Getting path integrals physically and technically right

TL;DR

This paper develops a rigorous method to derive the Schrödinger equation from Hamiltonian path integrals, clarifying the role of subpath endpoint constraints and providing a consistent quantization approach.

Contribution

It introduces a novel recasting of Hamiltonian phase-space path integrals as unconstrained functional integrals, ensuring unique quantization and resolving misconceptions about endpoint constraints.

Findings

Derived the Schrödinger equation from Hamiltonian path integrals with first-order accuracy.

Provided a method to evaluate quantum amplitudes for configuration and momentum paths as ordinary functional integrals.

Clarified the misconception about endpoint constraints in short-time path segments.

Abstract

Feynman's Lagrangian path integral was an outgrowth of Dirac's vague surmise that Lagrangians have a role in quantum mechanics. Lagrangians implicitly incorporate Hamilton's first equation of motion, so their use contravenes the uncertainty principle, but they are relevant to semiclassical approximations and relatedly to the ubiquitous case that the Hamiltonian is quadratic in the canonical momenta, which accounts for the Lagrangian path integral's "success". Feynman also invented the Hamiltonian phase-space path integral, which is fully compatible with the uncertainty principle. We recast this as an ordinary functional integral by changing direct integration over subpaths constrained to all have the same two endpoints into an equivalent integration over those subpaths' unconstrained second derivatives. Function expansion with generalized Legendre polynomials of time then enables the functional integral to be unambiguously evaluated through first order in the elapsed time, yielding the Schroedinger equation with a unique quantization of the classical Hamiltonian. Widespread disbelief in that uniqueness stemmed from the mistaken notion that no subpath can have its two endpoints arbitrarily far separated when its nonzero elapsed time is made arbitrarily short. We also obtain the quantum amplitude for any specified configuration or momentum path, which turns out to be an ordinary functional integral over, respectively, all momentum or all configuration paths. The first of these results is directly compared with Feynman's mistaken Lagrangian-action hypothesis for such a configuration path amplitude, with special heed to the case that the Hamiltonian is quadratic in the canonical momenta.