Stochastic perturbation theory: a prequel to Reptation Quantum Monte Carlo

TL;DR

This paper introduces a novel approach to Rayleigh-Schr"odinger perturbation theory using Laplace transforms and polynomial theory, leading to an iterative, symbolically computable expansion that connects to Reptation Quantum Monte Carlo.

Contribution

It presents a new perturbation theory method based on Laplace transforms, offering a stochastic interpretation and a re-summation scheme related to Reptation Quantum Monte Carlo.

Findings

Provides an iterative expression for energy perturbation expansion.

Establishes a stochastic interpretation leading to a re-summation scheme.

Connects the new approach to Reptation Quantum Monte Carlo.

Abstract

I present a different approach to Rayleigh-Schr\"odinger perturbation theory, based on Laplace transforms and polynomial theory, yielding an iterative expression for the perturbative expansion of the energy of the non-degenerate ground state of a quantum system, which easily lends itself to symbolic computation. A stochastic interpretation of the various perturbative corrections naturally leads to a re-summation scheme that is equivalent to Reptation Quantum Monte Carlo and that actually provided the original motivation to its development in the late nineties.