Many-Body Diffusion and Path Integrals for Identical Particles

TL;DR

This paper develops a method to compute path integrals for identical particles by separating potential effects from particle exchange symmetry, using Brownian motion and Feynman-Kac functionals, with applications to fermions and bosons.

Contribution

It introduces a novel approach that decomposes the path integral for identical particles into simpler 1D diffusion processes based on permutation symmetry.

Findings

Path integrals for identical particles can be expressed via 1D fermion and boson diffusion processes.

Permutation symmetry determines boundary conditions: absorption for fermions, reflection for bosons.

The method simplifies calculations of thermodynamical quantities for many-body quantum systems.

Abstract

For distinguishable particles it is well known that Brownian motion and a Feynman-Kac functional can be used to calculate the path integral (for imaginary times) for a general class of scalar potentials. In order to treat identical particles, we exploit the fact that this method separates the problem of the potential, dealt with by the Feynman-Kac functional, from the process which gives sample paths of a non-interacting system. For motion in 1 dimension, we emphasize that the permutation symmetry of the identical particles completely determines the domain of Brownian motion and the appropriate boundary conditions: absorption for fermions, reflection for bosons. Further analysis of the sample paths for motion in 3 dimensions allows us to decompose these paths into a superposition of 1-dimensional sample paths. This reduction expresses the propagator (and consequently the energy and other thermodynamical quantities) in terms of well-behaved 1-dimensional fermion and boson diffusion processes and the Feynman-Kac functional.