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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 10th 2018
• (edited Jan 10th 2018)

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:

1. 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.

2. 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:

$\mathcal{R} \circ \{-S',(-)\} \circ \mathcal{R}^{-1} \;=\; - \left( \left\{ S' + S_{int} \,,\, (-) \right\}_{\mathcal{T}} + i \hbar \Delta_{BV} \right)$