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...
'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 $C(W^{\\perp})$.
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...
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 ...
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)
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...
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
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...
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...
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
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
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.
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...
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.
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...
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 $\\omega$.
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...
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 $C(4,3)$.
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.
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 ...
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
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.
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...
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)
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 ...
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)].
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.
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.
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
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
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...
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.
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...
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.
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.
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.
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 ...
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.
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...
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
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
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 $B$.
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.
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)
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.
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.
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...
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.
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
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-...
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)
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.
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.
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
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.
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.)
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...
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.
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].
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?
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
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
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 ...
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.
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...
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
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.
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...
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)
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.
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.
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...
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.
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.)
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.
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.)
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...
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
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)
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
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.)
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)
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.
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...
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)
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
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...
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.)
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)
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.
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.
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.
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.)
'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 $C[V]$.
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.)
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
Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions
Gallier, Jean
2008-01-01
One of the main goals of these notes is to explain how rotations in reals^n are induced by the action of a certain group, Spin(n), on reals^n, in a way that generalizes the action of the unit complex numbers, U(1), on reals^2, and the action of the unit quaternions, SU(2), on reals^3 (i.e., the action is defined in terms of multiplication in a larger algebra containing both the group Spin(n) and reals^n). The group Spin(n), called a spinor group, is defined as a certain subgroup of units of a...
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 $q=0$.
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...
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)
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.)
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)
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.)
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.
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)
$Z_3$-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...
Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type
Ta Khongsap
2009-01-01
Full Text Available We introduce an odd double affine Hecke algebra (DaHa generated by a classical Weyl group $W$ and two skew-polynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by $W$ and two polynomial-Clifford subalgebras. There is yet a third algebra containing a spin Weyl group algebra which is Morita (superequivalent to the above two algebras. We establish the PBW properties and construct Verma-type representations via Dunkl operators for these algebras.
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.
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 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.
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...
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
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...
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.
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 ...
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
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
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.
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...
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.
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)
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.
A trial to find an elliptic quantum algebra for $sl_2$ using the Heisenberg and Clifford algebra
Shiraishi, Jun'ichi
1994-01-01
A Heisenberg-Clifford realization of a deformed $U(sl_{2})$ by two parameters $p$ and $q$ is discussed. The commutation relations for this deformed algebra have interesting connection with the theta functions.
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.
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
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...
Frobenius character formula and spin generic degrees for Hecke-Clifford algebra
Wan, Jinkui; Wang, Weiqiang
2012-01-01
The spin analogues of several classical concepts and results for Hecke algebras are established. A Frobenius type formula is obtained for irreducible characters of the Hecke-Clifford algebra. A precise characterization of the trace functions allows us to define the character table for the algebra. The algebra is endowed with a canonical symmetrizing trace form, with respect to which the spin generic degrees are formulated and shown to coincide with the spin fake degrees. We further provide a ...
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 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 $sl_{2}$.
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.
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.
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.
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
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.)
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 Algebras in relativistic quantum mechanics and in the gauge theory of electromagnetism
A Clifford Algebra is an algebra associated with a finite-dimensional vector space and a symmetric form on that space. It contains a multiplicative subgroup, the group of spinors, which is related to the group of orthogonal transformations of the vector space. This group may act on the algebra via multiplication on the left or right, or by the adjoint action. First, the author considers the problem of classifying the orbits of these actions in the algebras C(3,1) and C(3,2). For a ceratin subclass of orbit this problem is completely solved and the isotropy groups for elements in these orbits are determined. After writing the Dirac and Maxwell equations in terms of Clifford Albebras, the author shows how a classification of the solutions to these equations is related to the orbit and isotropy group calculations. Finally, he shows how Clifford algebras may be used to define spinor and r-vector fields on manifolds, gradients of such fields, and other more familiar concepts from differential geometry. The end result is that the calculations for C(3,1) and C(3,2) may be applied to fields on space-time and on the five-dimensional space of the gauge theory of electromagnetism, respectively. This gauge theory also allows us to relate Einstein's equations for free space to Maxwell's equations in a natural manner
Generalization of the twistor to Clifford algebras as a basis for geometry
The Penrose twistor theory to a Clifford algebra is generated. This allows basic geometric forms and relationships to be expressed purely algebraically. In addition, by means of an inner automorphism of this algebra, it is possible to regard these forms and relationships as emerging from a deeper pre-space, which it is calling an implicate order. The way is then opened up for a new mode of description, that does not start from continuous space-time, but which allows this to emerge as a limiting case. (Author)
On a Higher Order Cauchy-Pompeiu Formula for Functions with Values in a Universal Clifford Algebra
Zhongxiang, Zhang
2007-01-01
By constructing suitable kernel functions, a higher order Cauchy-Pompeiu formula for functions with values in a universal Clifford algebra is obtained, leading to a higher order Cauchy integral formula.
From Clifford Algebra of Nonrelativistic Phase Space to Quarks and Leptons of the Standard Model
Żenczykowski, Piotr
2015-01-01
We review a recently proposed Clifford-algebra approach to elementary particles. We start with: (1) a philosophical background that motivates a maximally symmetric treatment of position and momentum variables, and: (2) an analysis of the minimal conceptual assumptions needed in quark mass extraction procedures. With these points in mind, a variation on Born's reciprocity argument provides us with an unorthodox view on the problem of mass. The idea of space quantization suggests then the linearization of the nonrelativistic quadratic form ${\\bf p}^2 +{\\bf x}^2$ with position and momentum satisfying standard commutation relations. This leads to the 64-dimensional Clifford algebra ${Cl}_{6,0}$ of nonrelativistic phase space within which one identifies the internal quantum numbers of a single Standard Model generation of elementary particles (i.e. weak isospin, hypercharge, and color). The relevant quantum numbers are naturally linked to the symmetries of macroscopic phase space. It is shown that the obtained pha...
Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra
I.Yu. Krivsky
2010-01-01
Full Text Available We have proved on the basis of the symmetry analysis of the standard Dirac equation with nonzero mass that this equation may describe not only fermions of spin 1/2 but also bosons of spin 1. The new bosonic symmetries of the Dirac equation in both the Foldy-Wouthuysen and the Pauli-Dirac representations are found. Among these symmetries (together with the 32-dimensional pure matrix algebra of invariance the new, physically meaningful, spin 1 Poincare symmetry of equation under consideration is proved. In order to provide the corresponding proofs, a 64-dimensional extended real Clifford-Dirac algebra is put into consideration.
Relativistic quantum theory of fermions based on the Clifford algebra C7
A relativistic quantum theory of spin- 1/2 fermions is presented that includes a charge algebra, as well as an operator that distinguishes between leptons and baryons. This, in effect, extends the Clifford algebra C4 of Dirac's γ matrices to C7. Moreover, the particle states Psi are represented here by elements of C7 as products of projection operators, instead of column vectors. A number of important results are derived, and the theory serves as a foundation for constructing physical particle states as tensor products of the bare fermion states
A sequence of Clifford algebras and three replicas of Dirac particle
The embedding of Dirac algebra into a sequence N=1, 2, 3,... of Clifford algebras is discussed, leading to Dirac equations with N=1 additional, electromagnetically ''hidden'' spins 1/2. It is shown that there are three and only three replicas N=1, 3, 5 of Dirac particle if the theory of relativity together with the probability interpretation of wave function is applied both to the ''visible'' spin and ''hidden'' spins, and a new ''hidden exclusion principle''is imposed on the wave function (then ''hidden'' spins add up to zero). It is appealing to explore this idea in order to explain the puzzle of three generations of lepton and quarks. (author)
A novel Weyl-Heisenberg algebra in Clifford spaces is constructed that is based on a matrix-valued HAB extension of Planck's constant. As a result of this modified Weyl-Heisenberg algebra one will no longer be able to measure, simultaneously, the pairs of variables (x, px), (x, py), (x, pz), (y, px), ... with absolute precision. New Klein-Gordon and Dirac wave equations and dispersion relations in Clifford spaces are presented. The latter Dirac equation is a generalization of the Dirac-Lanczos-Barut-Hestenes equation. We display the explicit isomorphism between Yang's noncommutative spacetime algebra and the area-coordinates algebra associated with Clifford spaces. The former Yang's algebra involves noncommuting coordinates and momenta with a minimum Planck scale λ (ultraviolet cutoff) and a minimum momentum p ℎ/R (maximal length R, infrared cutoff). The double-scaling limit of Yang's algebra λ → 0, R → ∞, in conjunction with the large n → ∞ limit, leads naturally to the area quantization condition λR = L2 = nλ2 (in Planck area units) given in terms of the discrete angular-momentum eigenvalues n. It is shown how modified Newtonian dynamics is also a consequence of Yang's algebra resulting from the modified Poisson brackets. Finally, another noncommutative algebra which differs from Yang's algebra and related to the minimal length uncertainty relations is presented. We conclude with a discussion of the implications of noncommutative QM and QFT's in Clifford spaces
Schertzer, Daniel; Tchiguirinskaia, Ioulia
2014-05-01
A complex key feature of turbulence is that the velocity is a vector field, whereas intermittency, another key feature, has been mostly understood, analysed and simulated in scalar frameworks. This gap has prevented many developments. Some years ago, the general framework of 'Lie cascades' was introduced (Schertzer and Lovejoy, 1993) to deal with both features by considering cascades generated by stochastic Lie algebra. However, the theoretical efforts were mostly concentrated on the decomposition of this algebra into its radical and a semi-simple algebra and faced too many degrees of freedom. In this communication, we show that the class of Clifford algebra is already wide enough, very convenient and physically meaningful to understand, analyse and simulate intermittent vector fields.
Application of geometric algebra to electromagnetic scattering the Clifford-Cauchy-Dirac technique
Seagar, Andrew
2016-01-01
This work presents the Clifford-Cauchy-Dirac (CCD) technique for solving problems involving the scattering of electromagnetic radiation from materials of all kinds. It allows anyone who is interested to master techniques that lead to simpler and more efficient solutions to problems of electromagnetic scattering than are currently in use. The technique is formulated in terms of the Cauchy kernel, single integrals, Clifford algebra and a whole-field approach. This is in contrast to many conventional techniques that are formulated in terms of Green's functions, double integrals, vector calculus and the combined field integral equation (CFIE). Whereas these conventional techniques lead to an implementation using the method of moments (MoM), the CCD technique is implemented as alternating projections onto convex sets in a Banach space. The ultimate outcome is an integral formulation that lends itself to a more direct and efficient solution than conventionally is the case, and applies without exception to all types...
On Clifford-algebraic dimensional extension and SUSY holography
Gates, S. J.; Hübsch, T.; Stiffler, K.
2015-03-01
We analyze the group of maximal automorphisms of the N-extended worldline supersymmetry algebra, and its action on off-shell supermultiplets. This defines a concept of "holoraumy" that extends the notions of holonomy and curvature in a novel way and provides information about the geometry of the supermultiplet field-space. In turn, the "holoraumy" transformations of 0-brane dimensionally reduced supermultiplets provide information about Lorentz transformations in the higher-dimensional space-time from which the 0-brane supermultiplets are descended. Specifically, Spin(3) generators are encoded within 0-brane "holoraumy" tensors. Worldline supermultiplets are thus able to holographically encrypt information about higher-dimensional space-time geometry.
On Clifford-Algebraic "Holoraumy", Dimensional Extension, and SUSY Holography
Gates, S J; Stiffler, K
2014-01-01
We analyze the group of maximal automorphisms of the N-extended world-line supersymmetry algebra, and its action on off-shell supermultiplets. This defines a concept of "holoraumy" that extends the notions of holonomy and curvature in a novel way and provides information about the geometry of the supermultiplet field-space. In turn, the "holoraumy" transformations of 0-brane dimensionally reduced supermultiplets provide information about Lorentz transformations in the higher-dimensional spacetime from which the 0-brane supermultiplets are descended. World-line supermultiplets are thus able to holographically encrypt information about higher dimensional spacetime geometry.
The algebraic structure of the real Clifford algebras (CA) of vector spaces with non-degenerated scalar product of arbitrary signature is studied, and a classification formula for this is obtained. The latter is based on three sequences of integer numbers from which one is the Radon-Harwitz sequence. A new representation method of real CA is constructed. This leads to a geometrical representation of real Clifford algebras in which the representation spaces are subspaces of the CA itself (''spinor spaces''). One of these spinor spaces is a subalgebra of the original CA. The relation between CA and external algebras is studied. Each external algebra with a scalar product possesses the structure of a CA. From the geometric representation developed here then follows that spinors are inhomogeneous external forms. The transformation behaviour of spinors under the orthogonal, as well as under the general linear group is studied. By means of these algebraic results the spinor connexion and the covariant Dirac operator on a differential manifold are introduced. In the geometrical representation a further in ternal SL(2,R) symmetry of the Dirac equation (DE) is shown. Furthermore other equivalent formulations of the DE can be obtained. Of special interest is the tetrade formulation of the DE. A generalization of the DE is introduced. The equations of motion of the classical relativistic spin particle are derived by means of spinors and CA from a variational principle. From this interesting formal analogies with the supersymmetric theories of the spin particle result. Finally the DE in the curved space-time is established and studied in the tetrade formulation. Using the methods developed here a new exact solution of the coupled Einstein-Curtan-Dirac theory was found (massice ''Ghost-Dirac fields''). (orig.)
First order differential operator associated to the Cauchy-Riemann operator in a Clifford algebra
The complex differentiation transforms holomorphic functions into holomorphic functions. Analogously, the conjugate Cauchy-Riemann operator of the Clifford algebra transforms regular functions into regular functions. This paper determines more general first order operator L (matrix-type) for which Lu is regular provided u is regular. For such operator L, the initial value problem ∂u / ∂t = L (t, x, u, ∂u / ∂x) (1) u(0, x) = φ(x) (2) is solvable for an arbitrary regular function φ and the solution is regular in x for each t. (author)
A new linear Dirac-like spin-3/2 wave equation using Clifford algebra
A new linear Dirac-like wave equation for spin-3/2 is derived, employing four of the seven irreducible eight-dimensional matrices obeying the Clifford algebra C7 with the wave function having the needed eight components only. Though this wave equation is not manifestly covariant and the wave function employed is not locally covariant, it is relativistically invariant and by its very derivation is connected to the Weaver, Hammer and Good (Phys. Rev.; 135: B241 (1964)) formalism for spin-3/2 by a chain of transformations which can be arbitrarily chosen to be either unitary or non-unitary. (author)
Dechant, Pierre-Philippe
2013-01-01
Quaternionic representations of Coxeter (reflection) groups of ranks 3 and 4, as well as those of E_8, have been used extensively in the literature. The present paper analyses such Coxeter groups in the Clifford Geometric Algebra framework, which affords a simple way of performing reflections and rotations whilst exposing more clearly the underlying geometry. The Clifford approach shows that the quaternionic representations in fact have very simple geometric interpretations. The representations of the groups A_1 x A_1 x A_1, A_3, B_3 and H_3 of rank 3 in terms of pure quaternions are shown to be simply the Hodge dualised root vectors, which determine the reflection planes of the Coxeter groups. Two successive reflections result in a rotation, described by the geometric product of the two reflection vectors, giving a Clifford spinor. The spinors for the rank-3 groups A_1 x A_1 x A_1, A_3, B_3 and H_3 yield a new simple construction of binary polyhedral groups. These in turn generate the groups A_1 x A_1 x A_1 ...
The Clifford algebra of nonrelativistic phase space and the concept of mass
Prompted by a recent demonstration that the structure of a single quark-lepton generation may be understood via a Dirac-like linearization of the form p2 + x2, we analyse the corresponding Clifford algebra in some detail. After classifying all elements of this algebra according to their U(1) x SU(3) and SU(2) transformation properties, we identify the element which might be associated with the concept of lepton mass. This element is then transformed into a corresponding element for a single coloured quark. It is shown that-although none of the three thus obtained individual quark mass elements is rotationally invariant-the rotational invariance of the quark mass term is restored when the sum over quark colours is performed
A 64-dimensional extended real Clifford-Dirac algebra is introduced. On its basis, new pure matrix symmetries of the Dirac equation in the Foldy-Wouthuysen representation was found. Finally, spin 1 Poincare symmetries both for the Foldy-Wouthuysen and standard Dirac equations with nonzero mass are found.
The dynamics of split fields in one dimension are extended to three dimensions using Clifford algebra. The solutions of the resulting equations provide a unique insight into wave splitting and allow the construction of wave splittings in three dimensions that may be useful in solving the three-dimensional inverse scattering problem in the time domain
Using Clifford Algebra to Understand the Nature of Negative Pressure Waves
McClellan, Gene
2014-03-01
The geometric algebra of 3-D Euclidean space, a sub-discipline of Clifford algebra, is a useful tool for analyzing wave propagation. We use geometric algebra to explore the concept of negative pressure. In free space a straightforward extension of Maxwell's equations using geometric algebra yields a theory in which classical electromagnetic waves coexist with nonelectromagnetic waves having retrograde momentum. By retrograde momentum we mean waves carrying momentum pointing in the opposite direction of energy flow. If such waves exist, they would have negative pressure. In rebounding from a wall, they would pull rather than push. In this presentation we use standard methods of analyzing energy and momentum conservation and their flow through the surface of an enclosed volume to illustrate the properties of both the electromagnetic and nonelectromagnetic solutions of the extended Maxwell equations. The nonelectromagnetic waves consist of coupled scalar and electric waves and coupled magnetic and pseudoscalar waves. They superimpose linearly with electromagnetic waves. We show that the nonelectromagnetic waves, besides having negative pressure, propagate with the speed of light and do not interact with conserved electric currents. Hence, they have three properties in common with dark energy.
PT-symmetry, Cartan decompositions, Lie triple systems and Krein space-related Clifford algebras
Guenther, Uwe [Research Center Dresden-Rossendorf, PO Box 510119, D-01314 Dresden (Germany); Kuzhel, Sergii, E-mail: u.guenther@fzd.d, E-mail: kuzhel@imath.kiev.u [Institute of Mathematics of the NAS of Ukraine, 01601 Kyiv (Ukraine)
2010-10-01
Gauged PT quantum mechanics (PTQM) and corresponding Krein space setups are studied. For models with constant non-Abelian gauge potentials and extended parity inversions compact and noncompact Lie group components are analyzed via Cartan decompositions. A Lie-triple structure is found and an interpretation as PT-symmetrically generalized Jaynes-Cummings model is possible with close relation to recently studied cavity QED setups with transmon states in multilevel artificial atoms. For models with Abelian gauge potentials a hidden Clifford algebra structure is found and used to obtain the fundamental symmetry of Krein space-related J-self-adjoint extensions for PTQM setups with ultra-localized potentials. (fast track communication)
A Clifford Algebra Approach to the Classical Problem of a Charge in a Magnetic Monopole Field
Vaz, Jayme
2013-05-01
The motion of an electric charge in the field of a magnetic monopole is described by means of a Lagrangian model written in terms of the Clifford algebra of the physical space. The equations of motion are written in terms of a radial equation (involving r=| r|, where r( t) is the charge trajectory) and a rotor equation (written in terms of an unitary operator spinor R). The solution corresponding to the charge trajectory in the field of a magnetic monopole is given in parametric form. The model can be generalized in order to describe the motion of a charge in the field of a magnetic monopole and other additional central forces, and as an example, we discuss the classical ones involving linear and inverse square interactions.
New insights in the standard model of quantum physics in Clifford algebra
Daviau, Claude
2013-01-01
Why Clifford algebra is the true mathematical frame of the standard model of quantum physics. Why the time is everywhere oriented and why the left side shall never become the right side. Why positrons have also a positive proper energy. Why there is a Planck constant. Why a mass is not a charge. Why a system of particles implies the existence of the inverse of the individual wave function. Why a fourth neutrino should be a good candidate for black matter. Why concepts as “parity” and “reverse” are essential. Why the electron of a H atom is in only one bound state. Plus 2 very remarkable identities, and the invariant wave equations that they imply. Plus 3 generations and 4 neutrinos. Plus 5 dimensions in the space and 6 dimensions in space-time…
Families of isospectral matrix Hamiltonians by deformation of the Clifford algebra on a phase space
By using a recently developed method, we report five different families of isospectral 2 x 2 matrix Hamiltonians defined on a four-dimensional (4D) phase space. The employed method is based on a realization of the supersymmetry idea on the phase space whose complexified Clifford algebra structure is deformed with the Moyal star-product. Each reported family comprises many physically relevant special models. 2D Pauli Hamiltonians, systems involving spin-orbit interactions such as Aharonov-Casher-type systems, a supermembrane toy model and models describing motion in noncentral electromagnetic fields as well as Rashba- and Dresselhaus-type systems from semiconductor physics are obtained, together with their super-partners, as special cases. A large family of isospectral systems characterized by the whole set of analytic functions is also presented.
PT-symmetry, Cartan decompositions, Lie triple systems and Krein space-related Clifford algebras
Gauged PT quantum mechanics (PTQM) and corresponding Krein space setups are studied. For models with constant non-Abelian gauge potentials and extended parity inversions compact and noncompact Lie group components are analyzed via Cartan decompositions. A Lie-triple structure is found and an interpretation as PT-symmetrically generalized Jaynes-Cummings model is possible with close relation to recently studied cavity QED setups with transmon states in multilevel artificial atoms. For models with Abelian gauge potentials a hidden Clifford algebra structure is found and used to obtain the fundamental symmetry of Krein space-related J-self-adjoint extensions for PTQM setups with ultra-localized potentials. (fast track communication)
Vector coherent states from Plancherel's theorem, Clifford algebras and matrix domains
As a substantial generalization of the technique for constructing canonical and the related nonlinear and q-deformed coherent states, we present here a method for constructing vector coherent states (VCS) in the same spirit. These VCS may have a finite or an infinite number of components. The resulting formalism, which involves an assumption on the existence of a resolution of the identity, is broad enough to include all the definitions of coherent states existing in the current literature, subject to this restriction. As examples, we first apply the technique to construct VCS using the Plancherel isometry for groups and VCS associated with Clifford algebras, in particular quaternions. As physical examples, we discuss VCS for a quantum optical model and finally apply the general technique to build VCS over certain matrix domains
Clifford代数Clp，q的中心子代数%The Central Subalgebra of Clifford Algebra Clp,q
宋元凤; 李武明; 丁宝霞
2011-01-01
（p，q）型Minkowski空间Rp,q的Clifford代数Clp-q是一类2p＋q维的实结合代数，当p＋q〉1时是非交换代数．文中讨论了Clp-q的中心Cen（Clp,q）的相关性质，利用基元的Clifford积导出由p，q确定Clp,q中心的公式．%Clifford algebra Clp,q of （p,q） Minkowski space R p,qare a class of2p＋qdimension real associative algebra, as p ＋ q 〉 1 , they are non - commutative algebra. The properties of Clp,q center Cen （Clp,q） are discussed by using Clifford product to derive the formula of Clp,q center.
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
Dechant, Pierre-Philippe
2016-01-01
In this paper, we make the case that Clifford algebra is the natural framework for root systems and reflection groups, as well as related groups such as the conformal and modular groups: The metric that exists on these spaces can always be used to construct the corresponding Clifford algebra. Via the Cartan-Dieudonn\\'e theorem all the transformations of interest can be written as products of reflections and thus via `sandwiching' with Clifford algebra multivectors. These multivector groups can be used to perform concrete calculations in different groups, e.g. the various types of polyhedral groups, and we treat the example of the tetrahedral group $A_3$ in detail. As an aside, this gives a constructive result that induces from every 3D root system a root system in dimension four, which hinges on the facts that the group of spinors provides a double cover of the rotations, the space of 3D spinors has a 4D euclidean inner product, and with respect to this inner product the group of spinors can be shown to be cl...
Climate and weather across scales: singularities and stochastic Levy-Clifford algebra
Schertzer, Daniel; Tchiguirinskaia, Ioulia
2016-04-01
There have been several attempts to understand and simulate the fluctuations of weather and climate across scales. Beyond mono/uni-scaling approaches (e.g. using spectral analysis), this was done with the help of multifractal techniques that aim to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations of these equations (Royer et al., 2008, Lovejoy and Schertzer, 2013). However, these techniques were limited to deal with scalar fields, instead of dealing directly with a system of complex interactions and non trivial symmetries. The latter is unfortunately indispensable to answer to the challenging question of being able to assess the climatology of (exo-) planets based on first principles (Pierrehumbert, 2013) or to fully address the question of the relevance of quasi-geostrophic turbulence and to define an effective, fractal dimension of the atmospheric motions (Schertzer et al., 2012). In this talk, we present a plausible candidate based on the combination of Lévy stable processes and Clifford algebra. Together they combine stochastic and structural properties that are strongly universal. They therefore define with the help of a few physically meaningful parameters a wide class of stochastic symmetries, as well as high dimensional vector- or manifold-valued fields respecting these symmetries (Schertzer and Tchiguirinskaia, 2015). Lovejoy, S. & Schertzer, D., 2013. The Weather and Climate: Emergent Laws and Multifractal Cascades. Cambridge U.K. Cambridge Univeristy Press. Pierrehumbert, R.T., 2013. Strange news from other stars. Nature Geoscience, 6(2), pp.81-83. Royer, J.F. et al., 2008. Multifractal analysis of the evolution of simulated precipitation over France in a climate scenario. C.R. Geoscience, 340(431-440). Schertzer, D. et al., 2012. Quasi-geostrophic turbulence and generalized scale invariance, a theoretical reply. Atmos. Chem. Phys., 12, pp.327-336. Schertzer, D
Ablamowicz, Rafal
2011-01-01
We introduce on the abstract level in real Clifford algebras \\cl_{p,q} of a non-degenerate quadratic space (V,Q), where Q has signature \\epsilon=(p,q), a transposition anti-involution \\tp. In a spinor representation, the anti-involution \\tp gives transposition, complex Hermitian conjugation or quaternionic Hermitian conjugation when the spinor space \\check{S} is viewed as a \\cl_{p,q}-left and \\check{K}-right module with \\check{K} isomorphic to R or R^2, C, or, H or H^2. \\tp is a lifting to \\cl_{p,q} of an orthogonal involution \\tve: V \\rightarrow V which depends on the signature of Q. The involution is a symmetric correlatio \\tve: V \\rightarrow V^{*} \\cong V and it allows one to define a reciprocal basis for the dual space (V^{*},Q). The anti-involution \\tp acts as reversion on \\cl_{p,0} and as conjugation on \\cl_{0,q}. Using the concept of a transpose of a linear mapping one can show that if [L_u] is a matrix in the left regular representation of the operator L_u: \\cl_{p,q} \\rightarrow \\cl_{p,q} relative to ...
Clifford algebra-based structure filtering analysis for geophysical vector fields
Yu, Z.; Luo, W.; Yi, L.; Hu, Y.; Yuan, L.
2013-07-01
A new Clifford algebra-based vector field filtering method, which combines amplitude similarity and direction difference synchronously, is proposed. Firstly, a modified correlation product is defined by combining the amplitude similarity and direction difference. Then, a structure filtering algorithm is constructed based on the modified correlation product. With custom template and thresholds applied to the modulus and directional fields independently, our approach can reveal not only the modulus similarities but also the classification of the angular distribution. Experiments on exploring the tempo-spatial evolution of the 2002-2003 El Niño from the global wind data field are used to test the algorithm. The results suggest that both the modulus similarity and directional information given by our approach can reveal the different stages and dominate factors of the process of the El Niño evolution. Additional information such as the directional stability of the El Niño can also be extracted. All the above suggest our method can provide a new powerful and applicable tool for geophysical vector field analysis.
Clifford algebra-based structure filtering analysis for geophysical vector fields
Z. Yu
2013-07-01
Full Text Available A new Clifford algebra-based vector field filtering method, which combines amplitude similarity and direction difference synchronously, is proposed. Firstly, a modified correlation product is defined by combining the amplitude similarity and direction difference. Then, a structure filtering algorithm is constructed based on the modified correlation product. With custom template and thresholds applied to the modulus and directional fields independently, our approach can reveal not only the modulus similarities but also the classification of the angular distribution. Experiments on exploring the tempo-spatial evolution of the 2002–2003 El Niño from the global wind data field are used to test the algorithm. The results suggest that both the modulus similarity and directional information given by our approach can reveal the different stages and dominate factors of the process of the El Niño evolution. Additional information such as the directional stability of the El Niño can also be extracted. All the above suggest our method can provide a new powerful and applicable tool for geophysical vector field analysis.
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.
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
Quantum gravity, Clifford algebras, fuzzy set theory and the fundamental constants of nature
In a recent paper entitled 'Quantum gravity from descriptive set theory', published in Chaos, Solitons and Fractals, we considered following the P-adic quantum theory, the possibility of abandoning the Archimedean axiom and introducing a fundamental physical limitation on the smallest length in quantum spacetime. Proceeding that way we arrived at the conclusion that maximising the Hawking-Bekenstein informational content of spacetime makes the existence of a transfinite geometry for physical 'spacetime' plausible or even inevitable. Subsequently we introduced a mathematical description of a transfinite, non-Archimedean geometry using descriptive set theory where a similar conclusion regarding the transfiniteness of quantum spacetime may be drawn from the existence of the Unruh temperature. In particular we introduced a straight forward logarithmic gauge transformation linking, as far as we are aware for the first time, classical gravity with the electroweak via a version of informational entropy. That way we found using ε(∞) and complexity theory that α-barG=(2)α-barew-1=1.7x1038 where α-barG is the dimensionless Newton gravity constant and α-barew=128 is the fine structure constant at the electroweak unification scale. The present work is concerned with more or less the same category of fundamental questions pertinent to quantum gravity. However we switch our mathematical apparatus to a combination of Clifford algebras and set theory. In doing that, the central and vital role of the work of D. Finkelstein becomes much more tangible and clearer than in most of our previous works
We give an eight-dimensional realization of the Clifford algebra in the five-dimensional Galilean covariant spacetime by using a dimensional reduction from the (5 + 1) Minkowski spacetime to the (4 + 1) Minkowski spacetime which encompasses the Galilean covariant spacetime. A set of solutions of the Dirac-type equation in the five-dimensional Galilean covariant spacetime is obtained, based on the Pauli representation of 8 x 8 gamma matrices. In order to find an explicit solution, we diagonalize the Klein-Gordon divisor by using the Galilean boost
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
We investigate the two-dimensional version of the Chern-Simons action derived from the recently proposed even-dimensional generalized Chern-Simons action. We show that the two-dimensional topological gravity emerges if we choose the Clifford algebra as a nonstandard gauge symmetry algebra required from the generalized Chern-Simons action. We find a ''hidden order parameter'' which differentiates the gravity phase and nongravity phase
Henrik Aratyn
2007-02-01
Full Text Available We use a Grassmannian framework to define multi-component tau functions as expectation values of certain multi-component Fermi operators satisfying simple bilinear commutation relations on Clifford algebra. The tau functions contain both positive and negative flows and are shown to satisfy the $2n$-component KP hierarchy. The hierarchy equations can be formulated in terms of pseudo-differential equations for $nimes n$ matrix wave functions derived in terms of tau functions. These equations are cast in form of Sato-Wilson relations. A reduction process leads to the AKNS, two-component Camassa-Holm and Cecotti-Vafa models and the formalism provides simple formulas for their solutions.
Daviau, Claude
2014-01-01
A wave equation with mass term is studied for all particles and antiparticles of the first generation: electron and its neutrino, positron and antineutrino, quarks $u$ and $d$ with three states of color and antiquarks $\\overline{u}$ and $\\overline{d}$. This wave equation is form invariant under the $Cl_3^*$ group generalizing the relativistic invariance. It is gauge invariant under the $U(1)\\times SU(2) \\times SU(3)$ group of the standard model of quantum physics. The wave is a function of space and time with value in the Clifford algebra $Cl_{1,5}$. All features of the standard model, charge conjugation, color, left waves, Lagrangian formalism, are linked to the geometry of this extended space-time.
Daviau, Claude; Bertrand, Jacques
A wave equation with mass term is studied for all particles and antiparticles of the first generation: electron and its neutrino, positron and antineutrino, quarks $u$ and $d$ with three states of color and antiquarks $\\overline{u}$ and $\\overline{d}$. This wave equation is form invariant under the $Cl_3^*$ group generalizing the relativistic invariance. It is gauge invariant under the $U(1)\\times SU(2) \\times SU(3)$ group of the standard model of quantum physics. The wave is a function of space and time with value in the Clifford algebra $Cl_{1,5}$. All features of the standard model, charge conjugation, color, left waves, Lagrangian formalism, are linked to the geometry of this extended space-time.
Magnetic monopoles without string in the Kaehler-Clifford algebra: a geometrical interpretation
In substitution for Dirac monopoles with string (and for topological monopoles) we have recently introduced monopoles without string on the basis of a generalized potential, the sum of a vector A and a pseudo-vector sub(γ5)B potential. By making recourse to the Clifford bundle C (τ M,g) [ T sub(x) M,g) = IR sup(1,3); C (T sub(x) M,g) = IR sub(1,3)], which just allows adding together for each x ε M tensors of different ranks, in a previous paper we succeeded in constructing a lagrangian and hamiltonian formalism for interacting monopoles and charges that can be regarded as satisfactory from various points of view. In the present note, after having completed our formalism, we put forth a purely geometrical interpretation of it within the Kaehler-Clifford bundle K (τ sup(*) M
The classical problem of the charge and pole motion. A satisfactory formalism by Clifford algebras
In a previous paper of ours [Phys. Lett. B 173(1986) 233; B 188 (1987) 511E] we put forth a theoretical approach to magnetic monopoles without a string which is formulated in a Clifford bundle. To 'complete' our theory, we show that the Maxwell euqations - with magnetic monopoles - do imply the correct couplings of the electric current and magnetic current with the electromagnetic field (and, therefore, imply the Lorentz forces and the correct motion equations). In other words, within our Clifford approach to classical electromagnetism, the motion equations and the couplings are derivable from the field equations, without any further recourse to a variational principle and without any ad hoc postulate. The price to pay for that result seems merely to be a very natural assumption, analogous to a similar one that is quite standard in general relativity. (orig.)
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.
Understanding geometric algebra Hamilton, Grassmann, and Clifford for computer vision and graphics
Kanatani, Kenichi
2015-01-01
Introduction PURPOSE OF THIS BOOK ORGANIZATION OF THIS BOOK OTHER FEATURES 3D Euclidean Geometry VECTORS BASIS AND COMPONENTS INNER PRODUCT AND NORM VECTOR PRODUCTS SCALAR TRIPLE PRODUCT PROJECTION, REJECTION, AND REFLECTION ROTATION PLANES LINES PLANES AND LINES Oblique Coordinate Systems RECIPROCAL BASIS RECIPROCAL COMPONENTS INNER, VECTOR, AND SCALAR TRIPLE PRODUCTS METRIC TENSOR RECIPROCITY OF EXPRESSIONS COORDINATE TRANSFORMATIONSHamilton's Quaternion Algebra QUATERNIONS ALGEBRA OF QUATERNIONS CONJUGATE, NORM, AND INVERSE REPRESENTATION OF ROTATION BY QUATERNION Grassmann's Outer Product
Real representations of finite Clifford algebras. II. Explicit construction and pseudo-octonion
It has been shown that any real irreducible representation of any C(p,q) can be, in principle, explicitly constructed inductively. The relation between the pseudo-octonion algebra and C(8,0) as well as C(0,8) has also been investigated in some detail. Finally, dimensions of the real spinor representations of SO(p,q) have been studied
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
Pansart, Jean Pierre
2016-01-01
Gauge fields associated to the Dirac matrix algebra used with the standard quadratic gauge field Lagrangian lead to an extended gravitational Lagrangian which includes the Einstein-Hilbert one, plus quadratic, cosmological constant and torsion terms. This note looks at three cases : the static central symmetric field, the isotropic expanding universe, and the asymptotic field of a rotating body, and show that, in weak gravitational fields, there is no contradiction with General Relativity results.
Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra
I.Yu. Krivsky; Simulik, V. M.
2010-01-01
We have proved on the basis of the symmetry analysis of the standard Dirac equation with nonzero mass that this equation may describe not only fermions of spin 1/2 but also bosons of spin 1. The new bosonic symmetries of the Dirac equation in both the Foldy-Wouthuysen and the Pauli-Dirac representations are found. Among these symmetries (together with the 32-dimensional pure matrix algebra of invariance) the new, physically meaningful, spin 1 Poincare symmetry of equation under consideration ...
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)\
Analysis of two-player quantum games in an EPR setting using Clifford's geometric algebra.
James M Chappell
Full Text Available The framework for playing quantum games in an Einstein-Podolsky-Rosen (EPR type setting is investigated using the mathematical formalism of geometric algebra (GA. The main advantage of this framework is that the players' strategy sets remain identical to the ones in the classical mixed-strategy version of the game, and hence the quantum game becomes a proper extension of the classical game, avoiding a criticism of other quantum game frameworks. We produce a general solution for two-player games, and as examples, we analyze the games of Prisoners' Dilemma and Stag Hunt in the EPR setting. The use of GA allows a quantum-mechanical analysis without the use of complex numbers or the Dirac Bra-ket notation, and hence is more accessible to the non-physicist.
Analysis of Two-Player Quantum Games in an EPR Setting Using Clifford's Geometric Algebra
Chappell, James M.; Iqbal, Azhar; Abbott, Derek
2012-01-01
The framework for playing quantum games in an Einstein-Podolsky-Rosen (EPR) type setting is investigated using the mathematical formalism of geometric algebra (GA). The main advantage of this framework is that the players' strategy sets remain identical to the ones in the classical mixed-strategy version of the game, and hence the quantum game becomes a proper extension of the classical game, avoiding a criticism of other quantum game frameworks. We produce a general solution for two-player games, and as examples, we analyze the games of Prisoners' Dilemma and Stag Hunt in the EPR setting. The use of GA allows a quantum-mechanical analysis without the use of complex numbers or the Dirac Bra-ket notation, and hence is more accessible to the non-physicist. PMID:22279525
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.
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
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.
正交模上Clifford代数的支配权%Dominant Weights of the Clifford Algebra over an Orthogonal Module
何军华; 谭友军
2011-01-01
This paper deals with dominant weights of the Clifford algebra C(V) over an orthogonal g-module V, where the g-module C(V) is a multiple of Kostant's spin module Spin(V). Let △(V) be the set of nonzero weights of V. The half-sum of any positive convex half of △(V) is shown to be a dominant weight of C(V). Conversely, if a half-sum is a highest weight of C(V) with multiplicity 2mV(O)+dim V/2, then it is given by a positive convex half of △(V).%研究了正交g-模V上的Clifford代数C(V)的支配权,其中G-模C(V)是Kostant给出的旋模Spin(V)的倍数.设△(V)是V的非零权组成的集合.证明了△(V)任一正凸半的半和总是C(V)的一个支配权.反之,如果某一个半和是C(V)的重数为2 mV(O)+dim V/2 的最高权,那么该半和一定是△(V)的某个正凸半的半和.
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.
俞肇元; 袁林旺; 罗文; 易琳
2011-01-01
以Clifford代数为理论基础与数学工具,构建了时空分析原型系统：①在兼容多类常用GIS数据格式的基础上,根据Clifford代数空间构建的思想,对现有时空数据模型进行扩展,实现了时间、空间与属性的一体化表达;②定义了可支撑多维度时空分析的几何、度量等Clifford代数算子库;③基于插件的时空分析模型算法构建及集成框架,实现了高维邻域分析、网络分析以及时空栅格数据分析等地学分析算法。实验结果显示,根据Clifford代数所构建的时空分析系统可有效支撑多维时空分析。%By introducing Clifford Algebra as the theoretical foundation and mathematical tools,a prototype temporal spatial analysis software system was proposed.The characteristics of this system can be summarized as following.① Under the premise of keeping the compatibilities with the commonly used GIS data,a new type of temporal-spatial data model,which unified the expression of both temporal,spatial and attribution components within the multivector structure,was proposed.② Geometric and metric operator libraries were defined,which can support multidimensional temporal spatial analysis.③ Plugging based temporal spatial analysis model constructing and integrating framework was implemented.Typical GIS temporal spatial analysis algorithms like multidimensional V-neighbor analysis,minimum union analysis and unified spatial-temporal process analyses with spacetime algebra were implemented and integrated.Results suggest that the proposed system can support multidimensional temporal spatial analysis effectively,which can also provide reference on improving research on unified temporal spatial analysis methods and GIS systems.
Marchuk, Nikolay
2011-01-01
Exterior algebras and differential forms are widely used in many fields of modern mathematics and theoretical physics. In this paper we define a notion of $N$-metric exterior algebra, which depends on $N$ matrices of structure constants. The usual exterior algebra (Grassmann algebra) can be considered as 0-metric exterior algebra. Clifford algebra can be considered as 1-metric exterior algebra. $N$-metric exterior algebras for $N\\geq2$ can be considered as generalizations of the Grassmann alg...
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.
Clifford algebras in general relativity
Techniques for dealing with nonlinear partial differential equations of interest in physics have been widely developed in recent years. Generation and inverse scattering methods have been applied to a number of such equations, including the gravitational field equations when symmetries are present. Unfortunately, most of the relevant applications are restricted to cases with two independent variables, with the notable exception of the self-dual Yang-Mills equations in four-dimensional Euclidean space and a few other examples. This work is motivated by the questions of whether it is possible to use similar techniques in the general case (i.e., without symmetries) of the Einstein field equations in vacuum. As a preliminary step, it seems necessary to formulate the equations in a compact form, and in particular in such a way that integrability conditions are easily taken care of. To this end, it seems appropriate to use differential forms; the possibility of working with such conditions in a transparent way is a built-in feature of the differential form language
Cofree Hopf algebras on Hopf bimodule algebras
Fang, Xin; Jian, Run-Qiang
2013-01-01
We investigate a Hopf algebra structure on the cotensor coalgebra associated to a Hopf bimodule algebra which contains universal version of Clifford algebras and quantum groups as examples. It is shown to be the bosonization of the quantum quasi-shuffle algebra built on the space of its right coinvariants. The universal property and a Rota-Baxter algebra structure are established on this new algebra.
Some C*-algebras associated to quantum gauge theories
Hannabuss, Keith C.
2010-01-01
Algebras associated with Quantum Electrodynamics and other gauge theories share some mathematical features with T-duality Exploiting this different perspective and some category theory, the full algebra of fermions and bosons can be regarded as a braided Clifford algebra over a braided commutative boson algebra, sharing much of the structure of ordinary Clifford algebras.
Projector bases and algebraic spinors
In the case of complex Clifford algebras a basis is constructed whose elements satisfy projector relations. The relations are sufficient conditions for the elements to span minimal ideals and hence to define algebraic spinors
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.
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...
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 .
Algebraic formulation of duality
Two dimensional lattice spin (chiral) models over (possibly non-abelian) compact groups are formulated in terms of a generalized Pauli algebra. Such models over cyclic groups are written in terms of the generalized Clifford algebra. An automorphism of this algebra is shown to exist and to lead to the duality transformation
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.
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
Dirac matrices as elements of superalgebraic matrix algebra
Monakhov, V. V.
2016-01-01
The paper considers a Clifford extension of the Grassmann algebra, in which operators are built from Grassmann variables and by the derivatives with respect to them. It is shown that a subalgebra which is isomorphic to the usual matrix algebra exists in this algebra, the Clifford exten-sion of the Grassmann algebra is a generalization of the matrix algebra and contains superalgebraic operators expanding matrix algebra and produces supersymmetric transformations.
Tabak, John
2004-01-01
Looking closely at algebra, its historical development, and its many useful applications, Algebra examines in detail the question of why this type of math is so important that it arose in different cultures at different times. The book also discusses the relationship between algebra and geometry, shows the progress of thought throughout the centuries, and offers biographical data on the key figures. Concise and comprehensive text accompanied by many illustrations presents the ideas and historical development of algebra, showcasing the relevance and evolution of this branch of mathematics.
The quest for conformal geometric algebra Fourier transformations
Hitzer, Eckhard
2013-10-01
Conformal geometric algebra is preferred in many applications. Clifford Fourier transforms (CFT) allow holistic signal processing of (multi) vector fields, different from marginal (channel wise) processing: Flow fields, color fields, electro-magnetic fields, ... The Clifford algebra sets (manifolds) of √-1 lead to continuous manifolds of CFTs. A frequently asked question is: What does a Clifford Fourier transform of conformal geometric algebra look like? We try to give a first answer.
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.
无
2004-01-01
Through most of Greek history, mathematicians concentrated on geometry, although Euclid considered the theory of numbers. The Greek mathematician Diophantus (3rd century),however, presented problems that had to be solved by what we would today call algebra. His book is thus the first algebra text.
Flanders, Harley
1975-01-01
Algebra presents the essentials of algebra with some applications. The emphasis is on practical skills, problem solving, and computational techniques. Topics covered range from equations and inequalities to functions and graphs, polynomial and rational functions, and exponentials and logarithms. Trigonometric functions and complex numbers are also considered, together with exponentials and logarithms.Comprised of eight chapters, this book begins with a discussion on the fundamentals of algebra, each topic explained, illustrated, and accompanied by an ample set of exercises. The proper use of a
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...
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
Spinors in the hyperbolic algebra
Ulrych, S.
2006-01-01
The three-dimensional universal complex Clifford algebra is used to represent relativistic vectors in terms of paravectors. In analogy to the Hestenes spacetime approach spinors are introduced in an algebraic form. This removes the dependance on an explicit matrix representation of the algebra.
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 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.
Algebraic Apect of Helicities in Hadron Physics
An, Murat; Ji, Chueng
2015-04-01
We examined the relation of polarization vectors and spinors of (1 , 0) ⊕(0 , 1) representation of Lorentz group in Clifford algebra Cl1 , 3 , their relation with standard algebra, and properties of these spinors. Cl1 , 3 consists of different grades:e.g. the first and the second grades represent (1 / 2 , 1 / 2) and (1 , 0) ⊕(0 , 1) representation of spin groups respectively with 4 and 6 components. However, these Clifford numbers are not the helicity eigenstates and thus we transform them into combinations of helicity eigenstates by expressing them as spherical harmonics. We relate the spin-one polarization vectors and (1 , 0) ⊕(0 , 1) spinors under one simple transformation with the spin operators. We also link our work with Winnberg's work of a superfield of a spinors of Clifford algebra by giving a physical meaning to Grassmann variables and discuss how Grassman algebra is linked with Clifford algebra.
Quregisters, Symmetry Groups and Clifford Algebras
Cervantes, D.; Morales-Luna, G.
2016-03-01
Natural one-to-one and two-to-one homomorphisms from SO(3) into SU(2) are built conventionally, and the collection of qubits, is identified with a subgroup of SU(2). This construction is suitable to be extended to corresponding tensor powers. The notions of qubits, quregisters and qugates are translated into the language of symmetry groups. The corresponding elements to entangled states in the tensor product of Hilbert spaces reflect entanglement properties as well, and in this way a notion of entanglement is realised in the tensor product of symmetry groups.
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 ...
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
Decomposition numbers for Brauer algebras of type G(m,p,n) in characteristic zero
Bowman, C.; Cox, A.
2013-01-01
We introduce Brauer algebras associated to complex reflection groups of type $G(m,p,n)$, and study their representation theory via Clifford theory. In particular, we determine the decomposition numbers of these algebras in characteristic zero.
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...