Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 311 / 4 / a

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Solution

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 .

Acting with the creation operators builds the bosonic Fock space.

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