On the existence of the second Dirac operator in Riemannian space

TL;DR

This paper investigates the conditions under which a second Dirac operator exists in Riemannian spaces, demonstrating its equivalence to the standard Dirac operator, and explores implications for particle states and supersymmetry.

Contribution

It introduces a class of Riemannian spaces where the second Dirac operator appears and characterizes when it is equivalent to the standard Dirac operator, including specific examples.

Findings

Second Dirac operator exists in certain Riemannian spaces.

The second Dirac operator is always equivalent to the standard Dirac operator in these spaces.

Identified metrics with a five-dimensional motion group where the second Dirac operator exists.

Abstract

We describe a Riemannian space class where the second Dirac operator arises and prove that the operator is always equivalent to a standard Dirac one. The particle state in this gravitational field is degenerate to some extent and we introduce an additional value in order to describe a particle state completely. Some supersymmetry constructions are also discussed. As an example we study all Riemannian spaces with a five-dimentional motion group and find all metrics for which the second Dirac operator exists. On the basis of our discussed examples we hypothesize about the number of second Dirac operators in Riemannian space.