Variational Calculation of the Effective Action

TL;DR

This paper investigates spontaneous symmetry breaking in a two-dimensional $^4$ model by calculating an effective action using DLCQ, revealing a nonzero field configuration in the broken phase due to boundary effects.

Contribution

It introduces a method to compute the effective action in a quantum field theory using DLCQ and boundary conditions, highlighting boundary effects on symmetry breaking.

Findings

Existence of a nonzero field configuration in the broken phase.

Boundary effects influence symmetry breaking.

Effective action can be derived from ground state properties.

Abstract

An indication of spontaneous symmetry breaking is found in the two-dimensional $\lambda\phi^4$ model, where attention is paid to the functional form of an effective action. An effective energy, which is an effective action for a static field, is obtained as a functional of the classical field from the ground state of the hamiltonian $H[J]$ interacting with a constant external field. The energy and wavefunction of the ground state are calculated in terms of DLCQ (Discretized Light-Cone Quantization) under antiperiodic boundary conditions. A field configuration that is physically meaningful is found as a solution of the quantum mechanical Euler-Lagrange equation in the $J\to 0$ limit. It is shown that there exists a nonzero field configuration in the broken phase of $Z_2$ symmetry because of a boundary effect.