Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 305 / 3 / ii / Solution

A symmetry generated by $Q^a$ is spontaneously broken when the vacuum is not invariant:A local order parameter is obtained by finding a field ${\phi }_m$ for whichwith the sign depending on the commutator convention. This criterion is useful in infinite volume because it can remain finite even when the spatially integrated charge itself is not a well-defined normalizable operator.