On generalized Clifford algebras and their physical applications

TL;DR

This paper reviews the theory of generalized Clifford algebras (GCAs), their mathematical structures, and their diverse applications in quantum mechanics, group representations, and quantum groups, highlighting historical developments and key concepts.

Contribution

It provides a comprehensive overview of GCAs, including their representation theory, matrix decompositions, and connections to quantum groups, consolidating past research and applications.

Findings

GCAs relate to projective representations of finite abelian groups

Matrix decomposition theorems facilitate GCA analysis

GCAs connect to finite-dimensional quantum mechanics and quantum groups

Abstract

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 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.