A Probabilistic Representation of the Ground State Expectation of Fractional Powers of the Boson Number Operator

TL;DR

This paper introduces a probabilistic formula for calculating the ground state expectations of fractional powers of the boson number operator in the Nelson model, enabling precise bounds and applications to related models.

Contribution

It provides a novel probabilistic representation involving Gibbs measures and Poisson processes for boson number expectations, extending to fractional powers and various models.

Findings

Derived a formula using Gibbs measures and Poisson processes

Established tight two-sided bounds for expectations

Applied results to polaron and Nelson models

Abstract

We give a formula in terms of a joint Gibbs measure on Brownian paths and the measure of a random-time Poisson process of the ground state expectations of fractional (in fact, any real) powers of the boson number operator in the Nelson model. We use this representation to obtain tight two-sided bounds. As applications, we discuss the polaron and translation invariant Nelson models.