Named inverses and causal brackets

Let ${E_{\mu }}^{\mathrm{R}/A}$ be the retarded/advanced inverse of $\square +{\mu }^2$, and put $E_{\mu }={E_{\mu }}^{\mathrm{R}}-{E_{\mu }}^{\mathrm{A}}$. On compact sources,

These inverses have causal support. We use the standard distributional extension and smooth Cauchy theory BGP. The reduced causal operator is

It belongs to $\mathbf{P}$, not to ${\mathsf{K}}_{\mathrm{act}}$.

The retarded/advanced inverses of ${\mathsf{K}}_{\mathrm{act}}$ are

For example, for source $(U,v)$ the auxiliary solution is

Indeed $b=D\gamma -v/2$ because $D\mathcal{R}{E_0}^{\mathrm{R}/A}v=v/2$. Substitution verifies both inverse identities. The scalar and ghost inverse identities follow by direct multiplication.

The action causal matrix $\Delta ={G_{\mathrm{act}}}^{\mathrm{R}}-{G_{\mathrm{act}}}^{\mathrm{A}}$ therefore has blocks

All other blocks vanish. The causal algebra ${\mathcal{A}}_0(M)$ is the completed graded algebra generated by compact linear smearings, modulo the equations and