My initial plan was to have the second part of these notes be about gauge symmetry and the problems physicists have encountered in handling it, but as I started writing it quickly became apparent that explaining this in any detail would take me into various issues that are quite interesting, but far afield from what I want to get to. So, I hope to get back to this at some point, but for now will just assume that most of my readers know what gauge symmetry is, and that the rest just need to know that:
The actual situation is quite a bit more complicated than this, but for now we’ll focus on the simplest version of the mathematical problem that comes up here, and see how the BRST formalism deals with it. This posting will begin explaining one part of this story, starting with the simplest version of BRST cohomology, in a language familiar to physicists. The next posting will deal with Lie algebra cohomology in a more general mathematical context and work out some examples. For more about the material in this posting, see, for instance, Green, Schwarz and Witten, volume I, section 3.2.1, where they go on to apply this to the Virasoro algebra, or these lecture notes from Jose O’Figueroa-Farrill .
Physicists always begin by choosing a basis, in this case a basis of satisfying , where are called the structure constants of . A representation is then a set of linear operators on satisfying . Let be a basis of the dual space , dual to the basis .
Now, extend to , where is the exterior algebra on . On this space, define the “ghost” operator to be wedge-product with , and “anti-ghost” operator to be contraction (interior product) with . These operators satisfy “fermionic” anti-commutation relations
and one can get all vectors in from linear combinations of decomposable elements of (those given by repeated application of the to the “vacuum vector” ).
The ghost number operator on has eigenvectors the decomposable elements, with integer eigenvalues from 0 to dim , given by the number of ghost operators needed to produce the eigenvector from a vacuum vector.
The BRST operator is given by
which increases the ghost number by one, and has the crucial property of (this comes from the fact that the satisfy the Jacobi identity). The BRST cohomology is given by considering the space of elements of that are “BRST-closed”, i.e. satisfy , and identifying two such elements if they are “BRST-exact”, i.e. differ by for some . So BRST cohomology is defined by
with the component of the BRST cohomology of ghost number .
A vector of ghost number zero satisfies iff and only if for all i, so we can identify with the space of – invariant vectors in .
The essence of the BRST method is to replace the problem of finding the invariant subspace of a representation by the problem of finding the degree zero BRST cohomology .
There are two different ways of putting an inner product on and thus getting an inner product on ( is assumed to be unitary, so preserves a given inner product on ).
(this uses the “fermionic” or “Berezin” integral , although I have not properly dealt with signs here. ).
This inner product is indefinite, but it makes the BRST operator and ghost-operator self-adjoint.
(Note, here the integral sign is not Berezin integration, but the usual integration of differential forms over a compact manifold, in this case )
With this inner product and are not self-adjoint on . To get something self-adjoint, one can consider the operator where is the adjoint of , but this operator does not have a definite ghost-number.