Construction
1 Ghosts
One starts with the “classical action” 
which is invariant under some gauge symmetry.
Let 
be the “classical” space of fields, e.g. for Yang-Mills theory the fields are 
.
Suppose the gauge symmetry is 
, with the Lie algebra 
.
Then we should introduce ghost fields, geometrically:
where the action of
is via
right shift on
:
Notice that this commutes with
.
We can always find a representative with
(
i.e. choose
).
Then, with the standard notation
:
2 Integration measure
We assume that
comes with some integration measure:
This should be understood as an integration measure,
i.e. a density of weight
(rather than a function of
).
The product of this measure with the canonical measure on
gives us a measure on
which we will call
. Notice that
preserves this measure.
This can be proven as follows. For any function
:
because
comes from the canonical odd vector field on
.
3 Lift of symmetries to BRST configuration space
Original gauge symmetries can be lifted to the BRST field space as
left shifts on
:
Notice the following properties of left shifts:
on
(denoting “
” the coordinate on the fiber
) and
on
.
satisfy:
