The exact mode functions satisfy the Wronskian normalization
$$
f_kf_k^{*\prime}-f_k^*f_k'=i.
$$
Using the <Dirac delta function> generated by the ladder-operator commutator therefore gives the exact <canonical commutation relation>
$$
\boxed{[\hat f(\tau,\mathbf x),\hat\pi(\tau,\mathbf y)]
=i\delta^{(3)}(\mathbf x-\mathbf y)}.
$$
This remains true on <superhorizon scales>, so the exact operators do not literally commute. A claim of classical behavior must instead compare this fixed commutator with the growing statistical fluctuations, or appeal to <quantum decoherence>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 310 / 4 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-310/4/b/solution