Estimates on Functional Integrals of Quantum Mechanics and Non-Relativistic Quantum Field Theory

TL;DR

This paper introduces a unified method to estimate functional integrals in quantum mechanics and quantum field theory, providing bounds that help determine ground state energies for various models.

Contribution

It develops a novel approach to bound functional integrals using stochastic integral representations, applicable to multiple quantum systems and field theories.

Findings

Derived upper bounds for functional integrals in quantum models.

Established rigorous lower bounds for ground state energies.

Applied method successfully to models like the optical polaron and Nelson models.

Abstract

We provide a unified method for obtaining upper bounds for certain functional integrals appearing in quantum mechanics and non-relativistic quantum field theory, functionals of the form $E\left[\exp(A_T)\right]$, the (effective) action $A_T$ being a function of particle trajectories up to time $T$. The estimates in turn yield rigorous lower bounds for ground state energies, via the Feynman-Kac formula. The upper bounds are obtained by writing the action for these functional integrals in terms of stochastic integrals. The method is first illustrated in familiar quantum mechanical settings: for the hydrogen atom, for a Schr\"odinger operator with $1/|x|^2$ potential with small coupling, and, with a modest adaptation of the method, for the harmonic oscillator. We then present our principal applications of the method, in the settings of non-relativistic quantum field theories for particles moving in a quantized Bose field, including the optical polaron and Nelson models.