The Quantum Darboux Theorem,

TL;DR

This paper introduces a quantum Darboux theorem that simplifies complex quantum dynamics via gauge transformations on phase-spacetime, using a novel diagrammatic approach for anharmonic potentials.

Contribution

It develops a quantum analogue of the Darboux theorem, providing a method to trivialize local quantum dynamics through gauge transformations on contact manifolds.

Findings

Develops a diagrammatic method for asymptotic gauge transformations.

Applies the theorem to anharmonic quantum potentials.

Provides a framework for simplifying quantum propagators.

Abstract

The problem of computing quantum mechanical propagators can be recast as a computation of a Wilson line operator for parallel transport by a flat connection acting on a vector bundle of wavefunctions. In this picture the base manifold is an odd dimensional symplectic geometry, or quite generically a contact manifold that can be viewed as a "phase-spacetime", while the fibers are Hilbert spaces. This approach enjoys a "quantum Darboux theorem" that parallels the Darboux theorem on contact manifolds which turns local classical dynamics into straight lines. We detail how the quantum Darboux theorem works for anharmonic quantum potentials. In particular, we develop a novel diagrammatic approach for computing the asymptotics of a gauge transformation that locally makes complicated quantum dynamics trivial.