A symmetry generated by $Q^a$ is spontaneously broken when the vacuum is not invariant:
$$
Q^a|0\rangle\ne0.
$$
A local <order parameter> is obtained by finding a field $\varphi_m$ for which
$$
\boxed{\langle0|[Q^a,\varphi_m(0)]|0\rangle
=-(T^a)_{mn}\langle0|\varphi_n(0)|0\rangle\ne0},
$$
with 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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-305/3/ii/solution