Past exam of the theoretical physics major option course of the University of Oxford / 2022 / 5 / e / Solution

For the spatial momenta, integration by parts givesTheir Poisson bracket is thereforeFor the mixed case, $P_0=H$ and the spatial generator acts on both canonical fields by differentiation: $\{{\varphi }_a,P_i\}={\partial }_i{\varphi }_a$, $\{{\pi }_a,P_i\}={\partial }_i{\pi }_a$. Hence, for a translation-invariant local Hamiltonian density,Finally $\{H,H{\}}_{\mathrm{PB}}=0$ by antisymmetry. Together these proveThe proof requires commuting derivatives and no surviving boundary terms. It expresses the Abelian translation subgroup of Poincaré symmetry at the level of its canonical generators.