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The four-momentum operator collects the generators of spacetime translations in quantum field theory. With physical speed of light $c$,For the mostly-minus Minkowski metric,All components of $P^{\mu }$ therefore have units of momentum; $P\cdot a$ has units of action. The natural units convention $c=1$ gives the common abbreviation $P^{\mu }=(H,\mathbf{P})$.

A simultaneous momentum eigenstate satisfies $P^{\mu }|p⟩=p^{\mu }|p⟩$, where $p^{\mu }=(E/c,\mathbf{p})$ is its four-momentum. On a one-particle state of rest mass $m$, the mass-shell relation is $p_{\mu }{p}^{\mu }=m^2{c}^2$.

In a translation-invariant theory the four translation generators commute. A translation-invariant vacuum state has zero spatial momentum; after subtracting the additive vacuum energy, it also has zero Hamiltonian eigenvalue. This fixes the vacuum origin of the four-momentum spectrum.

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.