Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 311 / 3 / c

Let ${\Sigma }_t$ be a Cauchy hypersurface with induced metric $h_{ij}$, lapse $N$, shift $N^i$, and future unit normal $n^a$. For the normalization of the action in the question, the canonical momentum density isThe equal-time canonical commutation relations areandWith the conventional extra factor $1/2$ in the action, $\Pi$ loses the factor two.

For complex classical solutions, the Klein-Gordon inner product isThe integrand is a conserved current because both fields obey the Klein-Gordon equation. Applying the divergence theorem between two Cauchy hypersurfaces shows that the value is independent of the foliation, provided there is no boundary flux.

Solved by gpt-5.6-sol high.

A strictly stationary spacetime has an everywhere timelike Killing field $K$. Choose a complete orthonormal set of positive-frequency solutions satisfyingwith positive Klein-Gordon inner product. Expandwith the sum replaced by an integral for continuous labels. Equivalently, the coefficients are projections using the Klein-Gordon product: $a_j=(u_j,\hat{\Phi }{)}_{KG}$ and $a_j^{†}=-(u_j^{\ast },\hat{\Phi }{)}_{KG}$.

The stationary vacuum is uniquely selected, up to degeneracies and unitary changes of positive-frequency basis, byfor every $j$. Acting with the $a_j^{†}$ constructs the bosonic Fock space, the symmetric direct sum of all particle-number sectors.

In a nonstationary spacetime no preferred timelike Killing flow exists, so there is no canonical split into positive and negative frequencies. Different splits mix creation and annihilation operators by Bogoliubov transformation and lead to the vacuum ambiguity in a nonstationary spacetime.

Solved by gpt-5.6-sol high.