Timeless path integral for relativistic quantum mechanics

TL;DR

This paper rigorously derives a timeless path integral formulation for relativistic quantum mechanics, emphasizing its independence from parametrization and connecting it to classical action, thereby clarifying differences from nonrelativistic quantum mechanics.

Contribution

The paper introduces a rigorous derivation of the timeless path integral for relativistic quantum mechanics, extending Feynman's approach to a parametrization-independent framework.

Findings

Transition amplitude expressed as a sum over paths in the constraint surface.

Path integral is independent of path parametrization.

Special case yields a relativistic Feynman path integral over configuration space.

Abstract

Starting from the canonical formalism of relativistic (timeless) quantum mechanics, the formulation of timeless path integral is rigorously derived. The transition amplitude is reformulated as the sum, or functional integral, over all possible paths in the constraint surface specified by the (relativistic) Hamiltonian constraint, and each path contributes with a phase identical to the classical action divided by $\hbar$. The timeless path integral manifests the timeless feature as it is completely independent of the parametrization for paths. For the special case that the Hamiltonian constraint is a quadratic polynomial in momenta, the transition amplitude admits the timeless Feynman's path integral over the (relativistic) configuration space. Meanwhile, the difference between relativistic quantum mechanics and conventional nonrelativistic (with time) quantum mechanics is elaborated on in light of timeless path integral.