Four-momentum operator ($P^{\mu }=(H,\mathbf{P})$)

The generators of spacetime translations in a quantum field theory. With the mostly-minus Minkowski metric, $P\cdot a=H{a}^0-\mathbf{P}\cdot \mathbf{a}$. A momentum eigenstate satisfies $P^{\mu }|p⟩=p^{\mu }|p⟩$, where $p^{\mu }$ is its four-momentum. After subtracting the vacuum energy, a translation-invariant vacuum state has eigenvalue zero.