Wave operator and s-wave reduction

We use

For both the real system field $\varphi$ and real probe field $\psi$, the free action is

Both fields are massless.

Let $Y_0=(4\pi {)}^{-1/2}$. Writing

in the selected angular sector gives

For a real reduced test $f(t,x)$, define its bulk representative

Then

This follows from $\mathrm{d}{\mathrm{vol}}_g=r^{2}F\mathrm{d}t\mathrm{d}x\mathrm{d}\Omega$. Compact reduced support gives compact bulk support outside the horizon.

The causal propagator convention is

It is consistent with

For $\mathsf{D}={{\partial }_t}^2+\mathsf{L}$, the retarded solution has the usual positive sine kernel. The advanced-minus-retarded propagator has the corresponding negative sine kernel.