The Quantum Propagator for a Nonrelativistic Particle in the Vicinity of a Time Machine

TL;DR

This paper analyzes the quantum propagator of a non-relativistic particle in a spacetime with time machine features, demonstrating unitarity in regions without closed timelike curves and providing explicit expressions and proofs.

Contribution

It explicitly derives the propagator in a time-machine spacetime and proves unitarity in initial and final regions, extending classical evolution concepts to non-relativistic quantum mechanics.

Findings

Propagator satisfies unitarity outside the time machine

Explicit expression for the propagator in wormhole spacetime

Unitarity fails within the time machine region

Abstract

We study the propagator of a non-relativistic, non-interacting particle in any non-relativistic ``time-machine'' spacetime of the type shown in Fig.~1: an external, flat spacetime in which two spatial regions, $V_-$ at time $t_-$ and $V_+$ at time $t_+$, are connected by two temporal wormholes, one leading from the past side of $V_-$ to t the future side of $V_+$ and the other from the past side of $V_+$ to the future side of $V_-$. We express the propagator explicitly in terms of those for ordinary, flat spacetime and for the two wormholes; and from that expression we show that the propagator satisfies completeness and unitarity in the initial and final ``chronal regions'' (regions without closed timelike curves) and its propagation from the initial region to the final region is unitary. However, within the time machine it satisfies neither completeness nor unitarity. We also give an alternative proof of initial-region-to-final-region unitarity based on a conserved current and Gauss's theorem. This proof can be carried over without change to most any non-relativistic time-machine spacetime; it is the non-relativistic version of a theorem by Friedman, Papastamatiou and Simon, which says that for a free scalar field, quantum mechanical unitarity follows from the fact that the classical evolution preserves the Klein-Gordon inner product.