= Translation of a scalar creation operator
{title2=$e^{-i\mathbf P\cdot\mathbf y}a_{\mathbf p}^\dagger e^{i\mathbf P\cdot\mathbf y}=e^{-i\mathbf p\cdot\mathbf y}a_{\mathbf p}^\dagger$}
The <canonical commutation relation> gives $[P^\mu,a_p^\dagger]=p^\mu a_p^\dagger$ 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 $e^{-i\mathbf P\cdot\mathbf y}\phi(\mathbf x)e^{i\mathbf P\cdot\mathbf y}=\phi(\mathbf x+\mathbf y)$. The <spatial translation operator> translates a state's position wavefunction in the opposite argument direction.
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