Translations produce the conserved symmetric tensor
$$
T^{\mu\nu}
=\partial^\mu\phi\,\partial^\nu\phi-\eta^{\mu\nu}\mathcal L,
\qquad \partial_\mu T^{\mu\nu}=0.
$$
The conserved charges are
$$
\boxed{E=\int_{\mathbb R^3}d^3x\,T^{00}
=\int d^3x\left\{\frac12\dot\phi^2+\frac12|\nabla\phi|^2+V(\phi)\right\}},
$$
and
$$
\boxed{P^i=\int_{\mathbb R^3}d^3x\,T^{0i}}.
$$
Indeed,
$$
\frac d{dt}\int d^3x\,T^{0\nu}
=-\int d^3x\,\partial_iT^{i\nu}
=-\lim_{R\to\infty}\int_{S_R}T^{i\nu}n_i\,dS=0
$$
when the field and its derivatives decay sufficiently rapidly.


Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 301 / 1 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-301/1/i/solution