Unsafe operations
As we have previously explained, the
operation 
is ill-defined in infinite-dimensional BV formalism.Now we will show how this can lead to 
not being actually closed due to anomalies.
We have to prove that 
where 
where 
.
The computation goes as follows. Consider the evaluation of 
on a vector 
; the antifield expansion in the vicinity of 
is:
where where .
The computation goes as follows. Consider the evaluation of on a vector ; the antifield expansion in the vicinity of is:
|
Terms 
of the higher order in 
do not matter.
of the higher order in do not matter.
In passing from Line (44) to Line (45) we used the fact that 
is non-anomalous, in other words
. In passing from Line (45) to Line (46) we used that 
is non-anomalous
in the sense that 
. The assumption that is non-anomalous is quite reasonable, in fact
necessary for the consistency of the theory. But the ground for assuming
is much less solid. One possibility is to require:
is non-anomalous, in other words
. In passing from Line (45) to Line (46) we used that is non-anomalous
in the sense that . The assumption that is non-anomalous is quite reasonable, in fact
necessary for the consistency of the theory. But the ground for assuming
is much less solid. One possibility is to require:
|
which is more or less equivalent to:
It is quite possible that there are other possible ways to make calculations
meaningful, without imposing the ugly constraint (47).