Algebraic structure of the Feynman propagator and a new correspondence for canonical transformations

TL;DR

This paper explores the algebraic structure of the Feynman propagator for time-dependent quadratic Hamiltonians, revealing a new classical-quantum correspondence through a Lie-algebraic approach and the IWOP technique.

Contribution

It introduces a novel algebraic framework and a new classical-quantum correspondence for the Feynman propagator using Lie algebra and IWOP methods.

Findings

Derived a normal-ordered form of the time-evolution operator

Established a new classical-quantum correspondence via the propagator

Demonstrated the use of Lie algebraic techniques in quantum propagator analysis

Abstract

We investigate the algebraic structure of the Feynman propagator with a general time-dependent quadratic Hamiltonian system. Using the Lie-algebraic technique we obtain a normal-ordered form of the time-evolution operator, and then the propagator is easily derived by a simple ``Integration Within Ordered Product" (IWOP) technique.It is found that this propagator contains a classical generating function which demonstrates a new correspondence between classical and quantum mechanics.