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1.
It is known that Clifford (geometric) algebra offers a geometric interpretation for square roots of –1 in the form of blades
that square to –1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. 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 of
\mathbb R3{{\mathbb R^3}} . All these roots of –1 find immediate applications in the construction of new types of geometric Clifford Fourier transformations. 相似文献
2.
Bertfried Fauser 《Mathematical Methods in the Applied Sciences》2001,24(12):885-912
We propose a geometric method to parameterize inequivalent vacua by dynamical data. Introducing quantum Clifford algebras with arbitrary bilinear forms we distinguish isomorphic algebras—as Clifford algebras—by different filtrations (resp. induced gradings). The idea of a vacuum is introduced as the unique algebraic projection on the base field embedded in the Clifford algebra, which is however equivalent to the term vacuum in axiomatic quantum field theory and the GNS construction in C*‐algebras. This approach is shown to be equivalent to the usual picture which fixes one product but employs a variety of GNS states. The most striking novelty of the geometric approach is the fact that dynamical data fix uniquely the vacuum and that positivity is not required. The usual concept of a statistical quantum state can be generalized to geometric meaningful but non‐statistical, non‐definite, situations. Furthermore, an algebraization of states takes place. An application to physics is provided by an U (2)‐symmetry producing a gap equation which governs a phase transition. The parameterization of all vacua is explicitly calculated from propagator matrix elements. A discussion of the relation to BCS theory and Bogoliubov–Valatin transformations is given. Copyright © 2001 John Wiley & Sons, Ltd. 相似文献
3.
In this note, we establish the connection between certain quantum algebras and generalized Clifford algebras (GCA). Precisely,
we embed the quantum tori Lie algebra andU
q(sl (2)) in GCA. 相似文献
4.
G. Stacey Staples 《Advances in Applied Clifford Algebras》2005,15(2):213-232
The n-dimensional hypercube is a simple graph on 2n vertices labeled by binary strings, or words, of length n. Pairs of vertices are adjacent if and only if they differ in exactly one position as binary words; i.e., the Hamming distance
between the words is one. A discrete-time random walk is easily defined on the hypercube by “flipping” a randomly selected
digit from 0 to 1 or vice-versa at each time step. By associating the words as blades in a Clifford algebra of particular
signature, combinatorial properties of the geometric product can be used to represent this random walk as a sequence within
the algebra. A closed-form formula is revealed which yields probability distributions on the vertices of the hypercube at
any time k ≥ 0 by a formal power series expansion of elements in the algebra. Furthermore, by inducing a walk on a larger Clifford algebra,
probabilities of self-avoiding walks and expected first hitting times of specific vertices are recovered. Moreover, because the Clifford algebras used in
the current work are canonically isomorphic to fermion algebras, everything appearing here can be rewritten using fermion
creation/annihilation operators, making the discussion relevant to quantum mechanics and/or quantum computing. 相似文献
5.
6.
We introduce a generalization, called a skew Clifford algebra, of a Clifford algebra, and relate these new algebras to the notion of graded skew Clifford algebra that was defined in 2010. In particular, we examine homogenizations of skew Clifford algebras, and determine which skew Clifford algebras can be homogenized to create Artin-Schelter regular algebras. Just as (classical) Clifford algebras are the Poincaré-Birkhoff-Witt (PBW) deformations of exterior algebras, skew Clifford algebras are the -graded PBW deformations of quantum exterior algebras. We also determine the possible dimensions of skew Clifford algebras and provide several examples. 相似文献
7.
For the complex Clifford algebra (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.
The work of N.M. is supported in part by the Russian President’s grant NSh-6705.2006.1. 相似文献
8.
George Stacey Staples 《Journal of Theoretical Probability》2007,20(2):257-274
Given a fixed probability space (Ω,ℱ,ℙ) and m≥1, let X(t) be an L2(Ω) process satisfying necessary regularity conditions for existence of the mth iterated stochastic integral. For real-valued processes, these existence conditions are known from the work of D. Engel.
Engel’s work is extended here to L2(Ω) processes defined on Clifford algebras of arbitrary signature (p,q), which reduce to the real case when p=q=0. These include as special cases processes on the complex numbers, quaternion algebra, finite fermion algebras, fermion
Fock spaces, space-time algebra, the algebra of physical space, and the hypercube. Next, a graph-theoretic approach to stochastic
integrals is developed in which the mth iterated stochastic integral corresponds to the limit in mean of a collection of weighted closed m-step walks on a growing sequence of graphs. Combinatorial properties of the Clifford geometric product are then used to create
adjacency matrices for these graphs in which the appropriate weighted walks are recovered naturally from traces of matrix
powers. Given real-valued L2(Ω) processes, Hermite and Poisson-Charlier polynomials are recovered in this manner. 相似文献
9.
This paper is intended to investigate Grassmann and Clifford algebras over Peano spaces, introducing their respective associated
extended algebras, and to explore these concepts also from the counterspace viewpoint. The presented formalism explains how
the concept of chirality stems from the bracket, as defined by Rota et all [1]. The exterior (regressive) algebra is shown
to share the exterior (progressive) algebra in the direct sum of chiral and achiral subspaces. The duality between scalars
and volume elements, respectively under the progressive and the regressive products is shown to have chirality, in the case
when the dimension n of the Peano space is even. In other words, the counterspace volume element is shown to be a scalar or a pseudoscalar, depending
on the dimension of the vector space to be respectively odd or even. The de Rham cochain associated with the differential
operator is constituted by a sequence of exterior algebra homogeneous subspaces subsequently chiral and achiral. Thus we prove
that the exterior algebra over the space and the exterior algebra constructed on the counterspace are only pseudoduals each other, if we introduce chirality. The extended Clifford algebra is introduced in the light of the periodicity theorem
of Clifford algebras context, wherein the Clifford and extended Clifford algebras
can be embedded in
which is shown to be exactly the extended Clifford algebra. We present the essential character of the Rota’s bracket, relating
it to the formalism exposed by Conradt [25], introducing the regressive product and subsequently the counterspace. Clifford
algebras are constructed over the counterspace, and the duality between progressive and regressive products is presented using
the dual Hodge star operator. The differential and codifferential operators are also defined for the extended exterior algebras
from the regressive product viewpoint, and it is shown they uniquely tumble right out progressive and regressive exterior
products of 1-forms.
R. da Rocha is supported by CAPES 相似文献
10.
Bernd Schmeikal 《Advances in Applied Clifford Algebras》2001,11(1):63-80
Purpose of a minimal theory is to achief most with least. Least may be for example the spacetime algebra. But the symmetric
unitary groupSU(3) is not a part of any real Clifford algebra of 4-dimensional space, especially not of the algebraCl
1,3 of the Minkowski spacetime, nor of the algebraCl
3,1 in the opposite metric. Therefore we can ask how quantumchromodynamics enters into the theory. A first answer is that the
groupSU(3) is an object of both the complexified algebras C ⊕Cl
1,3 and C⊕Cl
3,1. To show this we first define six color spaces which are spanned by conjugate triples of commuting base elements. These contain
the six idempotent lattices that can be located inCl
3,1. Their images exist in both C⊕Cl
1,3 and C⊕Cl
3,1. Further in each color space there is defined an octahedral orientation stabilizer group which fixates one lepton and color
rotates the states in its quark family. Thus quantum numbers of strong interacting fields such as isospin, charge, hypercharge
and color turn out as geometric properties. Next we ask if the artificialty of complexification can be avoided. The answer
is yes. Defining the class of Clifford algebras with proper imaginary unit it turns out thatCl
1,3 andCl
3,1 do not belong to this class. ButCl
4,1 andCl
1,6 do. It is shown that in the latter algebra the whole color space Ansatz can be established and the generators ofSU(3) represented most naturally and without complexification. That the proposed theory becomes a physically true statement
requires that there exists a non rank preserving freedom of motion within the constituents of primitive idempotents, that
is, transpositions among conjugate triples in color space. 相似文献
11.
E. H. El Kinani 《Advances in Applied Clifford Algebras》2003,13(2):127-131
In this paper we construct the quantum Virasoro algebra
generators in terms of operators of the generalized Clifford algebras Cnk. Precisely, we show that
can be embedded into generalized Clifford algebras.
Junior Associate at The Abdus Salam ICTP, Trieste, Italy. 相似文献
12.
Bernd Schmeikal 《Advances in Applied Clifford Algebras》2006,16(2):159-165
The idempotent sets in sufficiently sophisticated algebras form manifolds and Hausdorff spaces. In this paper it is shown
how the idempotents in a real Clifford algebra Clp,q can be calculated by nilpotents and reflections. Minimal sets of nilpotents are given and generating relations are defined.
It is shown that the manifold thus constructed is complete. Every idempotent in the manifold can be calculated in the way
proposed here, namely by a nilpotent multinomial form. 相似文献
13.
Carlos Castro 《Advances in Applied Clifford Algebras》2014,24(1):55-69
An extended Orthogonal-Symplectic Clifford Algebraic formalism is developed which allows the novel construction of a graded Clifford gauge field theory of gravity. It has a direct relationship to higher spin gauge fields, bimetric gravity, antisymmetric metrics and biconnections. In one particular case it allows a plausible mechanism to cancel the cosmological constant contribution to the action. The possibility of embedding these Orthogonal-Symplectic Clifford algebras into an infinite dimensional algebra, coined Super-Clifford Algebra is described. Finally, some physical applications of the geometry of Super-Clifford spaces to Generalized Supergeometries, Double Field Theories, U-duality, 11D supergravity, M-theory, and E 7, E 8, E 11 algebras are outlined. 相似文献
14.
First, the basic concept of the vector derivative in geometric algebra is introduced. Second, beginning with the Fourier transform
on a scalar function we generalize to a real Fourier transform on Clifford multivector-valued functions
Third, we show a set of important properties of the Clifford Fourier transform on Cl3,0 such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties
for proving an uncertainty principle for Cl3,0 multivector functions. 相似文献
15.
For any finite commutative idempotent semigroup S, a semilattice, we show how to compute the amenability constant of its semigroup algebra ℓ
1(S). This amenability constant is always of the form 4n+1. We then show that these give lower bounds to amenability constants of certain Banach algebras graded over semilattices.
We also give example of a commutative Clifford semigroups G
n
whose semigroup algebras ℓ
1(G
n
) admit amenability constants of the form 41+4(n−1)/n. We also show there is no commutative semigroup whose semigroup algebra has an amenability constant between 5 and 9.
N. Spronk’s research was supported by NSERC Grant 312515-05. 相似文献
16.
Jacques Helmstetter 《Advances in Applied Clifford Algebras》2008,18(2):153-196
A meson algebra is involved in the Duffin wave equation for mesons in the same way as a Clifford algebra is involved in the
Dirac wave equation for electrons. Therefore meson algebras too should have geometrical properties after the manner of Grassmann.
Actually it is possible to define interior multiplications with similar properties, and deformations too. Every meson algebra
is a deformation of a neutral meson algebra, in the same way as (almost) every Clifford algebra is a deformation of an exterior
algebra. Some applications follow: the PBW-property is proved for all meson algebras, the injectiveness of Jacobson’s diagonal
morphism is proved with the minimal hypothesis, and the existence of Lipschitz monoids is established at least for meson algebras
over fields.
相似文献
17.
CLIFFORD performs various computations in Gra?mann and Clifford algebras. It can compute with quaternions, octonions, and
matrices with entries in C ℓ (B) - the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for the Clifford product are implemented: cmulNUM - based on Chevalley’s recursive formula, and cmulRS - based on a non-recursive Rota-Stein sausage. Gra?mann and Clifford bases can be used. Properties of reversion in undotted
and dotted wedge bases are discussed. 相似文献
18.
Kevin De Laet 《代数通讯》2013,41(10):4258-4282
In this article, we study graded Clifford algebras with a gradation preserving action of automorphisms given by H p , the Heisenberg group of order p 3 with p prime. After reviewing results in dimensions 3 and 4, we will determine the graded Clifford algebras that are AS-regular algebras of global dimension 5 and generalize certain results to arbitrary dimension p. 相似文献
19.
V. V. Varlamov 《Advances in Applied Clifford Algebras》2014,24(3):849-874
Supergroups are defined in the framework of \({\mathbb{Z}_2}\) 2-graded Clifford algebras over the fields of real and complex numbers, respectively. It is shown that cyclic structures of complex and real supergroups are defined by Brauer-Wall groups related with the modulo 2 and modulo 8 periodicities of the complex and real Clifford algebras. Particle (fermionic and bosonic) representations of a universal covering (spinor group Spin +(1, 3)) of the proper orthochronous Lorentz group are constructed via the Clifford algebra formalism. Complex and real supergroups are defined on the representation system of Spin +(1, 3). It is shown that a cyclic (modulo 2) structure of the complex supergroup is equivalent to a supersymmetric action, that is, it converts fermionic representations into bosonic representations and vice versa. The cyclic action of the real supergroup leads to a much more high-graded symmetry related with the modulo 8 periodicity of the real Clifford algebras. This symmetry acts on the system of real representations of Spin +(1, 3). 相似文献
20.
A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree, so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, z-Poisson algebras, Gerstenhaber algebras, and Schouten algebras among other classes of algebras. The present paper is devoted to the study of structure of Poisson color algebras of degree g0, where g0 is some element of the grading group G such that g0 = 0 or 4g0≠0, and with restrictions neither on the dimension nor the base field, by stating a second Wedderburn-type theorem for this class of algebras. 相似文献