Spin in an arbitrary gravitational field

TL;DR

This paper develops a quantum mechanical model for a Dirac fermion in any curved spacetime, deriving the Hamiltonian and equations of motion, and shows complete agreement with classical spin dynamics in gravitational fields.

Contribution

It provides a general Hermitian Dirac Hamiltonian and quantum equations of motion for spin in arbitrary gravitational fields, bridging quantum and classical descriptions.

Findings

Quantum and classical spin dynamics agree in arbitrary gravitational fields.

Derived a Hermitian Dirac Hamiltonian for arbitrary external fields.

Established consistency between quantum equations and classical spin theories.

Abstract

We study the quantum mechanics of a Dirac fermion on a curved spacetime manifold. The metric of the spacetime is completely arbitrary, allowing for the discussion of all possible inertial and gravitational field configurations. In this framework, we find the Hermitian Dirac Hamiltonian for an arbitrary classical external field (including the gravitational and electromagnetic ones). In order to discuss the physical content of the quantum-mechanical model, we further apply the Foldy-Wouthuysen transformation, and derive the quantum equations of motion for the spin and position operators. We analyse the semiclassical limit of these equations and compare the results with the dynamics of a classical particle with spin in the framework of the standard Mathisson-Papapetrou theory and in the classical canonical theory. The comparison of the quantum mechanical and classical equations of motion of a spinning particle in an arbitrary gravitational field shows their complete agreement.