Uniqueness and Self-Conjugacy of Dirac Hamiltonians in arbitrary Gravitational Fields

TL;DR

This paper proves that Dirac particles in arbitrary gravitational fields can be described using pseudo-Hermitian quantum mechanics, ensuring the Hamiltonians are unique and self-conjugate, with applications to various spacetime models.

Contribution

It demonstrates the use of the Parker weight operator and ta-representation to achieve self-conjugate Hamiltonians in arbitrary gravitational fields, including nonstationary metrics.

Findings

Dirac Hamiltonians are unique and self-conjugate in arbitrary gravitational fields.

The ta-representation yields flat scalar products, facilitating Hermitian quantum mechanics.

Applications to stationary, cosmologically flat, and open Friedmann models.

Abstract

Proofs of two statements are provided in this paper. First, the authors prove that the formalism of the pseudo-Hermitian quantum mechanics allows describing the Dirac particles motion in arbitrary stationary gravitational fields. Second, it is proved that using the Parker weight operator and the subsequent transition to the \eta -representation gives the transformation of the Schroedinger equation for nonstationary metric, when the evolution operator becomes self-conjugate. The scalar products in the \eta -representation are flat, which makes possible the use of a standard apparatus for the Hermitian quantum mechanics. Based on the results of this paper the authors draw a conclusion about solution of the problem of uniqueness and self-conjugacy of Dirac Hamiltonians in arbitrary gravitational fields including those dependent on time. The general approach is illustrated by the example of Dirac Hamiltonians for several stationary metrics, as well as for the cosmologically flat and the open Friedmann models.