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 5th 2018
   • (edited Jan 5th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am starting to write up at _[[BV-operator]]_ an account of the rigorous derivation/construction of the BV-operator and the BV quantum master equation in causal perturbation theory, due to Fredenhagen-Rejzner. As a first step, the statement and proof of the BV-operator arising as the difference of the plain and time-ordered BV-differential in free field theoy is now [here](https://ncatlab.org/nlab/show/BV-operator#InPerturbativeQuantumFieldTheory).

   I am starting to write up at BV-operator an account of the rigorous derivation/construction of the BV-operator and the BV quantum master equation in causal perturbation theory, due to Fredenhagen-Rejzner.

   As a first step, the statement and proof of the BV-operator arising as the difference of the plain and time-ordered BV-differential in free field theoy is now here.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 5th 2018
   • (edited Jan 5th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexWhile I am not concerned with the path integral picture for the moment, for completeness of the entry _[[BV-operator]]_ I have added a section _[For finite dimensional toy path integrals](https://ncatlab.org/nlab/show/BV-operator#ForFiniteDimensionalToyPathIntegrals)_. This contains now a brief explanation of how a BV-operator on finite dimensional manifolds induces a homological interpretation of integration of differential forms, and how one imagines this to yield a homological interpretation of path integrals. This is (and has been) discussed in more detail in the entry _[[BV-BRST formalism]]_ and so at _[[BV-operator]]_ I ended that section with a pointer to the relevant section [there](https://ncatlab.org/nlab/show/BV-BRST+formalism#HomologicalIntegration). In the course of this I did a little cleaning up at _[[BV-BRST formalism]]_. In particular I removed the previous sections "The standard construction" and "As a derived critical locus" (which were pretty rough and incomplete) with a new section "Classical BV as homological resolution of the reduced phase space" which I made a glorified pointer to the detailed polished discussion which is now at _[[A first idea of quantum field theory]]_, chapter _[11. Reduced phase space](https://ncatlab.org/nlab/show/BV-BRST+formalism#ClassicalBVAsHomologicalResolutionOfReducedPhaseSpace)_.

   While I am not concerned with the path integral picture for the moment, for completeness of the entry BV-operator I have added a section For finite dimensional toy path integrals.

   This contains now a brief explanation of how a BV-operator on finite dimensional manifolds induces a homological interpretation of integration of differential forms, and how one imagines this to yield a homological interpretation of path integrals. This is (and has been) discussed in more detail in the entry BV-BRST formalism and so at BV-operator I ended that section with a pointer to the relevant section there.

   In the course of this I did a little cleaning up at BV-BRST formalism. In particular I removed the previous sections “The standard construction” and “As a derived critical locus” (which were pretty rough and incomplete) with a new section “Classical BV as homological resolution of the reduced phase space” which I made a glorified pointer to the detailed polished discussion which is now at A first idea of quantum field theory, chapter 11. Reduced phase space.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 9th 2018
   • (edited Jan 9th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added to _[[BV-operator]]_ discussion of the relation to the antibracket, such as $$ \{A_1, A_2\} \;=\; (-1)^{deg(A_2)} \, \Delta_{BV}(A_1 \cdot A_2) - (-1)^{deg(A_2)} \, \Delta_{BV}(A_1) \cdot A_2 - A_1 \cdot \Delta_{BV}(A_2) $$ etc. ([here](https://ncatlab.org/nlab/show/BV-operator#InCausalPerturbationTheoryRelationToTimeOrderedAntibracket))

   I have added to BV-operator discussion of the relation to the antibracket, such as

   $$\{A_1, A_2\} \;=\; (-1)^{deg(A_2)} \, \Delta_{BV}(A_1 \cdot A_2) - (-1)^{deg(A_2)} \, \Delta_{BV}(A_1) \cdot A_2 - A_1 \cdot \Delta_{BV}(A_2)$$

   etc. (here)