# Vacuum and spacetime signature in the theory of superalgebraic spinors

## TL;DR

This paper explores the structure of gamma operators in superalgebraic spinors, revealing how vacuum conditions restrict spacetime signatures and the algebraic form of gamma matrices in a superalgebraic framework.

## Contribution

It introduces a superalgebraic approach to gamma matrices, deriving vacuum state formulas and Lorentz transformations, and shows how vacuum conditions constrain spacetime signatures and Clifford algebra variants.

## Key findings

- Existence of gamma operators from Grassmann variables and derivatives.
- Lorentz-invariant operators built from creation and annihilation operators.
- Spacetime signature restricted to (1, -1, -1, -1) under vacuum conditions.

## 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).