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Use the canonical commutation relation for the creation and annihilation operators and the transformation derived above. Directly,

[a_{i}^{'}, a_{k}^{'}^{†}] = \sum_{j} (\overline{A}_{ij} A_{kj} - \overline{B}_{ij} B_{kj}), [a_{i}^{'}, a_{k}^{'}] = \sum_{j} (- \overline{A}_{ij} \overline{B}_{kj} + \overline{B}_{ij} \overline{A}_{kj}) .

Requiring these to equal

δ_{ik}

and zero, respectively, and conjugating the equations yields the canonical identities for a bosonic Bogoliubov transformation:

A A^{†} - B B^{†} = I, A B^{T} = B A^{T} .

The commutator of the primed creation operators then vanishes by taking adjoints. These are exactly the corresponding positive-mode norm and positive-negative orthogonality conditions for the Bogoliubov transformation. They are sufficient for these algebraic commutators whenever the sums are well defined. Infinitely many modes introduce further questions of convergence and whether the two choices admit a common bosonic Fock space representation; the matrix identities alone do not settle unitary implementability.

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 311 / 4 / a / Solution Created 2026-10-03 Updated 2026-10-05

For a real free scalar field on a globally hyperbolic spacetime, take

S = - \frac{1}{2} \int d^{4} x - g (\nabla_{a} ϕ \nabla^{a} ϕ + (m^{2} + ξ R) ϕ^{2}) .

The Klein-Gordon equation has a well-posed initial-value problem on every Cauchy hypersurface. Its complex solution space carries the conserved Klein-Gordon inner product. Choose modes with

(u_{i}, u_{j})_{K G} = δ_{ij}, (u_{i}^{*}, u_{j}^{*})_{K G} = - δ_{ij}, (u_{i}, u_{j}^{*})_{K G} = 0,

and expand

ϕ = \sum_{i} (a_{i} u_{i} + a_{i}^{†} u_{i}^{*}), [a_{i}, a_{j}^{†}] = δ_{ij} .

(3)

The canonical commutation relations produce the bosonic Fock space, and

a_{i} ∣0 ⟩ = 0

defines the chosen vacuum.

The choice of positive-frequency solutions is extra structure. In a time-dependent geometry there is no preferred time translation, so different choices mix

u_{i}

with

u_{i}^{*}

and give the vacuum ambiguity in a nonstationary spacetime. In a stationary spacetime, a timelike Killing vector field defines positive frequency by

i L_{K} u_{i} = ω_{i} u_{i}

ω_{i} > 0

, and hence the preferred vacuum state in a stationary spacetime.

If the spacetime is stationary in the remote past and future, define normalized in- and out-modes there. Their Bogoliubov transformation is

u_{i}^{out} = \sum_{j} (α_{ij} u_{j}^{in} + β_{ij} u_{j}^{in *}) .

(4)

The expected out-particle number in the in-vacuum is

⟨ 0_{in} ∣ N_{i}^{out} ∣ 0_{in} ⟩ = j \sum ∣ β_{ij} ∣^{2} .

(5)

Thus nonzero beta coefficients measure particle creation by the intervening time dependence.

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 311 / 4 / a / Solution 2026-09-28

On a stationary spacetime, choose a complete set of complex solutions

u_{i}

of the massive Klein-Gordon equation that have positive frequency with respect to the timelike Killing vector field and are orthonormal in the conserved Klein-Gordon inner product:

(u_{i}, u_{j})_{K G} = δ_{ij}, (u_{i}^{*}, u_{j}^{*})_{K G} = - δ_{ij}, (u_{i}, u_{j}^{*})_{K G} = 0.

The existence of a Cauchy hypersurface in the globally hyperbolic spacetime makes the classical initial-value problem and this inner product well defined. Expand the real field as

ϕ = \sum_{i} (a_{i} u_{i} + a_{i}^{†} u_{i}^{*}), [a_{i}, a_{j}^{†}] = δ_{ij} .

The vacuum state in a stationary spacetime is defined by

a_{i} ∣0 ⟩ = 0 for every positive-frequency mode i .

(3)

Acting with the creation operators builds the bosonic Fock space.

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 311 / 3 / c / ii / Solution Created 2026-09-24 Updated 2026-09-25

A strictly stationary spacetime has an everywhere timelike Killing field

K

. Choose a complete orthonormal set of positive-frequency solutions satisfying

i L_{K} u_{j} = ω_{j} u_{j}, ω_{j} > 0,

with positive Klein-Gordon inner product. Expand

Φ = \sum_{j} (a_{j} u_{j} + a_{j}^{†} u_{j}^{*})

with the sum replaced by an integral for continuous labels. Equivalently, the coefficients are projections using the Klein-Gordon product:

a_{j} = (u_{j}, Φ)_{K G}

and

a_{j}^{†} = - (u_{j}^{*}, Φ)_{K G}

The stationary vacuum is uniquely selected, up to degeneracies and unitary changes of positive-frequency basis, by

a_{j} ∣0 ⟩ = 0

(3)

for 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.