Inifinite-dimensional case
1 Problem
We want to apply the BV formalism for a field-theoretic system (the string worldsheet theory).
Unfortunately, we do not know how to rigorously define 
in this case.
The main problem is the appearence of ill-defined expressions when acting by the “quantum part” 
on local functionals:
in this case.
The main problem is the appearence of ill-defined expressions when acting by the “quantum part”
on local functionals:
Sometimes the result of application of is “obviously zero”, for example with two flavours of
scalar fields:
is “obviously zero”, for example with two flavours of
scalar fields:
We feel that in such cases it is justified to assume that acts as zero.
When acting on local functionals, the result seems to be either undefined (as in Eq. (17))
or just zero (as in Eq. (18)). It seems to be impossible to obtain any meaningful
result other than zero.
acts as zero.
When acting on local functionals, the result seems to be either undefined (as in Eq. (17))
or just zero (as in Eq. (18)). It seems to be impossible to obtain any meaningful
result other than zero.At the same time, the application to multilocal functionals, generally speaking, does lead to meaningful
nonzero results, for example:
to multilocal functionals, generally speaking, does lead to meaningful
nonzero results, for example:
2 Potentially unsafe operations
Operators and 
are, generally speaking, unsafe, because they
contain .
The Lie derivative defined 
defined in Eq. (10)
is safe as a function of 
, but unsafe in 
.
We will give an example of a computation where this is important.