Functional Integral and Stochastic Representations for Ensembles of Identical Bosons on a Lattice

TL;DR

This paper develops regularized functional integral representations for ensembles of identical bosons on a lattice, demonstrating convergence in the continuum limit and connecting to stochastic models involving interacting random walks.

Contribution

It introduces a discretized Euclidean time regularization for bosonic lattice ensembles and establishes convergence and bounds, linking functional integrals to stochastic representations.

Findings

Proves convergence of discretized functional integrals in the continuum limit.

Provides uniform bounds for covariances to analyze thermodynamic limits.

Establishes explicit connection between functional integrals and stochastic random walk models.

Abstract

Regularized coherent-state functional integrals are derived for ensembles of identical bosons on a lattice, the regularization being a discretization of Euclidian time. Convergence of the time-continuum limit is shown for various discretized actions. The focus is on the integral representation for the partition function and expectation values in the canonical ensemble. The connection to the grand-canonical integral, and a number of differences, are discussed. Uniform bounds for covariances are proven, which simplify the analysis of the time-continuum limit and can also be used to analyze the thermodynamic limit. The relation to a stochastic representation by an ensemble of interacting random walks is made explicit, and its modifications in presence of a condensate are discussed.