Spatial translation operator ($U(\mathbf{y})=e^{-i\mathbf{P}\cdot \mathbf{y}}$)

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 $U(\mathbf{y})\varphi (\mathbf{x})U(\mathbf{y}{)}^{-1}=\varphi (\mathbf{x}+\mathbf{y})$.