Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 312 / 1 / ii

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- Solution ii

Solution

For

f (k, η) = \frac{Hη}{2 k} e^{- ik η},

the specified cosmological bulk-to-boundary propagator is

G_{R} (η, p) = f (p, η_{0}) f^{*} (p, η) = \frac{H ^{2} η _{0} η}{2 p} e^{- i p (η_{0} - η)} .

The right-right cosmological bulk-to-bulk propagator is time ordered:

G_{RR} (η_{1}, η_{2}, p) = \frac{H ^{2} η _{1} η _{2}}{2 p} e^{- i p ∣ η_{1} - η_{2} ∣} .

The mixed Wightman function in the order stated in the question is

G_{L R} (η_{1}, η_{2}, p) = f (p, η_{1}) f^{*} (p, η_{2}) = \frac{H ^{2} η _{1} η _{2}}{2 p} e^{- i p (η_{1} - η_{2})} .

The suppressed factor in each expression is

(2 π)^{3} δ^{(3)} (p + p^{'})

.

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