 Spatial translation operator

The spatial translation operator is the unitary operator generated by momentum. For displacement $\mathbf{y}$,where $\hslash$ is the reduced Planck constant. In a position representation in quantum mechanics, $\mathbf{P}=-i\hslash \nabla$, so its action on a wavefunction isA localized wave packet is therefore displaced by $+\mathbf{y}$. A momentum eigenstate of eigenvalue $\mathbf{p}$ instead acquires the phase $e^{-i\mathbf{p}\cdot \mathbf{y}/\hslash }$.

The canonical commutation relation givesThus the expectation value of the position operator in the translated quantum state increases by $\mathbf{y}$.

For an operator-valued scalar field, the compatible translation convention isInfinitesimally this means $[P_j,\varphi (\mathbf{x})]=i\hslash {\partial }_j\varphi (\mathbf{x})$. The field's coordinate argument labels where a local observable is evaluated; $\mathbf{X}$ is the position operator of a particle. These meanings explain their differently ordered conjugation formulas. ^[ref]