Discrete Symmetries and Clifford Algebras

TL;DR

This paper provides an algebraic framework for understanding discrete symmetries using Clifford algebras, classifies automorphism groups over real numbers, and explores their relation to orthogonal groups and periodicity phenomena.

Contribution

It offers a complete classification of automorphism groups of real Clifford algebras and links these to double coverings of orthogonal groups, including Lorentz groups, revealing new algebraic structures.

Findings

Classification of automorphism groups for real Clifford algebras.

Correspondence between Clifford algebras and orthogonal group coverings.

Connection to Atiyah-Bott-Shapiro periodicity.

Abstract

An algebraic description of basic discrete symmetries (space reversal P, time reversal T and their combination PT) is studied. Discrete subgroups of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. In accordance with a division ring structure, a complete classification of automorphisms groups is established for the Clifford algebras over the field of real numbers. The correspondence between eight double coverings (Dabrowski groups) of the orthogonal group and eight types of the real Clifford algebras is defined with the use of isomorphisms between the automorphism groups and finite groups. Over the field of complex numbers there is a correspondence between two nonisomorphic double coverings of the complex orthogonal group and two types of complex Clifford algebras. It is shown that these correspondences associate with a well-known Atiyah-Bott-Shapiro periodicity. Generalized Brauer-Wall groups are introduced on the extended sets of the Clifford algebras. The structure of the inequality between the two Clifford-Lipschitz groups with mutually opposite signatures is elucidated. The physically important case of the two different double coverings of the Lorentz groups is considered in details.