Vacuum and spacetime signature in the theory of superalgebraic spinors

Abstract

We investigated action of operator analogs of Dirac gamma matrices (we called them gamma operators) on a vacuum. We derived formulas for vacuum state vector and operators of the Lorentz transformations of spinors in the superalgebraic representation of spinors. Five operator analogs of five Dirac gamma matrices exist in the superalgebraic approach as well as two additional operator analogs of gamma matrices. Gamma operators are constructed from Grassmann densities and derivatives with respect to them. We have shown that there are operators which are built from creation and annihilation operators, and that they are also analogs of Dirac gamma matrices. However, unlike gamma operators of the first kind, they are Lorentz invariant. We have shown that the condition for the existence of spinor vacuum imposes restrictions on possible variants of Clifford algebras of gamma operators: only real algebra with one timelike basis Clifford vector corresponding to the zero gamma matrix in the Dirac representation can be realized. In this case, the signature of the four-dimensional spacetime, in which there is a vacuum state, can only be (1, -1, -1, -1), and there are two additional axes corresponding to the inner space of the spinor, with a signature (-1, -1).