Abstract
Real Clifford algebras for arbitrary numbers of space and time dimensions as well as their representations in terms of spinors are reviewed and discussed. The Clifford algebras are classified in terms of isomorphic matrix algebras of real, complex or quaternionic type. Spinors are defined as elements of minimal or quasi-minimal left ideals within the Clifford algebra and as representations of the pin and spin groups. Two types of Dirac adjoint spinors are introduced carefully. The relationship between mathematical structures and applications to describe relativistic fermions is emphasized throughout.
Highlights
As we will point out, the structure map plays an important role in defining a consistent action of the pin group on arbitrary elements of the Clifford algebra and in particular on spinor representations as left and right ideals
G can be realized as a linear map which may either be given by the product of all generators γ, complex conjugation, or the interchange of the two summands when the Clifford algebra has the structure of a direct sum
The inductive construction in the previous subsection has shown that the real Clifford algebras C l (r, d − r, R) can be constructed as different tensor products of the basis elements C l (0, 2, R) ∼
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. For even number of dimensions d, this structure map is just an element of the Clifford algebra itself and can be taken to be the product of all time-like and space-like generators. As we will point out, the structure map plays an important role in defining a consistent action of the pin group on arbitrary elements of the Clifford algebra and in particular on spinor representations as left and right ideals. For d − r even, space reversion (i.e., the reflection along all time-like coordinate axis) P does not connect different topologically disconnected elements of the group but for d − r odd this is the case.
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