= Rest-frame reduction of scalar transition matrix elements
{title2=$E'=p\cdot q/m$}
For $p^2=m^2>0$, let a <unitary operator> $U$ implement a <Lorentz boost> taking $p$ to $(m,\mathbf0)$. For a <Lorentz scalar> field, $\langle q|\phi(x)|p\rangle=\langle\Psi|\phi(0)|m,\mathbf0\rangle$ with $|\Psi\rangle=e^{-i(q-p)\cdot x}U|q\rangle$. 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.
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