Green operators and opposite causal conventions

For the componentwise massless wave equation on rank-$r$ tensors, write

Thus ${\mathsf{E}}_r$ is advanced minus retarded. If

then the time kernel of ${\mathsf{E}}_r$ is $S_{\mathrm{w}}(t)$.

For the massive scalar, use $G_R,G_A$, and set

The spatial Fourier kernels are

The scalar commutator below uses ${\Delta }_m$, not ${\mathsf{E}}_r$.

Compact distributions are allowed as Green-operator sources, using transposition. In particular,

for compact smooth tests $u$. For smooth spacelike-compact sources bounded below in time, the retarded solution is defined by zero data sufficiently far in the past. All support statements used below follow from finite propagation and these kernels.

We use the following compact-source consequence repeatedly:

In the forward direction the common support lies in $J^{+}(\mathrm{supp}C)\cap J^{-}(\mathrm{supp}C)$. In the reverse direction the Green identities on compact distributions give ${\mathsf{E}}_r\square A=0$.