Spacetime, tests, and propagator conventions

Take

All distributions act on compact smooth test densities; the fixed Minkowski volume identifies them with test functions. Set $\mathcal{D}(U)={C_c}^{\infty }(U)$, ${\mathcal{D}}^{\prime }(U)=\mathcal{D}(U{)}^{\prime }$, and ${\mathcal{E}}^{\prime }(U)$ for compactly supported distributions. For a finite species space, use componentwise distribution spaces.

The massless retarded fundamental solution is

The formula is understood as the standard cone measure, including its defined extension at the vertex, with $\square R_{\mathrm{ret}}={\delta }_0$. Use ${E_0}^{\mathrm{R}}$ for convolution with $R_{\mathrm{ret}}$ on compact or suitably past-compact inputs, and ${E_0}^{\mathrm{A}}$ for its transpose. Similarly ${E_m}^{\mathrm{R}/A}$ are the Green operators of $\square +m^2$. Existence, causal support, and Cauchy uniqueness are standard BGP.

The fixed smooth observation $f$ and interaction switch $g$ are compactly supported. The fixed smooth ramp ${\sigma }_{\epsilon }(t)$ vanishes for $t\le 0$, with fixed $\epsilon >0$; no sharp-ramp limit is used. Its precise transition profile is an input.

For a symmetric covariant tensor define trace reversal

In four dimensions ${\mathcal{P}}^2=\mathrm{id}$. The metric response operator and prescribed inverse are

They must not be confused with the coordinate-response kernel $L_{\mathrm{coord}}$ below, or with Track A's rate $R$.