Symbolic Computations in Higher Dimensional Clifford Algebras

International Journal of Theoretical Physics, 2001

The history and immediate future of the International Conferences on Clifford Algebras and Their Applications.

2010

Geometric algebra (also known as Clifford algebra) is a powerful mathematical tool that offers a natural and direct way to model geometric objects and their transformations. It is gaining growing attention in different research fields as physics, robotics, CAD/CAM and computer graphics. Clifford algebra makes geometric objects (points, lines and planes) into basic elements of computation and defines few universal operators that are applicable to all types of geometric elements. This paper provides an introduction to Clifford algebra elements and operators.

Advances in Applied Clifford Algebras, 2016

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 G 3 ≡ Cℓ3,0 as a computational model in the Maxima computer algebra system. We compare the geometric algebra-based approach with conventional symbolic tensor calculations supported by Maxima, based on the itensor package. The Clifford algebra functionality of Maxima is distributed as two new packages called clifford-for basic simplification of Clifford products, outer products, scalar products and inverses; and cliffordan-for applications of geometric calculus.

In this paper we focus on some combinatorial aspects of Clifford algebra and show how this algebra allows combinatorial theorems -like e.g. Sperner's lemma -to be "built into the algebraic background", and become part of the structure of the algebra itself. We also give an example of how cumbersome combinatorial proofs can be "mechanized" and carried out in a purely computational manner.

1987

Today certain computer software systems, commonly called Computer Algebra Systems or Symbolic Mathematical Systems, compute not only with numbers but also manipulate formulae and return answers in terms of symbols and algebraic or analytic expressions. They are characterized by exact computation rather than numerical approximation, and they can perform a large portion of the calculations usually done by hand.

Advances in Applied Clifford Algebras, 2014

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 and cmulRS from CLIFFORD. We comment on how to improve cmulWpar by taking advantage of multi-core processors and multithreading available in modern processors.

2008

Since the spinor groups are certain well chosen subgroups of units of Clifford algebras, it is necessary to investigate Clifford algebras to get a firm understanding of spinor groups. These notes provide a tutorial on Clifford algebra and the groups Spin and Pin, including a study of the structure of the Clifford algebra Cl_{p, q} associated with a nondegenerate symmetric bilinear form of signature (p, q) and culminating in the beautiful "8-periodicity theorem" of Elie Cartan and Raoul Bott (with proofs).

Clifford Algebras and their Applications in Mathematical Physics, 2000

Clifford algebras are naturally associated with quadratic forms. These algebras are Z2 -graded by construction. However, only a Zn -gradation induced by a choice of a basis, or even better, by a Chevalley vector space isomorphism Cℓ(V ) ↔ V and an ordering, guarantees a multivector decomposition into scalars, vectors, tensors, and so on, mandatory in physics. We show that the Chevalley isomorphism theorem cannot be generalized to algebras if the Zn -grading or other structures are added, e.g., a linear form. We work with pairs consisting of a Clifford algebra and a linear form or a Zn -grading which we now call Clifford algebras of multivectors or quantum Clifford algebras. It turns out, that in this sense, all multi-vector Clifford algebras of the same quadratic but different bilinear forms are non-isomorphic. The usefulness of such algebras in quantum field theory and superconductivity was shown elsewhere. Allowing for arbitrary bilinear forms however spoils their diagonalizability which has a considerable effect on the tensor decomposition of the Clifford algebras governed by the periodicity theorems, including the Atiyah-Bott-Shapiro mod 8 periodicity. We consider real algebras Cℓp,q which can be decomposed in the symmetric case into a tensor product Cℓp-1,q-1 ⊗ Cℓ1,1. The general case used in quantum field theory lacks this feature. Theories

2000

Clifford algebras and their applications in mathematical physics, p. cm.-(Progress in physics ; v. 18-19) Includes bibliographical references and indexes. Contents: v. 1. Algebra and physics / Rafal Ablamowicz, Bertfried Fauser, editors-v. 2. Clifford analysis / John Ryan, Wolfgang Sprößig, editors.

The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, 2010

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 σ-operation approach to the representation theory of Clifford algebra and GCAs, Dirac's positive energy relativistic wave equation, Weyl-Schwinger unitary basis for matrix algebra and Alladi Ramakrishnan's matrix decomposition theorem, finite-dimensional Wigner function, finite-dimensional canonical transformations, magnetic Bloch functions, finite-dimensional quantum mechanics, and the relation between GCAs and quantum groups.