A differential Galois approach to path integrals
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 [MoralesRuiz and Ramis, Methods Appl. Anal. 8, 3396 (2001); see also MoralesRuiz, in Differential Galois Theory and NonIntegrability of Hamiltonian Systems, Modern Birkhäuser Classics (Springer, Basel, 1999)]; 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 a 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.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 May 2020
 DOI:
 10.1063/1.5134859
 arXiv:
 arXiv:1910.06365
 Bibcode:
 2020JMP....61e2103M
 Keywords:

 Mathematical Physics;
 81S40;
 37J30;
 37J35
 EPrint:
 17 pages