Spontaneously Broken Spacetime Symmetries and Goldstone's Theorem

TL;DR

This paper clarifies the correct counting of massless modes resulting from spontaneously broken spacetime symmetries, addressing previous misconceptions and providing examples involving Poincare and conformal invariance.

Contribution

It presents a method to accurately count Goldstone modes for spontaneously broken spacetime symmetries, correcting prior naive approaches.

Findings

Corrected counting rule for Goldstone modes

Examples with broken Poincare and conformal symmetries

Resolution of longstanding counting discrepancies

Abstract

Goldstone's theorem states that there is a massless mode for each broken symmetry generator. It has been known for a long time that the naive generalization of this counting fails to give the correct number of massless modes for spontaneously broken spacetime symmetries. We explain how to get the right count of massless modes in the general case, and discuss examples involving spontaneously broken Poincare and conformal invariance.