What Are Large Gauge Transformations and Their Role in Physics?

[pinu]

Can some one explain what are the so called "large gauge transformations" and where do they play important role in physics? Explanations with less mathematical rigor will be greatly appreciated.

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

 

[Haelfix]

Large gauge transformations are gauge transformations that cannot be continously connected to the identity element homotopically. It's hard to explain this without math here.

Suffice it to say that they are quite different than usual small gauge transformations that act trivially on states in the Hilbert space. Here they are topologically non trivial (this is why its difficult to explain) and are associated with the actual global charges of a system.

I won't be able to explain further without getting into some formalism. Formalism which is explained in most textbooks on the geometrization of physics, like eg Nakahara. I think that Henneaux and Teitelboim also devote a section.

 

[pinu]

Thanks a lot for the reply!

Large gauge transformations are gauge transformations that cannot be continously connected to the identity element homotopically. It's hard to explain this without math here.

Can you please explain it in a bit detail (using required mathematical formalism).

 

[julian]

Yang-Mills theories are non-Abelian generalizations of Maxwell's theory. You start with a gauge group just like in Maxwell's theory, but instead of being a collection of numbers, it is a collection of matrices.

Small gauge transformation are defined with respect to group elements continuously connected with the unit matrix, as with a Taylor expansion of the exponential function about the unit matrix. If they are not `connected' to the unit matrix then they correspond to what are called large gauge transformations. Examples of `disconnected' gauge transformations occur with a non-Abelian gauge theory where you have a topologically non-trivial configuration space. Such disconnected gauge transformations do not occur with Abelian gauge theory - i.e. Maxwell's theory.

You want to look at the WKB approximation and "Instantons" (for a start) which can be used to probe the nonperturbative realm of gauge theories.

You can have a look at chapter 16 of "Quantum Field Theory" by Michio Kaku.

 

[torsten]

Yang-Mills theories are non-Abelian generalizations of Maxwell's theory. You start with a gauge group just like in Maxwell's theory, but instead of being a collection of numbers, it is a collection of matrices.

Small gauge transformation are defined with respect to group elements continuously connected with the unit matrix, as with a Taylor expansion of the exponential function about the unit matrix. If they are not `connected' to the unit matrix then they correspond to what are called large gauge transformations. Examples of `disconnected' gauge transformations occur with a non-Abelian gauge theory where you have a topologically non-trivial configuration space. Such disconnected gauge transformations do not occur with Abelian gauge theory - i.e. Maxwell's theory.

You want to look at the WKB approximation and "Instantons" (for a start) which can be used to probe the nonperturbative realm of gauge theories.

You can have a look at chapter 16 of "Quantum Field Theory" by Michio Kaku.

The explanation is perfect but let me add also the aspect of large diffeomorphisms (as large gauge transformations in GR). Here one can understand the problem geometrically.
A small diffeomorphism is a usual coordiante transformation. So, as an example let's take a torus i.e. a doughnut. Now cut this torus (to get a cylinder) then twist one side by at least 360° (or 2 Pi) and glue both ends together. You will get a torus but with a twist. This torus is diffeomorphic to the orginal torus but only by a large diffeomorphism (the procedure described above is called Dehn twist). If you choose an angle below 360° then you can describe it by a small diffeomorphism (coordiante transformation). Only the full twist can be described by a large diffeomorphism.