, at 2... factors are * * * considered in the disjunctive.'' Robert A. Leslie, M.D., 68 FR 15227, 15230 (2003). I... application under section 303. See The Lawsons, Inc., 72 FR 74334, 74337 (2007); Anthony D. Funches, 64...
78 FR 76322 - Thomas Neuschatz, M.D.; Decision and Order
2013-12-17
... of J.G.'s known addiction, ``but inaccurately represented this as a treatment for a chronic pain... her chronic pain rather than as treatment for her opioid addiction was patently false.'' Id. at 18...(f). ``These factors are . . . considered in the disjunctive.'' Robert A. Leslie, M.D., 68 FR...
Clifford algebras geometric modelling and chain geometries with application in kinematics
Klawitter, Daniel
2015-01-01
After revising known representations of the group of Euclidean displacements Daniel Klawitter gives a comprehensive introduction into Clifford algebras. The Clifford algebra calculus is used to construct new models that allow descriptions of the group of projective transformations and inversions with respect to hyperquadrics. Afterwards, chain geometries over Clifford algebras and their subchain geometries are examined. The author applies this theory and the developed methods to the homogeneous Clifford algebra model corresponding to Euclidean geometry. Moreover, kinematic mappings for special Cayley-Klein geometries are developed. These mappings allow a description of existing kinematic mappings in a unifying framework. Contents Models and representations of classical groups Clifford algebras, chain geometries over Clifford algebras Kinematic mappings for Pin and Spin groups Cayley-Klein geometries Target Groups Researchers and students in the field of mathematics, physics, and mechanical engineering About...
Steven Hawking with Robert Aymar
Maximilien Brice
2006-01-01
Steven Hawking is seen meeting with CERN's Director-General, Robert Aymar. Hawking visited CERN between 24 September and 1 October 2006. During his stay he gave two lectures and toured the LHC, which may provide insights into Hawking's most famous area of study, black holes.
Clifford algebra-valued orthogonal polynomials in the open unit ball of Euclidean space
Fred Brackx; Nele De Schepper; Frank Sommen
2004-01-01
A new method for constructing Clifford algebra-valued orthogonal polynomials in the open unit ball of Euclidean space is presented. In earlier research, we only dealt with scalar-valued weight functions. Now the class of weight functions involved is enlarged to encompass Clifford algebra-valued functions. The method consists in transforming the orthogonality relation on the open unit ball into an orthogonality relation on the real axis by means of the so-called Clifford-Heaviside functions. C...
Riemann surfaces, Clifford algebras and infinite dimensional groups
We introduce of class of Riemann surfaces which possess a fixed point free involution and line bundles over these surfaces with which we can associate an infinite dimensional Clifford algebra. Acting by automorphisms of this algebra is a 'gauge' group of meromorphic functions on the Riemann surface. There is a natural Fock representation of the Clifford algebra and an associated projective representation of this group of meromorphic functions in close analogy with the construction of the basic representation of Kac-Moody algebras via a Fock representation of the Fermion algebra. In the genus one case we find a form of vertex operator construction which allows us to prove a version of the Boson-Fermion correspondence. These results are motivated by the analysis of soliton solutions of the Landau-Lifshitz equation and are rather distinct from recent developments in quantum field theory on Riemann surfaces. (orig.)
m-Qubit states embedded in Clifford algebras CL2m
The quantum theory of a finite quantum system with L degrees of freedom is usually set up by associating it with a Hilbert space H of dimension d(L) and identifying observables and states in the matrix algebra Md(C). For the case d = 2m, m integer, this algebra can be identified with the Clifford algebra CL2m. The case of d = 2m dimensions is simply realized by a system with m dichotomic degrees of freedom, an m-qubit system for instance. The physically relevant new point is the appearance of a new (symmetry-?)group SO(2m). A possible interpretation of the space in which this group operates is proposed. It is shown that the eigenvalues of m-qubit-type states only depend on SO(2m)-invariants. We use this fact to determine state parameter domains (generalized Bloch spheres) for states classified as SO(2m)-tensors. The classification of states and interactions of components of a physical m-qubit system as k-tensors and pseudotensors (0 ≤ k ≤ m) leads to rules similar to those found in elementary quantum mechanics. The question of electromagnetic interactions is shortly broached. We sketch, pars pro toto, a graphical interpretation of tensor contractions appearing in perturbative expansions
$Z_3$-graded analogues of Clifford algebras and generalization of supersymmetry
Abramov, V.
1996-01-01
We define and study the ternary analogues of Clifford algebras. It is proved that the ternary Clifford algebra with $N$ generators is isomorphic to the subalgebra of the elements of grade zero of the ternary Clifford algebra with $N+1$ generators. In the case $N=3$ the ternary commutator of cubic matrices induced by the ternary commutator of the elements of grade zero is derived. We apply the ternary Clifford algebra with one generator to construct the $Z_3$-graded generalization of the simpl...
Lie algebras for the Dirac-Clifford ring
It is shown in a general way that the Dirac-Clifford ring formed by the Dirac matrices and all their products, for all even and odd spacetime dimensions D, span the cumulation algebras SU(2D/2) for even D and SU(2(D-1)/2) + SU(2(D-1)/2) for odd D. Some physical consequences of these results are discussed. (author)
Reflections in Conics, Quadrics and Hyperquadrics via Clifford Algebra
Klawitter, Daniel
2014-01-01
In this article we present a new and not fully employed geometric algebra model. With this model a generalization of the conformal model is achieved. We discuss the geometric objects that can be represented. Furthermore, we show that the Pin group of this geometric algebra corresponds to inversions with respect to axis aligned quadrics. We discuss the construction for the two- and three-dimensional case in detail and give the construction for arbitrary dimension. Key Words: Clifford algebra, ...
A satisfactory formalism for magnetic monopoles by Clifford algebras
The problem of electromagnetism with magnetic monopoles is approached by the physically interesting and mathematically powerful formalism of Clifford algebras, which provides a natural language for Minkowski space-time (Dirac algebra) and euclidean space (Pauli algebra). A lagrangian and hamiltonian formalism is constructed for interacting monopoles, which overcomes many of the long-standing difficulties that are known to plague the approaches developed till now. (orig.)
Unified theories for quarks and leptons based on Clifford algebras
The general standpoint is presented that unified theories arise from gauging of Clifford algebras describing the internal degrees of freedom (charge, color, generation, spin) of the fundamental fermions. The general formalism is presented and the ensuing theories for color and charge (with extension to N colors), and for generations, are discussed. The possibility of further including the spin is discussed, also in connection with generations. (orig.)
Clifford algebra and solution of Bargmann-Michel-Telegdi equation
The Clifford algebra structure of the Minkowski space is presented in the article. The method of solving motion equations within the frames of formalism of this algebra is described. The solution of the spin motion equation (the Bargmann-Michel-Telegdi equations) is obtained by means of the plotted algorithm for the case of the magnetic gyroscope motion in the constant and homogenous electromagnetic field
Hochschild Cohomology and Deformations of Clifford-Weyl Algebras
Ian M. Musson
2009-03-01
Full Text Available We give a complete study of the Clifford-Weyl algebra C(n,2k from Bose-Fermi statistics, including Hochschild cohomology (with coefficients in itself. We show that C(n,2k is rigid when n is even or when k ≠ 1. We find all non-trivial deformations of C(2n+1,2 and study their representations.
Quantum spinors and spin groups from quantum Clifford algebras
A general construction of multiparametric quantum spinors and corresponding quantum Spinμ(2ν-h,h) groups associated to 2ν-(Pseudo)-Euclidean spaces is presented, and their homomorphism to the respective SOμ groups is discussed. This construction is based on a quantum Clifford algebra and is described in detail for involutive (pure twists) intertwining braids. For general braid operators that admit abstract 'volume elements', a procedure is also given for deriving quantum analogues of these groups. (author)
Clifford support vector machines for classification, regression, and recurrence.
Bayro-Corrochano, Eduardo Jose; Arana-Daniel, Nancy
2010-11-01
This paper introduces the Clifford support vector machines (CSVM) as a generalization of the real and complex-valued support vector machines using the Clifford geometric algebra. In this framework, we handle the design of kernels involving the Clifford or geometric product. In this approach, one redefines the optimization variables as multivectors. This allows us to have a multivector as output. Therefore, we can represent multiple classes according to the dimension of the geometric algebra in which we work. We show that one can apply CSVM for classification and regression and also to build a recurrent CSVM. The CSVM is an attractive approach for the multiple input multiple output processing of high-dimensional geometric entities. We carried out comparisons between CSVM and the current approaches to solve multiclass classification and regression. We also study the performance of the recurrent CSVM with experiments involving time series. The authors believe that this paper can be of great use for researchers and practitioners interested in multiclass hypercomplex computing, particularly for applications in complex and quaternion signal and image processing, satellite control, neurocomputation, pattern recognition, computer vision, augmented virtual reality, robotics, and humanoids. PMID:20876017
Robert Zajonc: The Complete Psychologist
Berridge, Kent C.
2010-01-01
This article joins with others in the same issue to celebrate the career of Robert B. Zajonc who was a broad, as well as deeply talented, psychologist. Beyond his well-known focus in social psychology, the work of Zajonc also involved, at one time or another, forays into nearly every other subfield of psychology. This article focuses specifically on his studies that extended into biopsychology, which deserve special highlighting in order to be recognized alongside his many major achievements ...
Two-sided Clifford Fourier transform with two square roots of -1 in Cl(p,q)
Hitzer, Eckhard
2013-01-01
We generalize quaternion and Clifford Fourier transforms to general two-sided Clifford Fourier transforms (CFT), and study their properties (from linearity to convolution). Two general \\textit{multivector square roots} $\\in \\cl{p,q}$ \\textit{of} -1 are used to split multivector signals, and to construct the left and right CFT kernel factors. Keywords: Clifford Fourier transform, Clifford algebra, signal processing, square roots of -1 .
Shirokov, Dmitry
2009-01-01
In this paper we further develop the method of quaternion typification of Clifford algebra elements suggested by the author in the previous paper. On the basis of new classification of Clifford algebra elements it is possible to find out and prove a number of new properties of Clifford algebra. We use k-fold commutators and anticommutators.
The Application of Aristotle's Theory of Tragedy in the Analysis of Clifford
陈美林
2005-01-01
Clifford, a wealthy but paralyzed baronet, is always criticized without any compromise and is regarded hypocritical, cruel, in a word, a half-human and half-machine monster. However, the author believes that Clifford is a tragic character. This paper will analyze this character according to Aristotle's theory on tragedy, which is presented in his masterpiece Poetics.
The Riesz-Clifford Functional Calculus for Non-Commuting Operators and Quantum Field Theory
Kisil, Vladimir V.; de Arellano, Enrique Ramírez
1995-01-01
We present a Riesz-like hyperholomorphic functional calculus for a set of non-commuting operators based on the Clifford analysis. Applications to the quantum field theory are described. Keywords: Functional calculus, Weyl calculus, Riesz calculus, Clifford analysis, quantization, quantum field theory. AMSMSC Primary:47A60, Secondary: 81T10
A GENERALIZED WINDOWED FOURIER TRANSFORM IN REAL CLIFFORD ALGEBRA CL0;N
Bahri, Mawardi
2011-01-01
The Clifford Fourier transform in Cl0;n (CFT) can be regarded as a generalization of the two-dimensional quaternionic Fourier transform (QFT), which is introduced from the mathematical aspect by Brackx at first. In this research paper, we propose the Clifford windowed Fourier transform using the kernel of the CFT. Some important properties of the transform are investigated.
A classification of Lie algebras of pseudounitary groups in the techniques of Clifford algebras
Shirokov, Dmitry
2007-01-01
In this paper we present new formulas, which represent commutators and anticommutators of Clifford algebra elements as sums of elements of different ranks. Using these formulas we consider subalgebras of Lie algebras of pseudounitary groups. Our main techniques are Clifford algebras. We have find 12 types of subalgebras of Lie algebras of pseudounitary groups.
The Hidden Quantum Group of the 8-vertex Free Fermion Model: q-Clifford Algebras
Cuerno, Rodolfo; Gómez, César; López Manzanares, Esperanza; Sierra, Germán
1993-01-01
We prove in this paper that the elliptic $R$--matrix of the eight vertex free fermion model is the intertwiner $R$--matrix of a quantum deformed Clifford--Hopf algebra. This algebra is constructed by affinization of a quantum Hopf deformation of the Clifford algebra.
Between Quantum Virasoro Algebra \\cal{L}_c and Generalized Clifford Algebras
Kinani, E. H. El
2003-01-01
In this paper we construct the quantum Virasoro algebra ${\\mathcal{L}}_{c}$ generators in terms of operators of the generalized Clifford algebras $C_{n}^{k}$. Precisely, we show that ${\\mathcal{L}}_{c}$ can be embedded into generalized Clifford algebras.
Real Clifford Algebra Cln,0, n = 2, 3(mod 4) Wavelet Transform
We show how for n = 2, 3(mod 4) continuous Clifford (geometric) algebra (GA)Cln-valued admissible wavelets can be constructed using the similitude group SIM(n). We strictly aim for real geometric interpretation, and replace the imaginary unit i is an element of C therefore with a GA blade squaring to -1. Consequences due to non-commutativity arise. We express the admissibility condition in terms of a Cln Clifford Fourier Transform and then derive a set of important properties such as dilation, translation and rotation covariance, a reproducing kernel, and show how to invert the Clifford wavelet transform. As an example, we introduce Clifford Gabor wavelets. We further invent a generalized Clifford wavelet uncertainty principle.
THE DEMOKTESIS OF ROBERT NOZICK A DEMOKTESIS DE ROBERT NOZICK
Luiz Felipe Netto de Andrade e Silva Sahd
2008-04-01
Full Text Available the present article aims at reconstructing Robert Nozick’s cen-tral arguments about the extreme positions held by North American li-bertarians who do not distinguish between Welfare state and totalitarian state. Despite divergences on a pivotal question, that of the state, there are some affinities between Nozick and this current of thought. Contrary to the anarchist theory, the Minimal state is preferable to the state of nature as described by John locke.O presente artigo tem como objetivo reconstruir argumentos cen-trais, desenvolvidos por Robert Nozick, acerca das posições extremas que não diferenciam Estado-providência e Estado totalitário na política dos libertarianos norte-americanos, isto é, sobre as afinidades percebidas por Nozick com as teses desta corrente de pensamento, embora se afastando num ponto essencial: a questão do Estado. Ao contrário da teoria anar-quista, o Estado mínimo é preferível ao estado de natureza, tal como John locke o descreve.
Priebe, A; Dehning, B; Redaelli, S; Salvachua Ferrando, BM; Sapinski, M; Solfaroli Camillocci, M; Valuch, D
2013-01-01
The fast beam losses in the order of 1 ms are expected to be a potential major luminosity limitation for higher beam energies after the LHC long shutdown (LS1). Therefore a Quench Test is planned in the winter 2013 to estimate the quench limit in this timescale and revise the current models. This experiment was devoted to determination the LHC Transverse Damper (ADT) as a system for fast losses induction. A non-standard operation of the ADT was used to develop the beam oscillation instead of suppressing them. The sign flip method had allowed us to create the fast losses within several LHC turns at 450 GeV during the previous test (26th March 2012). Thus, the ADT could be potentially used for the studies of the UFO ("Unidentied Falling Object") impact on the cold magnets. Verification of the system capability and investigations of the disturbed beam properties were the main objectives of this MD. During the experiment, the pilot bunches of proton beam were excited independently in the horizontal and vertical ...
Connections on Clifford bundles and the Dirac operator
It is shown, how - in the setting of Clifford bundles - the spin connection (or Dirac operator) may be obtained by averaging the Levi-Civita connection (or Kaehler-Dirac operator) over the finite group generated by an orthonormal frame of the base-manifold. The familiar covariance of the Dirac equation under a simultaneous transformation of spinors and matrix-representations emerges very naturally in this scheme, which can also be applied when the manifold does not possess a spin-structure. (Author)
Local unitary versus local Clifford equivalence of stabilizer states
We study the relation between local unitary (LU) equivalence and local Clifford (LC) equivalence of stabilizer states. We introduce a large subclass of stabilizer states, such that every two LU equivalent states in this class are necessarily LC equivalent. Together with earlier results, this shows that LC, LU, and stochastic local operation with classical communication equivalence are the same notions for this class of stabilizer states. Moreover, recognizing whether two given stabilizer states in the present subclass are locally equivalent only requires a polynomial number of operations in the number of qubits
Galilean-covariant Clifford algebras in the phase space representation
We apply the Galilean covariant formulation of quantum dynamics to derive the phase-space representation of the Pauli-Schroedinger equation for the density matrix of spin-1/2 particles in the presence of an electromagnetic field. The Liouville operator for the particle with spin follows from using the Wigner-Moyal transformation and a suitable Clifford algebra constructed on the phase space of a (4+1)-dimensional spacetime with Galilean geometry. Connections with the algebraic formalism of thermofield dynamics are also investigated. (author)
Relativistic Electrodynamics without Reference Frames. Clifford Algebra Formulation
Ivezic, Tomislav
2002-01-01
In the usual Clifford algebra formulation of electrodynamics the Faraday bivector field $F$ is expressed in terms of \\QTR{em}{the observer dependent} relative vectors $\\QTR{bf}{E}$ and $\\QTR{bf}{B.}$ In this paper we present \\QTR{em}{the observer independent}decomposition of $F$ by using the vectors (grade-1) of electric $E$ and magnetic $B$ fields and we develop the formulation of relativistic electrodynamics which is independent of the reference frame and of the chosen coordinatization. We ...
张裔智; 赵毅; 汤小斌
2008-01-01
随着网络技术的迅速发展,信息加密技术已成为保障网络安全的一种重要手段,加密算法已经成为人们的一个研究热点.本文对MD5算法进行了深入研究,介绍MD5算法的产生背景、应用及其算法流程,并提出了MD5算法的一个改进方案.
Bilinear covariants and spinor fields duality in quantum Clifford algebras
Abłamowicz, Rafał; Gonçalves, Icaro; da Rocha, Roldão
2014-10-01
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.
Topological classification with additional symmetries from Clifford algebras
Morimoto, Takahiro; Furusaki, Akira
2013-09-01
We classify topological insulators and superconductors in the presence of additional symmetries such as reflection or mirror symmetries. For each member of the 10 Altland-Zirnbauer symmetry classes, we have a Clifford algebra defined by operators of the generic (time-reversal, particle-hole, or chiral) symmetries and additional symmetries, together with gamma matrices in Dirac Hamiltonians representing topological insulators and superconductors. Following Kitaev's approach, we classify gapped phases of noninteracting fermions under additional symmetries by examining all possible distinct Dirac mass terms which can be added to the set of generators of the Clifford algebra. We find that imposing additional symmetries in effect changes symmetry classes and causes shifts in the periodic table of topological insulators and superconductors. Our results are in agreement with the classification under reflection symmetry recently reported by Chiu, Yao, and Ryu [Phys. Rev. B1098-012110.1103/PhysRevB.88.075142 88, 075142 (2013)]. Several examples are discussed including a topological crystalline insulator with mirror Chern numbers and mirror superconductors.
Bilinear covariants and spinor fields duality in quantum Clifford algebras
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
Bilinear covariants and spinor fields duality in quantum Clifford algebras
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.
Clifford Space as a Generalization of Spacetime: Prospects for Unification in Physics
Pavsic, Matej
2004-01-01
The geometric calculus based on Clifford algebra is a very useful tool for geometry and physics. It describes a geometric structure which is much richer than the ordinary geometry of spacetime. A Clifford manifold ($C$-space) consists not only of points, but also of 1-loops, 2-loops, etc.. They are associated with multivectors which are the wedge product of the basis vectors, the generators of Clifford algebra. We assume that $C$-space is the true space in which physics takes place and that p...
Structure of inverse elements of Clifford algebra of dimension at most two to the fifth power
Suzuki, Yuka; Yamaguchi, Naoya
2016-01-01
A Clifford algebra of two to the fifth power dimension is applicable to computer graphics. In particular, one needs to compute inverse elements of the Clifford algebra. The inverse formulas for elements of Clifford algebras of dimensions less than or equal to two the fifth power are known by direct calculation. The formulas have some structure. However, we did not know the reason for the formulas have the structure. In this paper, we give the reason by some anti-isomorphisms on subspaces of C...
2013-03-06
... Praxedes E. Alverez Santiago, M.D., Daniel Perez Brisebois, M.D., Jorge Grillasca Palou, M.D., Rafael Garcia Nieves, M.D., Francis M. Vazques Roura, M.D., Angel B. Rivera Santos, M.D., Cosme D. Santos Torres, M.D., and Juan L. Vilaro Chardon, M.D.; Analysis of Agreement Containing Consent Order To Aid...
The many faces of Maxwell, Dirac and Einstein equations a Clifford bundle approach
Rodrigues, Jr, Waldyr A
2016-01-01
This book is an exposition of the algebra and calculus of differential forms, of the Clifford and Spin-Clifford bundle formalisms, and of vistas to a formulation of important concepts of differential geometry indispensable for an in-depth understanding of space-time physics. The formalism discloses the hidden geometrical nature of spinor fields. Maxwell, Dirac and Einstein fields are shown to have representatives by objects of the same mathematical nature, namely sections of an appropriate Clifford bundle. This approach reveals unity in diversity and suggests relationships that are hidden in the standard formalisms and opens new paths for research. This thoroughly revised second edition also adds three new chapters: on the Clifford bundle approach to the Riemannian or semi-Riemannian differential geometry of branes; on Komar currents in the context of the General Relativity theory; and an analysis of the similarities and main differences between Dirac, Majorana and ELKO spinor fields. The exercises with solut...
Minimal surfaces in the three-Sphere by doubling the Clifford Torus
Kapouleas, Nicolaos; Yang, Seong-Deog
2007-01-01
We construct embedded closed minimal surfaces in the round three-sphere, resembling two parallel copies of the Clifford torus, joined by m^2 small catenoidal bridges symmetrically arranged along a square lattice of points on the torus.
ON CAUCHY-POMPEIU FORMULA FOR FUNCTIONS WITH VALUES IN A UNIVERSAL CLIFFORD ALGEBRA
无
2003-01-01
This paper obtains the Cauchy-Pompeiu formula on certain distinguishedboundary for functions with values in a universal Clifford algebra. This formula is just anextension of the Cauchy's integral formula obtained in [11].
A list of identities made with products between two different generators of the Clifford algebra
Formiga, J. B.
2012-01-01
Here I present a full list with all possibles products between the generators of the Clifford algebra in a four-dimensional spacetime. The resulting expressions turned out to be very simple and easy to deal with.
A new description of space and time using Clifford multivectors
Chappell, James M; Iqbal, Azhar; Abbott, Derek
2012-01-01
Following the development of the special theory of relativity in 1905, Minkowski sought to provide a physical basis for Einstein's two fundamental postulates of special relativity, proposing a four dimensional spacetime structure consisting of three space and one time dimension, with the relativistic effects then being straightforward consequences of this spacetime geometry. As an alternative to Minkowski's approach, we produce the results of special relativity directly from three space ($ \\Re_3 $) without the addition of an extra dimension, through identifying the local time with the three rotational degrees of freedom of this space. The natural mathematical formalism within which to describe this definition of spacetime is found to be Clifford's geometric algebra, and specifically a three-dimensional multivector. With time now identified with the three rotational degrees of freedom of space, time becomes three-dimensional, which provides a natural symmetry between space and time in the form of a complex-typ...
Quantum ring in the eyes of geometric (Clifford) algebra
Dargys, A.
2013-01-01
The quantum ring with spin-orbit interaction included is analyzed in a nonstandard way using Clifford or geometric algebra (GA). The solution of the Schrödinger-Pauli equation is presented in terms of rotors having clear classical mechanics interpretation, i.e., in GA the rotors act in 3D Euclidean space rather than as operators in an abstract Hilbert space. This classical-like property of spin control in GA provides a more transparent approach in designing and understanding spintronic devices. The aim of the paper is to attract readers attention to new possibilities in spin physics and to demonstrate how the quantum ring problem can be solved by GA methods.
Bilinear Covariants and Spinor Fields Duality in Quantum Clifford Algebras
Ablamowicz, Rafal; da Rocha, Roldao
2014-01-01
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 is thus discussed. Hence, by endowing the underlying spacetime with an arbitrary bilinear form with a antisymmetric part in addition to a symmetric spacetime metric, quantum algebraic spinor fields and deformed bilinear covariants can be constructed. They are therefore compared to the classical (non quantum) ones. Classes of quantum spinor fields 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. Here, it is shown further to play a prominent role in the structure of D...
Elementary particle states based on the Clifford algebra C7
The lepton isodoublet (e-,νsub(e)), the ''bare'' nucleon is isodoublet (n,p), and their antiparticles are shown to constitute a basis of the irreducible representation of the Clifford algebra C7. The excited states of these doublets, i.e., (μ-,νsub(μ)), (tau-,νsub(tau)),..., and (s0,c+),(b0,t+) are generated by the products (e-,νsub(e))sup(x)a and (n,p)sup(x)a, where a is identical to 2sup(-1/2)(e-e+ + νsub(e)ν-barsub(e)) has the same quantum numbers as the photon state. The bare baryons s,c,b,t carry the strangeness, charm, bottom, and top quantum numbers. These lepton and bare baryon states are in one-to-one correspondence with the integrally charged colored Han-Nambu quarks, and generate all the observed su(3) and su(4) hadron multiplets. (author)
Magnetic monopoles without strings by Kaeler-Clifford algebra
In substitution for Dirac monopoles with strings, we have recently introduced monopoles without string on the basis of a generalized potential, the sum of vector A and a psudovector *g*L5 B potential. By making recourse to the (graded) Clifford algebra, which just allows adding together tensors of different rank (e.g., scalars + psudoscalars + vectors + pseudovectors + ...), in a previous paper we succeeded in constructing a lagrangian and hamiltonian formalism for interacting monopoles that can be regarded as satisfactory from various points of view. In the present note, after having completed that formalism, we put forth a purely geometrical interpretation of it within the Kaeler algebra on differential forms, essential ingredients being the natural introduction of a generalized curvature and the Hodge decomposition. We thus pave the way for the extension of our monopoles without string to non-abelian gauge groups. The analogies of this approach with supersymmetric theories are apparent
How to efficiently select an arbitrary Clifford group element
Koenig, Robert [Institute for Quantum Computing and Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1 (Canada); Smolin, John A. [IBM T.J. Watson Research Center, Yorktown Heights, New York 10598 (United States)
2014-12-15
We give an algorithm which produces a unique element of the Clifford group on n qubits (C{sub n}) from an integer 0≤i<|C{sub n}| (the number of elements in the group). The algorithm involves O(n{sup 3}) operations and provides, in addition to a canonical mapping from the integers to group elements g, a factorization of g into a sequence of at most 4n symplectic transvections. The algorithm can be used to efficiently select random elements of C{sub n} which are often useful in quantum information theory and quantum computation. We also give an algorithm for the inverse map, indexing a group element in time O(n{sup 3})
Magnetic monopoles without strings by Kaehler-Clifford algebra
In place of Dirac monopoles with string, this paper presents monopoles without string on the basis of a generalized potential, the sum of a vector A and a pseudovector γB potential. By having recourse to the (graded) Clifford algebra which allows adding together tensors of different ranks (e.g., scalars + pseudoscalars + vectors + pseudovectors + . . .), in a previous paper we succeeded in constructing a Lagrangian and Hamiltonian formalism for interacting monopoles that can be regarded as satisfactory from various points of view. In the present note, after having completed that formalism, the authors put forth a purely geometrical interpretation of it within the Kahler algebra on differential forms, essential ingredients being the natural introduction of a generalized curvature and the Hodge decomposition. The authors thus pave the way for the extension of monopoles without string to non-abelian gauge groups. The analogies of this approach with supersymmetric theories are apparent
Z2-gradings of Clifford algebras and multivector structures
Let Cl(V, g) be the real Clifford algebra associated with the real vector space V, endowed with a nondegenerate metric g. In this paper, we study the class of Z2-gradings of Cl(V, g) which are somehow compatible with the multivector structure of the Grassmann algebra over V. A complete characterization for such Z2-gradings is obtained by classifying all the even subalgebras coming from them. An expression relating such subalgebras to the usual even part of Cl(V, g) is also obtained. Finally, we employ this framework to define spinor spaces, and to parametrize all the possible signature changes on Cl(V, g) by Z2-gradings of this algebra
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.
Clifford algebra-based spatio-temporal modelling and analysis for complex geo-simulation data
Luo, Wen; Yu, Zhaoyuan; Hu, Yong; Yuan, Linwang
2013-10-01
The spatio-temporal data simulating Ice-Land-Ocean interaction of Antarctic are used to demonstrate the Clifford algebra-based data model construction, spatio-temporal query and data analysis. The results suggest that Clifford algebra provides a powerful mathematical tool for the whole modelling and analysis chains for complex geo-simulation data. It can also help implement spatio-temporal analysis algorithms more clearly and simply.
Pauli theorem in the description of n-dimensional spinors in the Clifford algebra formalism
Shirokov, D. S.
2013-04-01
We discuss a generalized Pauli theorem and its possible applications for describing n-dimensional (Dirac, Weyl, Majorana, and Majorana-Weyl) spinors in the Clifford algebra formalism. We give the explicit form of elements that realize generalizations of Dirac, charge, and Majorana conjugations in the case of arbitrary space dimensions and signatures, using the notion of the Clifford algebra additional signature to describe conjugations. We show that the additional signature can take only certain values despite its dependence on the matrix representation
We construct the Clifford-space tensorial-gauge fields generalizations of Yang-Mills theories and the Standard Model that allows to predict the existence of new particles (bosons, fermions) and tensor-gauge fields of higher-spins in the 10 Tev regime. We proceed with a detailed discussion of the unique D 4 - D 5 - E 6 - E 7 - E 8 model of Smith based on the underlying Clifford algebraic structures in D = 8, and which furnishes all the properties of the Standard Model and Gravity in four-dimensions, at low energies. A generalization and extension of Smith's model to the full Clifford-space is presented when we write explicitly all the terms of the extended Clifford-space Lagrangian. We conclude by explaining the relevance of multiple-foldings of D = 8 dimensions related to the modulo 8 periodicity of the real Cliford algebras and display the interplay among Clifford, Division, Jordan, and Exceptional algebras, within the context of D = 26, 27, 28 dimensions, corresponding to bosonic string, M and F theory, respectively, advanced earlier by Smith. To finalize we describe explicitly how the E 8 x E 8 Yang-Mills theory can be obtained from a Gauge Theory based on the Clifford (16) group
Undergraduate Research at Oral Roberts University.
Couch, Richard; Thurman, Duane
1981-01-01
Explains Oral Roberts University's undergraduate requirement for research proficiency and how this requirement is fulfilled by biology majors. Topics of the required courses include: introduction to biological research; research techniques; independent research and senior paper; and senior seminar. (DS)
Robert Walters named vice president for research
Trulove, Susan
2007-01-01
Robert Walters, interim associate vice president of research at Virginia Tech, has been named vice president for research effective immediately for a three-year period, announced Mark McNamee, university provost and vice president for academic affairs.
Clifford Algebra Cℓ 3(ℂ) for Applications to Field Theories
Panicaud, B.
2011-10-01
The multivectorial algebras present yet both an academic and a technological interest. Difficulties can occur for their use. Indeed, in all applications care is taken to distinguish between polar and axial vectors and between scalars and pseudo scalars. Then a total of eight elements are often considered even if they are not given the correct name of multivectors. Eventually because of their simplicity, only the vectorial algebra or the quaternions algebra are explicitly used for physical applications. Nevertheless, it should be more convenient to use directly more complex algebras in order to have a wider range of application. The aim of this paper is to inquire into one particular Clifford algebra which could solve this problem. The present study is both didactic concerning its construction and pragmatic because of the introduced applications. The construction method is not an original one. But this latter allows to build up the associated real algebra as well as a peculiar formalism that enables a formal analogy with the classical vectorial algebra. Finally several fields of the theoretical physics will be described thanks to this algebra, as well as a more applied case in general relativity emphasizing simultaneously its relative validity in this particular domain and the easiness of modeling some physical problems.
On n-ary algebras, branes and poly-vector gauge theories in noncommutative Clifford spaces
In this paper, poly-vector-valued gauge field theories in noncommutative Clifford spaces are presented. They are based on noncommutative (but associative) star products that require the use of the Baker-Campbell-Hausdorff formula. Using these star products allows the construction of actions for noncommutative p-branes (branes moving in noncommutative spaces). Noncommutative Clifford-space gravity as a poly-vector-valued gauge theory of twisted diffeomorphisms in Clifford spaces would require quantum Hopf algebraic deformations of Clifford algebras. We proceed with the study of n-ary algebras and find an important relationship among the n-ary commutators of the noncommuting spacetime coordinates [X1, X2, ..., Xn] with the poly-vector-valued coordinates X123...n in noncommutative Clifford spaces given by [X1, X2, ..., Xn] = n!X123...n. The large N limit of n-ary commutators of n hyper-matrices Xi1i2...in leads to Eguchi-Schild p-brane actions for p + 1 = n. A noncomutative n-ary . product of n functions is constructed which is a generalization of the binary star product * of two functions and is associated with the deformation quantization of n-ary structures and deformations of the Nambu-Poisson brackets.
On n-ary algebras, branes and poly-vector gauge theories in noncommutative Clifford spaces
Castro, Carlos, E-mail: perelmanc@hotmail.co [Center for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta, GA 30314 (United States)
2010-09-10
In this paper, poly-vector-valued gauge field theories in noncommutative Clifford spaces are presented. They are based on noncommutative (but associative) star products that require the use of the Baker-Campbell-Hausdorff formula. Using these star products allows the construction of actions for noncommutative p-branes (branes moving in noncommutative spaces). Noncommutative Clifford-space gravity as a poly-vector-valued gauge theory of twisted diffeomorphisms in Clifford spaces would require quantum Hopf algebraic deformations of Clifford algebras. We proceed with the study of n-ary algebras and find an important relationship among the n-ary commutators of the noncommuting spacetime coordinates [X{sup 1}, X{sup 2}, ..., X{sup n}] with the poly-vector-valued coordinates X{sup 123...n} in noncommutative Clifford spaces given by [X{sup 1}, X{sup 2}, ..., X{sup n}] = n!X{sup 123...n}. The large N limit of n-ary commutators of n hyper-matrices X{sub i{sub 1i{sub 2...i{sub n}}}} leads to Eguchi-Schild p-brane actions for p + 1 = n. A noncomutative n-ary . product of n functions is constructed which is a generalization of the binary star product * of two functions and is associated with the deformation quantization of n-ary structures and deformations of the Nambu-Poisson brackets.
MD 751: Train Instability Threshold
Carver, Lee Robert; Metral, Elias; Salvant, Benoit; Levens, Tom; Nisbet, David; Zobov, M; CERN. Geneva. ATS Department
2016-01-01
The purpose of this MD is to measure the octupole current thresholds for stability for a single bunch, and then make an immediate comparison (with the same operational settings) for a train of 72 bunches separated by 25ns. From theory, the expected thresholds should be similar. Any discrepancy between the two cases will be of great interest as it could indicate the presence of additional mechanisms that contribute to the instability threshold, for example electron cloud.
On supergroups with odd Clifford parameters and non-anticommutative supersymmetry
We investigate super groups with Grassmann parameters replaced by odd Clifford parameters. The connection with non-anti commutative supersymmetry is discussed. A Berezin-like calculus for odd Clifford variables is introduced. Fermionic covariant derivatives for super groups with odd Clifford variables are derived. Applications to supersymmetric quantum mechanics are made. Deformations of the original supersymmetric theories are encountered when the fermionic covariant derivatives do not obey the graded Leibniz property. The simplest non-trivial example is given by the N = 2 SQM with a real (1, 2, 1) multiplet and a cubic potential. The action is real. Depending on the overall sign ('Euclidean' or 'Lorentzian') of the deformation, a Bender-Boettcher pseudo-hermitian Hamiltonian is encountered when solving the equation of motion of the auxiliary field. A possible connection of our framework with the Drinfeld twist deformation of supersymmetry is pointed out. (author)
A symplectic subgroup of a pseudounitary group as a subset of Clifford algebra
Marchuk, Nikolai; Dyabirov, Roman
2008-01-01
Let Cl1(1,3) and Cl2(1,3) be the subsets of elements of the Clifford algebra Cl(1,3) of ranks 1 and 2 respectively. Recently it was proved that the subset Cl2(p,q)+iCl1(p,q) of the complex Clifford algebra can be considered as a Lie algebra. In this paper we prove that for p=1, q=3 the Lie algebra Cl2(p,q)+iCl1(p,q) is isomorphic to the well known matrix Lie algebra sp(4,R) of the symplectic Lie group Sp(4,R). Also we define the so called symplectic group of Clifford algebra and prove that th...
In order to realize supersymmetric quantum mechanics methods on a four-dimensional classical phase space, the complexified Clifford algebra of this space is extended by deforming it with the Moyal star product in composing the components of Clifford forms. Two isospectral matrix Hamiltonians having a common bosonic part but different fermionic parts depending on four real-valued phase-space functions are obtained. The Hamiltonians are doubly intertwined via matrix-valued functions which are divisors of zero in the resulting Moyal-Clifford algebra. Two illustrative examples corresponding to Jaynes-Cummings-type models of quantum optics are presented as special cases of the method. Their spectra, eigenspinors and Wigner functions as well as their constants of motion are also obtained within the autonomous framework of deformation quantization.
Real Clifford Algebra Cl(n,0), n=2,3(mod 4) Wavelet Transform
Hitzer, Eckhard
2013-01-01
We show how for $n=2,3 (\\mod 4)$ continuous Clifford (geometric) algebra (GA) $Cl_n$-valued admissible wavelets can be constructed using the similitude group $SIM(n)$. We strictly aim for real geometric interpretation, and replace the imaginary unit $i \\in \\C$ therefore with a GA blade squaring to $-1$. Consequences due to non-commutativity arise. We express the admissibility condition in terms of a $Cl_{n}$ Clifford Fourier Transform and then derive a set of important properties such as dila...
Clifford algebra and the structure of point groups in higher-dimensional spaces
With the basic Clifford units being identified as mirrors, it is demonstrated how proper and improper symmetry operations of point groups in spaces of arbitrary dimensions can be parametrized. In such an approach consistency with parametrizations for groups in three dimensions can be achieved even if double groups are considered. The conversion of Clifford parameters into Cartesian matrices and vice versa is discussed and, for rotations in R4, also the parametrization in terms of pairs of rotations in R3. The formalism is illustrated by a number of examples
The Inharmonious Conflicts in Robert Frost's Poetry
杨苗
2015-01-01
Robert Frost is one of the most distinguished poets,most of his poems are about the inharmonious relationship between nature and men,Conflicts are like a“thread”appearing in his poems.Frost’s true philosophy on men and life contributes to his wisdom and artistic poems.Frost tries to illustrate the conflict between nature and men in philosophy concern.
Robert Aymar, Director-General of CERN
Patrice Loïez
2003-01-01
Robert Aymar, photographed in 2003 before taking his position as Director-General at CERN, succeeding Luciano Maiani in 2004. At this time, Aymar was director of the International Thermonuclear Experimental Reactor (ITER) although he had already been involved with developments at CERN, chairing the External Review Committee, set up in 2001 in response to the increased cost of the LHC.
Robert Lee Pyle honored with emeritus status
Douglas, Jeffrey S.
2007-01-01
Dr. Robert Lee Pyle of Blacksburg, professor of cardiology in the Department of Small Animal Clinical Science at the Virginia-Maryland Regional College of Veterinary Medicine at Virginia Tech, was conferred the "professor emeritus" title by the Virginia Tech Board of Visitors during the board's quarterly meeting August 27.
Hansen, Pelle Guldborg
2007-01-01
In this interview Nobel Prize Winner Robert Aumann talks about how he was initially drawn into game theory, when he came to think of formalizing the folk-theorem, the proper role of game theory in relation to other disciplines and why behavioral game theory probably won't last long....
Beyond War Stories: Clifford G. Christians' Influence on the Teaching of Media Ethics, 1976-1984.
Peck, Lee Anne
Clifford Glenn Christians' work in the area of media ethics education from 1976 through 1984 has influenced the way media ethics is taught to many college students today. This time period includes, among his other accomplishments, Christians' work on an extensive survey of how media ethics was taught in the late 1970s, his work on the Hastings…
A Clifford Algebra approach to the Discretizable Molecular Distance Geometry Problem
Andrioni, Alessandro
2013-01-01
The Discretizable Molecular Distance Geometry Problem (DMDGP) consists in a subclass of the Molecular Distance Geometry Problem for which an embedding in ${\\mathbb{R}^3}$ can be found using a Branch & Prune (BP) algorithm in a discrete search space. We propose a Clifford Algebra model of the DMDGP with an accompanying version of the BP algorithm.
Advancing Scholarship and Intellectual Productivity: An Interview with Clifford A. Lynch
Hawkins, Brian L.
2006-01-01
In this second part of a two-part interview with Clifford A. Lynch, Executive Director of the Coalition for Networked Information, Lynch talks to Hawkins about the most provocative and exciting projects that are being developed in the field of networked information worldwide. He also talks on how institutional repositories are being currently…
The Actions Of Subgroups Of $SL$ For The Clifford Algebra In EPH Cases
Biswas, Debapriya
2011-01-01
We study the action of each subgroup $A$, $N$ and $K$ of the group $SL_2(\\mathbb{R})$ for the Clifford algebra $\\mathcal{C}\\ell(a)$ and calculate their vector fields, using the derived representation of the Lie algebra $sl_{2}$.
Analytic Properties of the Conformal Dirac Operator on the Sphere in Clifford Analysis
Pansano, Brett
2015-01-01
In this paper the conformal Dirac operator on the sphere is defined to be operating on the space of square-integrable Clifford algebra-valued functions. The spinorial Laplacian of order d>0 is defined and used to establish Sobolev embedding theorems.
Fierz identities for real Clifford algebras and the number of supercharges
One considers supersymmetric gauge theories in quantum mechanics with the bosons and fermions belonging to the adjoint representation of the gauge group. One shows that the supersymmetry constraints are related to the existence of certain Fierz identities for real Clifford algebras. These identities are valid when one has 2, 4, 8, and 16 supercharges
The peak algebra and the Hecke-Clifford algebras at $q=0$
Bergeron, Nantel; Hivert, Florent; Thibon, Jean-Yves
2003-01-01
Using the formalism of noncommutative symmetric functions, we derive the basic theory of the peak algebra of symmetric groups and of its graded Hopf dual. Our main result is to provide a representation theoretical interpretation of the peak algebra and its graded dual as Grothendieck rings of the tower of Hecke-Clifford algebras at $q=0$.
We introduce q-analogues of Clifford and Weyl algebras. Using these, we construct spinor and oscillator representations of quantum enveloping algebras of type AN-1, BN, CN, DN and AN-1(1). Also we discuss the irreducibility and the unitarity of these representations. (orig.)
Causal phase-space approach to fermion theories understood through Clifford algebras
A Wigner-Moyal phase-space approach is developed for the Dirac and Feynman-Gell-Mann equations. The role of spinors as primitive elements of the spacetime and phase-space Clifford algebras is emphasized. A conserved phase-space current is constructed. (orig.)
CONSTITUTIONAL TRADITIONALISM IN THE ROBERTS COURT
Louis J Virelli III
2011-05-01
Full Text Available The debate over the role of traditionalism in constitutional interpretation has itself become a tradition. It remains a popular and controversial topic among constitutional scholars and presents normative questions that are as divisive, difficult, and important today as at the Founding. Missing from the discussion, however, is a comprehensive account of how the Supreme Court has employed traditionalism-an approach that looks for meaning in present manifestations of longstanding practices or beliefs-in its constitutional jurisprudence. This project is the first to fill this gap by providing an exhaustive and systematic analysis of the Court's use of constitutional traditionalism. This article focuses on the Roberts Court's first five terms to provide an empirical foundation that will not only offer previously unavailable insights into the Court's current traditionalist practices, but will also set forth a useful framework for the ongoing normative debate over traditionalism. This project uses content analysis of key terms to identify every instance in which the Roberts Court employed traditionalism to interpret the Constitution. More specifically, this project set out to answer the following three questions: First, how frequently does the Roberts Court employ traditionalism in its constitutional jurisprudence? Second, how robust is the Court's use of traditionalism (i.e., is it used to interpret a broad or narrow range of constitutional provisions? And finally, how often and in what contexts do individual Justices on the Roberts Court rely on traditionalism in their own constitutional opinions? The research provided here suggests answers to all three of these questions. First, the data indicate that traditionalism has been relied upon regularly by the Roberts Court, appearing in nearly half of the Court's constitutional cases. Second, traditionalism is frequently applied to a wide variety of constitutional provisions: Two-thirds of the
Cavity Voltage Phase Modulation MD
Mastoridis, T; Butterworth, A; Molendijk, J; Tuckmantel, J
2012-01-01
The LHC RF/LLRF system is currently setup for extremely stable RF voltage to minimize transient beam loading eects. The present scheme cannot be extended beyond nominal beam current since the demanded power would push the klystrons to saturation. For beam currents above nominal (and possibly earlier), the cavity phase modulation by the beam will be not be corrected (transient beam loading), but the strong RF feedback and One-Turn Delay feedback will still be active for loop and beam stability in physics. To achieve this, the voltage set point will be adapted for each bunch. The goal of this MD was to test an iterative algorithm that would adjust the voltage set point to achieve the optimal phase modulation for klystron forward power considerations.
Robert Bellah, religion og menneskelig evolution
Jensen, Hans Jørgen Lundager
2013-01-01
in the middle of 1st mill. BC, where new radical and intellectual ideas and practices, sceptial or world renouncing, appeared in China, India and Greece. Hopefully, Bellah's book will be a standard reference work in the academic study of religion and an inspiration for the history of religion in the future......ENGLISH ABSTRACT: Introduction to and discussion of Robert Bellah's major book, Religion in Human Evolution (2011). which defines and describes tribal religion (religion in pre-state societies), archaic religion (religion in early states) and religious currents in the axial age, the period...... to engage in historical and comparative studies. DANSK RESUMÉ: Introduktion til og diskussion af Robert Bellahs hovedværk fra 2011, Religion in Human Evolution, der definerer og beskriver tribal religion, dvs. religion i før-statslige samfund, arkaisk religion, dvs. religion i tidlig-statslige kulturer samt...