Translation covariance of a quantum field ($\Phi (x)=e^{iP\cdot x}\Phi (0)e^{-iP\cdot x}$)

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.

For $p^2=m^2>0$, let a unitary operator $U$ implement a Lorentz boost taking $p$ to $(m,0)$. For a Lorentz scalar field, $⟨q|\varphi (x)|p⟩=⟨\Psi |\varphi (0)|m,0⟩$ with $|\Psi ⟩=e^{-i(q-p)\cdot x}U|q⟩$. The phase follows from translation covariance of a quantum field; its complex conjugate enters the bra. The transformed energy eigenvalue is $(\Lambda q{)}^0=p\cdot q/m$ by invariance of the Minkowski inner product. The reduction requires a massive initial state.