Classical BV formalism
1 Expansion in Planck constant
Let us substitute the following ansatz for 
:
:
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where 
was defined previously.
In this case:
was defined previously.
In this case:
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In the “classical” BV formalism, we want to just drop the first term 
, but we have to
be careful. Notice that we are dealing with multilocal expressions, basically products
of integrals. By definition, we will consider such expression to be classical,
if the power of 
is equal to the number of integrals:
, but we have to
be careful. Notice that we are dealing with multilocal expressions, basically products
of integrals. By definition, we will consider such expression to be classical,
if the power of is equal to the number of integrals:
where each 
is a local expression of the form:
is a local expression of the form:
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On such expressions, we define the classical action of 
as follows:
as follows:
Notice that:
Both terms on the right hand side are equally classical: the power of
is the same as the number of integrals.The term with
came from the application of . The lesson is that we can (and should) only neglect
when acting on expressions with single integration. When acting on the multilocal expression is essential.
2 Generalization and algebraic interpretation
We might need a small generalization (basically, a completion) of the space of expressions of the form (20):
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The generalization of Eq. (21) is straightforward:
This construction has the following Lie-algebraic interpretation. Notice that the space of expressions of the form
is closed under the odd Poisson bracket. Let us call this space 
.
The odd Poisson bracket determines therefore the Lie superalgebra structure on 
.
The standard Lie algebra homology complex of has the canonical BV structure with the BV generator
given by Eq. (22).
is closed under the odd Poisson bracket. Let us call this space .
The odd Poisson bracket determines therefore the Lie superalgebra structure on .
The standard Lie algebra homology complex of has the canonical BV structure with the BV generator
given by Eq. (22).Consideration of arbitrary smooth functions
requires some completion of the tensor product
. Simple linear-algebraic tensor product leads to finite sums of products of functions of
, … ,
. We feel that this is actually even enough for our purposes.