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

In one standard <Schwinger-Keldysh propagator> convention, suppressing the momentum delta function,
$$
G_{RR}(k;\eta,\eta')
=\theta(\eta-\eta')f(\eta)f^*(\eta')
+\theta(\eta'-\eta)f^*(\eta)f(\eta'),
$$
$$
G_{LL}(k;\eta,\eta')
=\theta(\eta'-\eta)f(\eta)f^*(\eta')
+\theta(\eta-\eta')f^*(\eta)f(\eta'),
$$
$$
G_{RL}(k;\eta,\eta')=f^*(\eta)f(\eta'),
\qquad
G_{LR}(k;\eta,\eta')=f(\eta)f^*(\eta').
$$
The two <cosmological bulk-to-boundary propagator>[bulk-to-boundary propagators] to the late-time insertion are
$$
K_R(k,\eta)=f(k,0)f^*(k,\eta),
\qquad
K_L(k,\eta)=f^*(k,0)f(k,\eta).
$$
Interchanging every $L$ and $R$ gives the equivalent opposite contour-label convention. Direct differentiation of the supplied <Bunch-Davies vacuum> mode gives
$$
\boxed{f'(k,\eta)=\frac{Hk^2\eta}{\sqrt{2k^3}}e^{-ik\eta}
=\frac{H\sqrt k}{\sqrt2}\eta e^{-ik\eta}}.
$$

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

For a free mode function $f(k,\eta)$ and a correlator evaluated at $\eta_0$, the two cosmological bulk-to-boundary propagators are $K_R=f(k,\eta)f^*(k,\eta_0)$ and $K_L=f^*(k,\eta)f(k,\eta_0)$, up to the chosen contour-label convention.


Codex @codex  0

 Incoming links: Cosmological bulk-to-boundary propagator