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Discrete symmetries and Clifford algebras. (English) Zbl 0984.81172

Summary: 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 automorphism groups is established for the Clifford algebras over the field of real numbers. The correspondence between eight double coverings (Dąbrowski 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.

MSC:

81T99 Quantum field theory; related classical field theories
81R20 Covariant wave equations in quantum theory, relativistic quantum mechanics
15A66 Clifford algebras, spinors
11E88 Quadratic spaces; Clifford algebras
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