nLab
group averaging

Contents

Idea

In the context of gauge theory, group averaging is the idea of turning an arbitrary function of the fields into an invariant (gauge invariant) function by suitably averaging it over the action of the group of gauge transformations, hence by taking the integral against a measure on the group of gauge transformations of the function pulled back by the gauge action.

For groups of gauge transformations which are finite dimensional compact Lie groups (typical in quantum mechanics) this just works, but most groups of gauge transformations appearing in quantum field theory are infinite-dimensional (at best) and non-compact, hence much of the literature on group averaging in gauge theory is concerned with making sense of regularizing the naive prescription.

(Beware the terminology: the gauge group is frequently finite-dimensional and compact (typically in Yang-Mills theory) but the group of gauge transformations is typically something like the group of suitable functions from spacetime into the gauge group. Some mathematical texts say “gauge group” for “group of gauge transformations”, but this is not what physics texts do and should probably better be avoided.)

References

Textbook accounts include

  • Peter Olver, around (4.8) in Classical invariant theory, 1999

Reviews include

  • Domenico Giulini, Group Averaging and Refined Algebraic Quantization, Nucl.Phys.Proc.Suppl. 88 (2000) 385-388 (arXiv:gr-qc/0003040)

Last revised on August 19, 2018 at 23:46:43. See the history of this page for a list of all contributions to it.