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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 301 / 1 / ii / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-301/1/ii/e/solution