Convergent perturbative series via finite path integral limits: application to energy at strong coupling of the anharmonic oscillator

TL;DR

This paper introduces a method using finite path integral limits to obtain convergent perturbative series that accurately compute energies at strong coupling in anharmonic oscillators, overcoming divergence issues of traditional series.

Contribution

It demonstrates that finite integration limits in path integrals produce absolutely convergent series applicable at strong coupling, validated through quantum mechanical models.

Findings

Convergent series match exact energies within 0.1% at strong coupling.

Finite path integral limits yield reliable results where traditional series diverge.

Method applicable to various anharmonic oscillators and basic integrals.

Abstract

Solving quantum field theories at strong coupling remains a challenging task. The main issue is that the usual perturbative series are asymptotic series which can be useful at weak coupling but break down completely at strong coupling. In this work, we show that if the limits of integration in the path integral are finite, the perturbative series is remarkably an absolutely convergent series which works well at strong coupling. For now, we apply this perturbative approach to $\lambda \phi^4$ theory in 0+0 dimensions (a basic integral) and 0+1 dimensions (quartic anharmonic oscillator). As a further application, we also consider the sextic anharmonic oscillator. For the basic integral, we show that finite integral limits yields a convergent series whose values are in agreement with exact analytical results at any coupling. This worked even when the asymptotic series was not Borel summable. It is well known that the perturbative series expansion in powers of the coupling for the energy of the anharmonic oscillator yields an asymptotic series and hence fails at strong coupling. In quantum mechanics, if one is interested in the energy, it is often easier to use Schr\"odinger's equation to develop a perturbative series than path integrals. Finite path integral limits are then equivalent to placing infinite walls at positions -L and L in the potential where L is positive, finite and can be arbitrarily large. With walls, the series expansion for the energy is now convergent and approaches the energy of the anharmonic oscillator as the walls are moved further apart. We use the convergent series to calculate the ground state energy at weak, intermediate and strong coupling. At strong coupling, the result from the series agrees with the exact energy to within $0.1\%$, a remarkable result in light of the fact that at strong coupling the usual perturbative series diverges badly immediately.