Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 312 / 1 / ii

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- Solution ii

Solution

In one standard Schwinger-Keldysh propagator convention, suppressing the momentum delta function,

G_{RR} (k; η, η^{'}) = θ (η - η^{'}) f (η) f^{*} (η^{'}) + θ (η^{'} - η) f^{*} (η) f (η^{'}),

G_{LL} (k; η, η^{'}) = θ (η^{'} - η) f (η) f^{*} (η^{'}) + θ (η - η^{'}) f^{*} (η) f (η^{'}),

G_{R L} (k; η, η^{'}) = f^{*} (η) f (η^{'}), G_{L R} (k; η, η^{'}) = f (η) f^{*} (η^{'}) .

The two bulk-to-boundary propagators to the late-time insertion are

K_{R} (k, η) = f (k, 0) f^{*} (k, η), K_{L} (k, η) = f^{*} (k, 0) f (k, η) .

Interchanging every

L

and

R

gives the equivalent opposite contour-label convention. Direct differentiation of the supplied Bunch-Davies vacuum mode gives

f^{'} (k, η) = \frac{H k ^{2} η}{2 k ^{3}} e^{- ik η} = \frac{H k}{2} η e^{- ik η} .

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