Solution of the problem of uniqueness and hermiticity of hamiltonians for Dirac particles in gravitational fields

TL;DR

This paper demonstrates that the dynamics of spin 1/2 particles in stationary gravitational fields can be consistently described using pseudo-Hermitian Hamiltonians, showing their equivalence to the Parker scalar product approach and addressing Hamiltonian non-uniqueness.

Contribution

It proves the equivalence of pseudo-Hermitian Hamiltonian formalism and Parker scalar product for Dirac particles in gravitational fields, resolving non-uniqueness issues.

Findings

Pseudo-Hermitian Hamiltonians yield the same spectrum as Dirac Hamiltonians.

Different tetrad choices lead to non-unique Hamiltonians.

The Parker scalar product approach is equivalent to pseudo-Hermitian formalism.

Abstract

The authors prove that the dynamics of spin 1/2 particles in stationary gravitational fields can be described using an approach, which builds upon the formalism of pseudo-Hermitian Hamiltonians. The proof consists in the analysis of three expressions for Hamiltonians, which are derived from the Dirac equation and describe the dynamics of spin 1/2 particles in the gravitational field of the Kerr solution. The Hamiltonians correspond to different choices of tetrad vectors and differ from each other. The differences between the Hamiltonians confirm the conclusion known from many studies that the Hamiltonians derived from the Dirac equation are non-unique. Application of standard pseudo-Hermitian quantum mechanics rules to each of these Hamiltonians produces the same Hermitian Hamiltonian. The eigenvalue spectrum of the resulting Hamiltonian is the same as that of the Hamiltonians derived from the Dirac equation with any chosen system of tetrad vectors. For description of the dynamics of spin 1/2 particles in stationary gravitational fields can be used not only the formalism of pseudo-Hermitian Hamiltonians, but also an alternative approach, which employs the Parker scalar product. The authors show that the alternative approach is equivalent to the formalism of pseudo-Hermitian Hamiltonians.