An introduction to Goldstone boson physics and to the coset construction

TL;DR

This paper provides an accessible introduction to Goldstone boson physics, emphasizing the universality of spontaneous symmetry breaking, the derivation of NG modes, and the coset construction as an effective field theory tool, illustrated with ferromagnetism.

Contribution

It offers a comprehensive, pedagogical overview of Goldstone theorem, NG modes, and the coset construction, including recent developments and limitations, suitable for graduate students in high-energy physics.

Findings

Derivation of a counting rule for NG modes

Discussion of limitations and extensions of the rule

Application of concepts to ferromagnetism

Abstract

These lecture notes are based on a six-hour series of lectures given at the XVII Modave summer school in mathematical physics, aimed at Ph.D. students in high-energy theoretical physics. The manuscript starts by briefly stating Goldstone's theorem and emphasises the motivations behind Goldstone physics; the main asset being the universality of spontaneous symmetry breaking (SSB) which is the fundamental hypothesis of Goldstone's theorem. Once the different notions of SSB will be clarified/reviewed, Goldstone's theorem will be stated and proved. A prediction of this theorem is the existence of gapless particles, called Nambu-Goldstone modes (NG modes). From the discussion on Goldstone's results, some aspects of the NG modes will emerge. Besides to be gapless, they are systematically weakly coupled at low energy. Therefore, an effective field theory (EFT) building tool called ``coset construction'' will be presented to explicitly display these specific features of the NG modes. The coset construction suits our goal since it is mainly based on the symmetry realisation of the perturbed theory around the background inducing SSB. From the general obtained EFT, a counting rule for the NG modes will be derived. The limitations of this rule as well as the still ongoing generalisation will be discussed (e.g. spacetime symmetry breaking). The tools developed during this course will be illustrated with a concrete example in physics: ferromagnetism. The notes end with a brief state of the art of Goldstone physics. This provides some directions into which the interested reader could investigate to expand his knowledge on the subject. N.B.: No prerequisites are required beside the standard courses of a Master in theoretical physics.