Past exam of the advanced quantum field theory course of the University of Oxford / 2022 / 3 / a / i / Solution

The Goldstone theorem states that spontaneous breaking of a continuous internal global symmetry in field theory in a local relativistic quantum field theory produces a zero-mass Goldstone boson for each independent broken generator, under the usual positive-state-space and conserved-current hypotheses. For $G\to H$ there are $\dim G-\dim H$ symmetry-required Goldstone bosons. This count assumes a Lorentz-invariant vacuum state; nonrelativistic counting can differ.

At the classical level, invariance of the scalar potential gives ${\partial }_{i}V(T^a\varphi {)}_i=0$. Differentiating at a stationary vacuum expectation value ${\varphi }_0$ yieldsThese are the Goldstone directions in the scalar mass matrix. Broken generators have $T^a{\varphi }_0\ne 0$, so their independent directions have zero quadratic mass with a canonical kinetic term. The theorem guarantees these modes; it does not exclude further accidental zero-mass modes. A local gauge symmetry is a redundancy and does not meet the physical global-symmetry hypothesis: the Higgs mechanism can instead supply longitudinal particle polarizations of massive gauge bosons.