# Dirac's formalism for time-dependent Hamiltonian systems in the extended phase space

## TL;DR

This paper applies Dirac's formalism to analyze time-dependent Hamiltonian systems in extended phase space, demonstrating invariance properties and deriving the Feynman propagator with a focus on boundary terms and canonical transformations.

## Contribution

It introduces a novel application of Dirac's formalism to time-dependent systems in extended phase space, highlighting invariance and deriving the Feynman propagator.

## Key findings

- Lewis invariant is reparametrization invariant
- Feynman propagator calculated in extended phase space
- Quantum phase linked to boundary terms of canonical transformation

## Abstract

The Dirac's formalism for constrained systems is applied to the analysis of time-dependent Hamiltonians in the extended phase space. We show that the Lewis invariant is a reparametrization invariant and we calculate the Feynman propagator using the extended phase description. We show that the quantum phase of the Feynman propagator is given by the boundary term of the canonical transformation of the extended phase space.