Guest Editor's Note: Clifford Algebras and Their Applications (original) (raw)

On Generalized Clifford Algebras and their Physical Applications

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.

Clifford Algebras and their Applications in Mathematical Physics

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.

Signature Change and Clifford Algebras

International Journal of Theoretical Physics - INT J THEOR PHYS, 2001

Given the real Clifford algebra of a quadratic space with a given signature, we define a new product in this structure such that it simulates the Clifford product of a quadratic space with another signature different from the original one. Among the possible applications of this new product, we use it in order to write the Minkowskian Dirac equation over the Euclidean spacetime and to define a new duality operation in terms of which one can find self-dual and anti-self-dual solutions of gauge fields over Minkowski spacetime analogous to the ones over Euclidean spacetime and without needing to complexify the original real algebra.

An Introduction to Clifford Algebras and Spinors

2016

This book is unique in the literature on spinors and Clifford algebras in that it is accessible to both students and researchers while maintaining a formal approach to these subjects. Besides thoroughly introducing several aspects of Clifford algebras, it provides the geometrical aspects underlying the Clifford algebras, as well as their applications, particularly in physics. Previous books on spinors and Clifford algebras have either required the reader to have some prior expertise in these subjects, and thus were difficult to access, or did not provide a deep approach. In contrast, although this book is mathematically complete and precise, it demands little in the way of prerequisites—indeed, a course in linear algebra is the sole prerequisite. This book shows how spinors and Clifford algebras have fuelled interest in the no man’s land between physics and mathematics, an interest resulting from the growing awareness of the importance of algebraic and geometric properties in many p...

ON CLIFFORD SUBALGEBRAS, SPACETIME SPLITTINGS AND APPLICATIONS

International Journal of Geometric Methods in Modern Physics, 2006

Z 2 -gradings of Clifford algebras are reviewed and we shall be concerned with an αgrading based on the structure of inner automorphisms, which is closely related to the spacetime splitting, if we consider the standard conjugation map automorphism by an arbitrary, but fixed, splitting vector. After briefly sketching the orthogonal and parallel components of products of differential forms, where we introduce the parallel [orthogonal] part as the space [time] component, we provide a detailed exposition of the Dirac operator splitting and we show how the differential operator parallel and orthogonal components are related to the Lie derivative along the splitting vector and the angular momentum splitting bivector. We also introduce multivectorial-induced α-gradings and present the Dirac equation in terms of the spacetime splitting, where the Dirac spinor field is shown to be a direct sum of two quaternions. We point out some possible physical applications of the formalism developed.