Classical Mechanics in Hilbert Space: Path Integral Formulation, and a Quantum Correction

TL;DR

This paper demonstrates a path integral formulation for classical mechanics based on Koopman-von Neumann theory, introduces a quantum correction for semi-classical analysis, and proposes an efficient numerical implementation.

Contribution

It extends the path integral approach to classical mechanics using Hilbert space formalism and derives a semi-classical correction incorporating quantum effects.

Findings

Path integral representation of classical propagator derived

Efficient numerical implementation using Fourier transforms proposed

First quantum correction to classical mechanics formulated

Abstract

While it is well-known that quantum mechanics can be reformulated in terms of a path integral representation, it will be shown that such a formulation is also possible in the case of classical mechanics. From Koopman-von Neumann theory, which recasts classical mechanics in terms of a Hilbert space wherein the Liouville operator acts as the generator of motion, we derive a path integral representation of the classical propagator and suggest an efficient numerical implementation using fast fourier transform techniques. We then include a first quantum correction to derive a revealing expression for the semi-classical path integral, which augments the classical picture of a single trajectory through phase space with additional wave-like spreading.