Translation of a scalar creation operator ($e^{-i\mathbf{P}\cdot \mathbf{y}}{a}_{\mathbf{p}}^{†}{e}^{i\mathbf{P}\cdot \mathbf{y}}=e^{-i\mathbf{p}\cdot \mathbf{y}}{a}_{\mathbf{p}}^{†}$)

The canonical commutation relation gives $[P^{\mu },a_p^{†}]=p^{\mu }{a}_p^{†}$ for a free real scalar field. Differentiate the exponential conjugation: its derivative is a scalar multiple of the same operator, yielding the displayed phase. The annihilation operator receives the opposite phase, so at equal time $e^{-i\mathbf{P}\cdot \mathbf{y}}\varphi (\mathbf{x})e^{i\mathbf{P}\cdot \mathbf{y}}=\varphi (\mathbf{x}+\mathbf{y})$. The spatial translation operator translates a state's position wavefunction in the opposite argument direction.