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

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 $(+---)$.

Insert the mode expansion of a free field into the classical expression for $\mathbf{P}$. The $aa$ and $a^{†}{a}^{†}$ terms cancel after $\mathbf{p}\mapsto -\mathbf{p}$, while the mixed terms give, after the stated normal ordering,Thus the momentum operator counts each occupied mode with weight $\mathbf{p}$.

Using $[AB,C]=A[B,C]+[A,C]B$ and the canonical commutator,

The iterated commutators areso the Baker--Campbell--Hausdorff formula gives

The vacuum has zero momentum and is invariant under translations. HenceThus a one-particle momentum eigenstate remains the same ray and acquires the translation phase appropriate to its momentum.

Similarly $[\mathbf{P},a_{\mathbf{p}}]=-\mathbf{p},a_{\mathbf{p}}$. Transforming both terms of the field expansion givesTherefore the spatial translation operator generated by $\mathbf{P}$ translates the field argument. Equivalently, its action on states translates a wavefunction in the opposite argument convention.