Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Geometry of gauge fields

  1. Jun 3, 2009 #1
    Recently I've been trying to understand the geometric formulation of gauge field theory.

    The mantra I've been hearing is that a gauge field is a connection for a principal bundle where the structure group corresponds to the gauge group. Fields which are charged under the gauge group form sections of associated vector bundles.

    John Baez's book has a slight different take, however, in which he essentially defines the gauge field as the connection for a G-bundle, which is vector bundle obtained by gluing together trivial bundles subject to G-identifications on the overlaps, what need then for the principal bundle?

    I'm also grappling with the idea that a choice of gauge corresponds to a local trivialization. The idea seems to be that sufficiently small regions of the bundle are diffeomorphic to the product [itex]U \times F[/itex] where F is the typical fiber. I don't quite see how the different gauge possibilities arise out of this, however.

    Thanks.
     
  2. jcsd
  3. Jun 3, 2009 #2

    tom.stoer

    User Avatar
    Science Advisor

    Regarding you last remark "I don't quite see how the different gauge possibilities arise out of this". I think one can't see that. One has to chose a global gauge function and check if it induces the correct local gauge sections. The other way round, namely to start with a local condition and try to "globalize" it may work if you have a trivial bundle,but in general it will not work.

    Regarding Baez's book: I am not familiar with these ideas.
     
  4. Jun 3, 2009 #3

    garrett

    User Avatar
    Gold Member

  5. Jun 3, 2009 #4
    Well, in my opinion it would make more sense to define a gauge choice on some spacetime patch as a section of a local trivialization of the principal fiber bundle.

    Is this the correct way of thinking about it?
     
  6. Jun 3, 2009 #5
    I think I can see where this is going.

    It makes sense to me that changing the local trivialization corresponds to a gauge transformation since it is assumed from the outset that the fibers are sewn together using smooth (group-valued) transition maps.

    If I want to do a gauge tranformation on some specified patch, then I need to figure out another way of assigning a local trivialization to that patch. But since I am working with a vector bundle with structure group G, the only way I know how to do this is to do a continuous G-tranformation of the fibers.

    Suppose that I want to do a gauge transformation on
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook