Canonical quantization of the relativistic particle in static spacetimes

TL;DR

This paper performs canonical quantization of a relativistic particle in static curved spacetime, revealing classical particle-antiparticle duality and constructing a two-particle Hilbert space, with implications for quantum field theory in curved backgrounds.

Contribution

It introduces a canonical quantization framework for relativistic particles in static spacetimes, addressing constraints and Hilbert space construction, and clarifies ordering ambiguities.

Findings

Classical theory describes both particle and antiparticle.

Constructed two-particle Hilbert space in static spacetime.

Identified ordering ambiguities similar to non-relativistic case.

Abstract

We perform the canonical quantization of a relativistic spinless particle moving in a curved and static spacetime. We show that the classical theory already describes at the same time both particle and antiparticle. The analyses involves time-depending constraints and we are able to construct the two-particle Hilbert space. The requirement of a static spacetime is necessary in order to have a well defined Schr\"odinger equation and to avoid problems with vacuum instabilities. The severe ordering ambiguities we found are in essence the same ones of the well known non-relativistic case.