nLab
gauge fixing

Context

\infty-Chern-Weil theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

Configuration spaces in physics are typically not just plain spaces, but are groupoids and generally ∞-groupoids. This is traditionally most familiar for configuration spaces of gauge theories:

The operation called gauge fixing can essentially be understood as the passage to a more skelatized ∞-groupoid: in the simplest case, it amounts to picking one representative configuration from each equivalence class.

Generally, since the configuration ∞-groupoid typically carries extra structure in that it is a topological ∞-groupoid or a Lie ∞-groupoid or the like, the choice of gauge slice as it is called is similarly required to preserve that structure.

Examples

Standard textbook examples of gauge fixings include the following:

  • in the gauge theory of the electromagnetic field, a field configuration is, on a given pseudo-Riemannian manifold a line bundle with connection. Often the special case is considered where the underlying manifold is just Minkowski space and the bundle is assumed to be trivial, in which case a configuration of the gauge field configuration is just a 1-form AA on Minkowski space, and a gauge transformation λ:AA\lambda: A \to A' is a 0-form, i.e. a function, such that A=A+dλA' = A + d \lambda.

    • in the Lorenz gauge? the gauge field AA is taken to be a harmonic 1-form ddA=0d \star d \star A = 0.

Category-theoretic description

Here are more details on how one may think of gauge fixing from the nPOV.

In the Freed and Alm-Schreiber approach to quantization, the action functional is a functor

e iS:XnVect, e^{\mathrm{i}S}:X \to nVect,

where XX is some (,n)(\infty,n)-groupoid called the space of fields. The space of fields comes equipped with a projection π:XM\pi:X\to M to an (,n)(\infty,n)-groupoid MM called the moduli space of the quantum theory. Then the (path integral) quantization is, if it exists, the Kan extension Z:MnVectZ:M\to nVect of e iSe^{\mathrm{i}S} along π\pi. The functor ZZ is customary called the partition function of the theory.

A gauge fixing is a choice of a subgroupoid X gfX_{gf} of XX such that the inclusion X gfXX_{gf}\hookrightarrow X is an equivalence. The basic idea of gauge fixing is that the gauge fixed partition functions Z gfZ_{gf}, when they exist, are independent of the particular gauge fixing (since gauge fixing are all equivalent each other) and are equivalent to the original partition function ZZ (since X gfX_{gf} is equivalent to XX).

A classical instance of gauge fixing is when X=X˜//GX=\tilde{X}//G is an action groupoid, for the action of some group GG (the gauge group) on a manifold X˜\tilde{X}. In this case a classical gauge fixing is the choice of a slice SS in X˜\tilde{X} intersecting each orbit of GG exactly once. If the action of GG on X˜\tilde{X} is not free, there still will be nontrivial automorphisms in the groupoid S//GS//G; these residual internal symmetries are sometimes called ghost symmetries

Classical gauge fixings

Examples

Revised on December 28, 2013 14:30:06 by Urs Schreiber (89.204.130.243)