= Spatial translation operator
{title2=$U(\mathbf y)=e^{-i\mathbf P\cdot\mathbf y/\hbar}$}
The <spatial translation operator> is the <unitary operator> generated by <momentum>. For displacement $\mathbf y$,
$$
U(\mathbf y)=e^{-i\mathbf P\cdot\mathbf y/\hbar},
$$
where $\hbar$ is the <reduced Planck constant>. In a <position representation in quantum mechanics>, $\mathbf P=-i\hbar\nabla$, so its action on a <wavefunction> is
$$
(U(\mathbf y)\psi)(\mathbf x)=\psi(\mathbf x-\mathbf y).
$$
A localized <wave packet> is therefore displaced by $+\mathbf y$. A <momentum eigenstate> of <eigenvalue> $\mathbf p$ instead acquires the phase $e^{-i\mathbf p\cdot\mathbf y/\hbar}$.
The <canonical commutation relation> gives
$$
U^\dagger\mathbf XU=\mathbf X+\mathbf yI.
$$
Thus 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 is
$$
U(\mathbf y)\phi(\mathbf x)U(\mathbf y)^\dagger=\phi(\mathbf x+\mathbf y).
$$
Infinitesimally this means $[P_j,\phi(\mathbf x)]=i\hbar\partial_j\phi(\mathbf x)$. 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. <Quantum Field Theory by David Tong>[Author supplement More translation invariance, formulas for translated position eigenstates, the position operator and a local field; hbar restored here]{ref}
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