Superalgebraic structure of Lorentz transformations

TL;DR

This paper develops a superalgebraic framework for Lorentz transformations and Dirac matrices, providing explicit algebraic forms and revealing asymmetries related to the time axis in relativistic quantum theory.

Contribution

It introduces a superalgebraic representation of second quantization for spinors, expressing Dirac matrices and Lorentz generators via Grassmann densities and derivatives.

Findings

Dirac matrices expressed in terms of Grassmann densities

Superalgebraic form of the Dirac equation constructed

Asymmetry in Clifford algebra generators related to time axis

Abstract

Modern relativistic theory of the second quantization of fermion and boson fields is based on the use of the mathematical apparatus of C*-algebras and Lie superalgebras. In this case, for fermions, the Lorentz transformations are considered as Bogolyubov transformations of creation and annihilation operators. However, in this approach one can not obtain an explicit form of the Dirac gamma-matrices. The mathematical apparatus of the superalgebraic representation of the algebra of the second quantization of spinors is developed in the article. It is based on the use of density in the impulse space of Grassmann variables and their derivatives. It is shown that the Dirac matrices and the Lorentz transformation generators can be expressed in terms of such densities. A superalgebraic form of the Dirac equation and the vacuum state vector are constructed. It is shown that in the superalgebraic form of the complex Clifford algebra the generators corresponding to the Dirac gamma matrices are not equivalent. Clifford vector corresponding to diagonal matrix annihilates the vacuum, and the remaining ones give nonzero values. This means that there is asymmetric direction corresponding to the time axis