 Four-momentum operator

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.