The constant <scalar-field shift symmetry> has momentum-space action
$$
\boxed{\Delta\phi(\mathbf k)=c(2\pi)^3\delta^{(3)}(\mathbf k)}.
$$
For the <canonical commutation relation> $[\phi(\mathbf x),\Pi(\mathbf y)]=i\delta^{(3)}(\mathbf x-\mathbf y)$, its <Noether charge> can be written
$$
\boxed{Q=c\int d^3x\,\Pi(\mathbf x)=c\,\Pi(\mathbf k=0)}.
$$
Indeed, $i[Q,\phi(\mathbf x)]=c$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 312 / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-312/2/a/solution