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

A spacetime translation $x^{\mu }\mapsto x^{\mu }+{ϵ}^{\mu }$ gives, by Noether theorem, the canonical stress-energy tensorThe Klein-Gordon equation implies ${\partial }_{\mu }{T}^{\mu \nu }=0$. With canonical momentum $\pi =\dot{\varphi }$, the conserved physical three-momentum isThe minus sign follows from ${\partial }^i=-{\partial }_i$ for metric signature $(+---)$.