Effective Classical Hamiltonian from Perturbatively Defined Path Integral

TL;DR

This paper introduces a perturbative approach to phase space path integrals that avoids slicing, enabling efficient calculation of short-time quantum amplitudes and effective classical Hamiltonians, demonstrated on the harmonic oscillator.

Contribution

It presents a novel perturbative formalism for phase space path integrals that simplifies calculations of quantum amplitudes and effective Hamiltonians without slicing.

Findings

Derived a short-time expansion for quantum amplitudes

Calculated the effective classical Hamiltonian for the harmonic oscillator

Provided a new method for high-temperature density matrix approximation

Abstract

Introducing a perturbative definition, phase space path integrals can be calculated without slicing. This leads to a short-time expansion of the quantum-mechanical path amplitude, or a high-temperature expansion of the unnormalized density matrix, respectively. We use the proposed formalism to calculate the effective classical Hamiltonian for the harmonic oscillator.