A Simple Bound on the Error of Perturbation Theory in Quantum Mechanics

TL;DR

This paper presents a simple, assumption-free bound on the error of perturbation series in quantum mechanics, applicable to a broad class of systems, including quantum field theory.

Contribution

It provides a straightforward proof that the partial sums of perturbation series approximate the exact result within a bound equal to the next term, without convergence assumptions.

Findings

Bound applies to finite-time Euclidian transition amplitudes.

Proof extends to a wide class of quantum systems.

Open questions remain for renormalized perturbation theory.

Abstract

I provide a straightforward proof that a simple harmonic oscillator perturbed by an (almost) arbitrary positive interaction has a perturbative expansion for any finite-time Euclidian transition amplitude which obeys the following result: the difference of the sum of the first N terms of the series and the exact result is bounded in absolute value by the next term in the series. The proof makes no assumptions about either the strength of the interactions or the convergence of the perturbation series. I then argue that the result generalizes immediately to a much broader class of quantum mechanical systems, including bare perturbation theory in quantum field theory. The case of renormalized perturbation theory is more subtle and remains open, as does the generalization to energy levels and connected S-matrix elements.