A spacetime translation $x^\mu\mapsto x^\mu+\epsilon^\mu$ gives, by <Noether theorem>, the <canonical stress-energy tensor>
$$
\boxed{T^{\mu\nu}=\partial^\mu\phi\,\partial^\nu\phi
-\eta^{\mu\nu}\mathcal L.}
$$
The <Klein-Gordon equation> implies $\partial_\mu T^{\mu\nu}=0$. With canonical momentum $\pi=\dot\phi$, the conserved physical three-momentum is
$$
\boxed{\mathbf P=\int d^3x\,T^{0i}\mathbf e_i
=-\int d^3x\,\pi(\mathbf x)\boldsymbol\nabla\phi(\mathbf x).}
$$
The minus sign follows from $\partial^i=-\partial_i$ for metric signature $(+---)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 301 / 1 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-301/1/i/solution