= Translation covariance of a quantum field
{title2=$\Phi(x)=e^{iP\cdot x}\Phi(0)e^{-iP\cdot x}$}
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 $\langle q|\Phi(x)|p\rangle=e^{i(q-p)\cdot x}\langle q|\Phi(0)|p\rangle$. 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.
Back to article page
