Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 10th 2018
   • (edited Jan 10th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have started spelling out details at _[[quantum master equation]]_, following the rigorous derivation in causal perturbation theory due to [Fredenhagen-Rejzner 11b](https://ncatlab.org/nlab/show/quantum+master+equation#FredenhagenRejzner11b), [Rejzner 11](https://ncatlab.org/nlab/show/quantum+master+equation#Rejzner11). So far I have added some backgound infrastructure and then the proof of [this theorem](https://ncatlab.org/nlab/show/quantum+master+equation#QuantumMasterEquation) ((Rejzner 11, (5.35) - (5.38)): *** Consider an [[adiabatic switching|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]]. 1. 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 product|time-ordered]] [[antibracket]] with the _total_ [[action functional]] $S' + S_{int}$ and $i \hbar$ times the [[BV-operator]]: $$ \mathcal{R} \circ \{-S',(-)\} \circ \mathcal{R}^{-1} \;=\; - \left( \left\{ S' + S_{int} \,,\, (-) \right\}_{\mathcal{T}} + i \hbar \Delta_{BV} \right) $$

   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:

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