Paths of Canonical Transformations and their Quantization

TL;DR

This paper explores the deep connections between classical Hamiltonian flows and quantum propagators, emphasizing the role of the metaplectic representation and providing explicit formulas for flow factorization and their quantum analogs.

Contribution

It offers a detailed analysis of the analogy between Hamiltonian flows and quantum propagators, including explicit formulas and the quantum counterpart of flow factorizations.

Findings

Explicit formulas for Hamiltonian flow factorization

Quantum counterparts of classical flow decompositions

Clarification of the relationship between classical and quantum mechanics

Abstract

In their simplest formulations, classical dynamics is the study of Hamiltonian flows and quantum mechanics that of propagators. Both are linked, and emerge from the datum of a single classical concept, the Hamiltonian function. We study and emphasize the analogies between Hamiltonian flows and quantum propagators; this allows us to verify G. Mackey's observation that quantum mechanics (in its Weyl formulation) is a refinement of Hamiltonian mechanics. We discuss in detail the metaplectic representation, which very explicitly shows the close relationship between classical mechanics and quantum mechanics, the latter emerging from the first by lifting Hamiltonian flows to the double covering of the symplectic group. We also give explicit formulas for the factorization of Hamiltonian flows into simpler flows, and prove a quantum counterpart of these results.