Path integration in relativistic quantum mechanics

TL;DR

This paper calculates the exact propagator for a free relativistic particle, revealing nonlocal features and complexities in the path integral formulation of relativistic quantum mechanics, especially outside the light cone.

Contribution

It provides the exact relativistic propagator including spacelike trajectories and analyzes its relation to the WKB approximation, highlighting unique relativistic features.

Findings

Propagator is nonzero outside the light cone.

Short-time limit differs from WKB approximation.

Spacelike geodesics contribute significantly to the path integral.

Abstract

The simple physics of a free particle reveals important features of the path-integral formulation of relativistic quantum theories. The exact quantum-mechanical propagator is calculated here for a particle described by the simple relativistic action proportional to its proper time. This propagator is nonvanishing outside the light cone, implying that spacelike trajectories must be included in the path integral. The propagator matches the WKB approximation to the corresponding configuration-space path integral far from the light cone; outside the light cone that approximation consists of the contribution from a single spacelike geodesic. This propagator also has the unusual property that its short-time limit does not coincide with the WKB approximation, making the construction of a concrete skeletonized version of the path integral more complicated than in nonrelativistic theory.