Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 305 / 1 / Solution

A global symmetry is spontaneously broken when it preserves the action but does not preserve a chosen ground state. If $G$ is broken to the stabilizer $H$, the degenerate vacua form a vacuum manifold $G/H$. The Goldstone theorem states that a relativistic theory has one massless scalar mode for each broken continuous internal generator, so the standard counting gives $\dim G-\dim H$ Goldstone bosons.

Solved by gpt-5.6-sol high.