4

The physical minimal coupling procedure is usually expressed mathematically in the language of fibre bundles where instead of local presentations we deal with global objects - gauge fields are connections on a principal bundle with structure group of symmetries, matter fields are sections of bundles associated to the principal bundle. In order to come back to the usual physical version, we just present those object locally via a local section of the principal bundle (and often in the image of some coordinate maps). Now, the usual gauge transformations in the physical sense may be mathematically exercised in two, equivalent, ways:

  1. By changing the local section of the principal bundle

  2. By transforming the global connection form and matter fields with an element of the gauge group (isomorphic to sections of the adjoint bundle)

I know that these two produce the same change in local presentations (given that they are induced by the same section of the adjoint bundle) and my question regards possible interpretations of them.

  1. About the change of local section I think as something like "coordinate change", i.e. something not physically important.

  2. Transforming the global object seems more physical, in the sense that it links different fields of our field theory which produce the same value of the Lagrangian.

My questions are: do my interpretations agree with the usual view on this topic? which one of those two transformations physicists mean when they say "gauge transformations"?

szantag
  • 87
  • 4

1 Answers1

5

That's just the active/passive distinction you have for all transformations: Is a "rotation" an actual, physical motion of an object or is it rotating your coordinate system? It depends on the context - and often it doesn't actually matter!

There's nothing special about gauge transformations in this context, sometimes people will mean the active version and sometimes they will mean the passive version and sometimes they won't have thought about which they mean at all.

However, in terms of "bundle theory" it is not always sufficient to think about gauge transformations as happening on a fixed principal bundle: There are "gauge transformations" in physics that change the bundle and that's one meaning of the phrase "large gauge transformation", see this question and this question.

ACuriousMind
  • 124,833