For
$$
f(k,\eta)=\frac{H\eta}{\sqrt{2k}}e^{-ik\eta},
$$
the specified <cosmological bulk-to-boundary propagator> is
$$
\boxed{G_R(\eta,p)=f(p,\eta_0)f^*(p,\eta)
=\frac{H^2\eta_0\eta}{2p}e^{-ip(\eta_0-\eta)}}.
$$
The right-right <cosmological bulk-to-bulk propagator> is time ordered:
$$
\boxed{G_{RR}(\eta_1,\eta_2,p)
=\frac{H^2\eta_1\eta_2}{2p}
e^{-ip|\eta_1-\eta_2|}}.
$$
The mixed <Wightman function> in the order stated in the question is
$$
\boxed{G_{LR}(\eta_1,\eta_2,p)
=f(p,\eta_1)f^*(p,\eta_2)
=\frac{H^2\eta_1\eta_2}{2p}
e^{-ip(\eta_1-\eta_2)}}.
$$
The suppressed factor in each expression is $(2\pi)^3\delta^{(3)}(\mathbf p+\mathbf p')$.

Solution

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

For a free mode $f(k,\eta)$, the right-right bulk propagator is time ordered,
$$
G_{RR}(\eta_1,\eta_2;k)=
\theta(\eta_1-\eta_2)f(\eta_1)f^*(\eta_2)
+\theta(\eta_2-\eta_1)f(\eta_2)f^*(\eta_1),
$$
whereas the mixed propagators are Wightman functions.


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