Spinors as fundamental objects

Abstract

We define, on the algebraic Dirac spinor space Ψ, some operators D± and T±. By means of them we show how the fundamental operations of Hermitian conjugation, complex conjugation, bar conjugation, and so on may be introduced in the Clifford algebra approach. These definitions depend on the geometrical property of the pure imaginary unit i of the Dirac algebra Ds,t and are the same only for mod 4 dimensions of vector space-times Rs,t. Furthermore, on the set Ψ×Ψ we introduce equivalence relations R± and define bijections χ± between Ψ×Ψ/R± and Ds,t. We investigate some properties of χ± and give the necessary and sufficient conditions for u∈Ds,t to belong to some minimal left ideal of Ds,t. Next we use the decomposition of the Dirac algebra Ds,t onto the Dirac spinor spaces to demonstrate two different ways of an action of any element s∈Spin(x,t). These considerations throw a new light onto the problem of the covariant derivative on the bundle of algebraic spinors over a space-time manifold.