A Note on a Differential Galois Approach to Path Integrals

TL;DR

This paper explores how Differential Galois Theory can determine the exactness of semiclassical path integral computations in quantum mechanics, especially for integrable Hamiltonian systems, by linking integrability to Galois group properties.

Contribution

It establishes a necessary condition connecting the integrability of classical Hamiltonian systems with the abelian nature of the Galois group, enabling closed-form semiclassical calculations.

Findings

Galois group of variational equations must be abelian for integrable systems

Semiclassical path integrals are computable in closed form for integrable systems

Provides a theoretical foundation for the success of semiclassical methods in quantum mechanics

Abstract

We point out the relevance of the Differential Galois Theory of linear differential equations for the exact semiclassical computations in path integrals in quantum mechanics. The main tool will be a necessary condition for complete integrability of classical Hamiltonian systems obtained by Ramis and myself : if a finite dimensional complex analytical Hamiltonian system is completely integrable with meromorphic first integrals, then the identity component of the Galois group of the variational equation around any integral curve must be abelian. A corollary of this result is that, for finite dimensional integrable Hamiltonian systems, the semiclassical approach is computable in closed form in the framework of the Differential Galois Theory. This explains in a very precise way the success of quantum semiclassical computations for integrable Hamiltonian systems.