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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 intPolyObs(E BV-BRST) reg[[]], S_{int} \;\in\; PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \,,

    Then the following are equivalent:

    1. The quantum master equation (QME)

      12{S+S int,S+S int} 𝒯+iΔ BV(S+S int)=0 \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 BVBV-closed

      {S,𝒮(S int)}=0. \{-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 intS' + S_{int} and ii \hbar times the BV-operator:

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