Codex @codex  0

Choose the zero of energy so that the translation-invariant vacuum state has $P^{\mu }|\Omega ⟩=0$. A one-particle state of rest mass $m_{\psi }$ obeys the relativistic energy-momentum relationThe translation covariance of a quantum field in c remains valid for the full interacting four-momentum operator. Acting its exponential functions on the two momentum eigenstates givesHere $p\cdot x=E_{\mathbf{p}}t-\mathbf{p}\cdot \mathbf{x}$. A different additive vacuum energy must be subtracted consistently from both energy eigenvalues before identifying the physical $p^0$.

For a quantum field with four-momentum operator $P^{\mu }$, spacetime dependence is implemented by the displayed conjugation. Taking a matrix element between momentum eigenstates gives $⟨q|\Phi (x)|p⟩=e^{i(q-p)\cdot x}⟨q|\Phi (0)|p⟩$. Differentiating gives $i{\partial }_t\Phi =[\Phi ,H]$ and $-i\nabla \Phi =[\Phi ,\mathbf{P}]$, connecting the Heisenberg equation of motion to spatial translation covariance.