Rest-frame reduction of scalar transition matrix elements ($E^{\prime }=p\cdot q/m$)

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.