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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.
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.
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)
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