Generating Non-perturbative Physics from Perturbation Theory

Abstract

In a large variety of quantum mechanical systems, we show that the full non-perturbative expression for energy eigenvalues, containing all orders of perturbative, non-perturbative and quasi-zero-mode terms, may be generated directly from the perturbative expansion about the perturbative vacuum, combined with a single global boundary condition. This provides a dramatic realization of the principle of "resurgence", that the fluctuations about different semiclassical saddle points are related to one another in a precise quantitative manner. The analysis of quantum mechanics also generalizes to certain calculable regimes of quantum field theory.

Figures (1)

• Figure 1: Plots of the ratio of the exact perturbative coefficients to the large-order growth (\ref{['eq:large-order-N']}), for the double-well potential, for the $N=0$ ground state (upper plot) and $N=1$ first excited state (lower plot). The solid (blue) circles, (red) squares and (gold) diamonds denote the leading, sub-leading and sub-sub-leading large $n$ behavior, respectively, including increasing information from (\ref{['eq:h0']}, \ref{['eq:imag']}, \ref{['eq:large-order-N']}) concerning the fluctuations about the instanton/anti-instanton sector.