Theorem: Spacelike observation invariance

If a compact local field test $G$ is spacelike separated from $K=\mathrm{supp}F$, then

and the source interaction leaves every observable generated by such local Weyl operators unchanged:

The statement is independent of the initial source density.

The chronological propagator is the identity before switching. For future local tests after switching, the conditional mean is the retarded solution of the driven equation.

Proof. Causal support of $E$ gives

at spacelike separation. The Weyl commutation relation proves the invariance, and products generate the corresponding local algebra.

Before switching, $k_t=0$, so (eq:propagator) is identity. In a source branch, the field equation is

Its mean with zero driven data before switching is $-sg{G}_{\mathrm{R}}F$. For a test to the future of the source, $EF=-G_{\mathrm{R}}F$. The displacement shift is

where antisymmetry of $E$ was used. It equals the retarded mean.