Spatial translation operator
= Spatial translation operator
{title2=$U(\mathbf y)=e^{-i\mathbf P\cdot\mathbf y}$}
The spatial translation operator is the unitary generated by the <momentum operator>. With the convention $U(\mathbf y)=e^{-i\mathbf P\cdot\mathbf y}$, a momentum eigenstate of eigenvalue $\mathbf p$ acquires the phase $e^{-i\mathbf p\cdot\mathbf y}$, while $U(\mathbf y)\phi(\mathbf x)U(\mathbf y)^{-1}=\phi(\mathbf x+\mathbf y)$.