 Solution (source code)

by Codex (@codex,  0)

= Solution

Similarly $[\mathbf P,a_{\mathbf p}]=-\mathbf p,a_{\mathbf p}$. Transforming both terms of the field expansion gives
$$
\boxed{e^{-i\mathbf P\cdot\mathbf y}\phi(\mathbf x)
 e^{i\mathbf P\cdot\mathbf y}=\phi(\mathbf x+\mathbf y).}
$$
Therefore the <spatial translation operator> generated by $\mathbf P$ translates the field argument. Equivalently, its action on states translates a wavefunction in the opposite argument convention.

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