The phase of a quantum mechanical particle in curved spacetime

TL;DR

This paper examines the quantum phase of particles with different spins in curved spacetime, finding that only spin 1/2 particles have an exact phase expression, with higher spins requiring approximations.

Contribution

It provides a detailed analysis of the quantum phase in curved spacetime for various spins, highlighting the exactness for spin 1/2 and approximations for others.

Findings

Spin 1/2 particles have an exact quantum phase in curved spacetime.

Spin 0 and 1 particles require WKB approximation for phase analysis.

Derived phase expressions applicable to neutrino oscillations.

Abstract

We investigate the quantum mechanical wave equations for free particles of spin 0,1/2,1 in the background of an arbitrary static gravitational field in order to explicitly determine if the phase of the wavefunction is $S/\hbar = \int p_{\mu} dx^{\mu} / \hbar$, as is often quoted in the literature. We work in isotropic coordinates where the wave equations have a simple managable form and do not make a weak gravitational field approximation. We interpret these wave equations in terms of a quantum mechanical particle moving in medium with a spatially varying effective index of refraction. Due to the first order spatial derivative structure of the Dirac equation in curved spacetime, only the spin 1/2 particle has \textit{exactly} the quantum mechanical phase as indicated above. The second order spatial derivative structure of the spin 0 and spin 1 wave equations yield the above phase only to lowest order in $\hbar$. We develop a WKB approximation for the solution of the spin 0 and spin 1 wave equations and explore amplitude and phase corrections beyond the lowest order in $\hbar$. For the spin 1/2 particle we calculate the phase appropriate for neutrino flavor oscillations.