BRST Quantization - Gauge Bundles and The Vertical Ideal

Gauge Bundles and The Vertical Ideal

In order to do the BRST method justice, we must switch from the "algebra-valued fields on Minkowski space" picture typical of quantum field theory texts (and of the above exposition) to the language of fiber bundles, in which there are two quite different ways to look at a gauge transformation: as a change of local section (also known in general relativity as a passive transformation) or as the pullback of the field configuration along a vertical diffeomorphism of the principal bundle. It is the latter sort of gauge transformation that enters into the BRST method. Unlike a passive transformation, it is well-defined globally on a principal bundle with any structure group over an arbitrary manifold; this is important in several approaches to a Theory of Everything. (However, for concreteness and relevance to conventional QFT, this article will stick to the case of a principal gauge bundle with compact fiber over 4-dimensional Minkowski space.)

A principal gauge bundle over a 4-manifold is locally isomorphic to, where the fiber is isomorphic to a Lie group, the gauge group of the field theory (this is an isomorphism of manifold structures, not of group structures; there is no special surface in corresponding to, so it is more proper to say that the fiber is a -torsor). Thus, the (physical) principal gauge bundle is related to the (mathematical) principal G-bundle but has more structure. Its most basic property as a fiber bundle is the "projection to the base space", which defines the "vertical" directions on (those lying within the fiber over each point ). As a gauge bundle it has a left action of on which respects the fiber structure, and as a principal bundle it also has a right action of on which also respects the fiber structure and commutes with the left action.

The left action of the structure group on corresponds to a mere change of coordinate system on an individual fiber. The (global) right action of a (fixed) corresponds to an actual automorphism of each fiber and hence to a map of to itself. In order for to qualify as a principal -bundle, the global right action of each must be an automorphism with respect to the manifold structure of with a smooth dependence on —i. e., a diffeomorphism from to .

The existence of the global right action of the structure group picks out a special class of right invariant geometric objects on —those which do not change when they are pulled back along for all values of . The most important right invariant objects on a principal bundle are the right invariant vector fields, which form an ideal of the Lie algebra of infinitesimal diffeomorphisms on . Those vector fields on which are both right invariant and vertical form an ideal of, which has a relationship to the entire bundle analogous to that of the Lie algebra of the gauge group to the individual -torsor fiber .

We suppose that the "field theory" of interest is defined in terms of a set of "fields" (smooth maps into various vector spaces) defined on a principal gauge bundle . Different fields carry different representations of the gauge group, and perhaps of other symmetry groups of the manifold such as the Poincaré group. One may define the space of local polynomials in these fields and their derivatives. The fundamental Lagrangian density of one's theory is presumed to lie in the subspace of polynomials which are real-valued and invariant under any unbroken non-gauge symmetry groups. It is also presumed to be invariant not only under the left action (passive coordinate transformations) and the global right action of the gauge group but also under local gauge transformations—pullback along the infinitesimal diffeomorphism associated with an arbitrary choice of right invariant vertical vector field .

Identifying local gauge transformations with a particular subspace of vector fields on the manifold equips us with a better framework for dealing with infinite-dimensional infinitesimals: differential geometry and the exterior calculus. The change in a scalar field under pullback along an infinitesimal automorphism is captured in the Lie derivative, and the notion of retaining only the term linear in the scale of the vector field is implemented by separating it into the inner derivative and the exterior derivative. (In this context, "forms" and the exterior calculus refer exclusively to degrees of freedom which are dual to vector fields on the gauge bundle, not to degrees of freedom expressed in (Greek) tensor indices on the base manifold or (Roman) matrix indices on the gauge algebra.)

The Lie derivative on a manifold is a globally well-defined operation in a way that the partial derivative is not. The proper generalization of Clairaut's theorem to the non-trivial manifold structure of is given by the Lie bracket of vector fields and the nilpotence of the exterior derivative. And we obtain an essential tool for computation: the generalized Stokes theorem, which allows us to integrate by parts and drop the surface term as long as the integrand drops off rapidly enough in directions where there is an open boundary. (This is not a trivial assumption, but can be dealt with by renormalization techniques such as dimensional regularization as long as the surface term can be made gauge invariant.)