 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2026

Paper 301

Let
$$
N=\int\frac{d^3q}{(2\pi)^3}a_{\mathbf q}^\dagger a_{\mathbf q},
\qquad
K=\int\frac{d^3q}{(2\pi)^3}a_{\mathbf q}^\dagger a_{-\mathbf q}.
$$
The canonical commutation relations give $[N,a_{\mathbf k}]=-a_{\mathbf k}$ and $[K,a_{\mathbf k}]=-a_{-\mathbf k}$. Hence exponentiating the adjoint action gives
$$
\mathcal P_1a_{\mathbf k}\mathcal P_1^{-1}
=e^{i\pi/2}a_{\mathbf k}=ia_{\mathbf k}.
$$
Writing $R a_{\mathbf k}=a_{-\mathbf k}$, one has $R^2=1$ and $\operatorname{ad}_K=-R$ on annihilation operators. Therefore
$$
\mathcal P_2a_{\mathbf k}\mathcal P_2^{-1}
=e^{-i\gamma_p\pi R/2}a_{\mathbf k}
=-i\gamma_p a_{-\mathbf k}.
$$

Solved by gpt-5.6-sol high.


Solution

Paper 301 / 2 / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 301 / 2 / c / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 301 / 2 / c / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)