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I have started spelling out details at quantum master equation, following the rigorous derivation in causal perturbation theory due to Fredenhagen-Rejzner 11b, Rejzner 11.
So far I have added some backgound infrastructure and then the proof of this theorem ((Rejzner 11, (5.35) - (5.38)):
Consider an adiabatically switched non-point-interaction action functional in the form of a regular polynomial observable
$S_{int} \;\in\; PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \,,$Then the following are equivalent:
The quantum master equation (QME)
$\tfrac{1}{2} \{ S' + S_{int}, S' + S_{int} \}_{\mathcal{T}} + i \hbar \Delta_{BV}( S' + S_{int} ) \;=\; 0$holds on regular polynomial observables.
The perturbative S-matrix on regular polynomial observables is $BV$-closed
$\{-S', \mathcal{S}(S_{int})\} = 0 \,.$Moreover, if these equivalent conditions hold, then the interacting quantum BV-differential is equal, up to a sign, to the sum of the time-ordered antibracket with the total action functional $S' + S_{int}$ and $i \hbar$ times the BV-operator:
$\mathcal{R} \circ \{-S',(-)\} \circ \mathcal{R}^{-1} \;=\; - \left( \left\{ S' + S_{int} \,,\, (-) \right\}_{\mathcal{T}} + i \hbar \Delta_{BV} \right)$1 to 1 of 1