A non-perturbative method of calculation of Green functions

TL;DR

This paper introduces a non-perturbative approach to calculating Green functions in quantum mechanics and field theory, using an approximation of Schwinger-Dyson equations that is exactly solvable.

Contribution

It proposes a novel non-perturbative method based on an approximation of Schwinger-Dyson equations, with solutions demonstrated for the $_d$-model and renormalization in various dimensions.

Findings

Ground state energy calculated for anharmonic oscillator at d=1

Renormalization performed for field theories at d=2 and 3

Trivialization of the theory at d=4 due to coupling renormalization

Abstract

A new method for non-perturbative calculation of Green functions in quantum mechanics and quantum field theory is proposed. The method is based on an approximation of Schwinger-Dyson equation for the generating functional by exactly soluble equation in functional derivatives. Equations of the leading approximation and the first step are solved for $\phi^4_d$-model. At $d=1$ (anharmonic oscillator) the ground state energy is calculated. The renormalization program is performed for the field theory at $d=2,3$. At $d=4$ the renormalization of the coupling involves a trivialization of the theory.