Goldstone Theorem in the Gaussian Functional Approximation to the Scalar $\phi^{4}$ Theory

TL;DR

This paper verifies the Goldstone theorem within the Gaussian functional approximation for the scalar  theory with O(2) symmetry, using Schwinger-Dyson equations to explicitly demonstrate the theorem.

Contribution

It reformulates the Gaussian approximation in terms of Schwinger-Dyson equations to explicitly prove the Goldstone theorem in  theory.

Findings

Goldstone theorem holds in the Gaussian approximation for  theory.

Reformulation using Schwinger-Dyson equations provides a direct proof.

Explicit demonstration confirms the validity of the approximation.

Abstract

We verify the Goldstone theorem in the Gaussian functional approximation to the $\phi^{4}$ theory with internal O(2) symmetry. We do so by reformulating the Gaussian approximation in terms of Schwinger-Dyson equations from which an explicit demonstration of the Goldstone theorem follows directly.