Gauge theory
Basically gauge theories are based on the idea that symmetry transformations can only be performed locally. So, if you try to "rotate" something in a certain region, this does not determine how objects are rotated in another regions. So, the best way to summarize it is to say it is symmetry transformations are localized.In Mathematics, a gauge is some degree of freedom within a theory that has no observable effect. In fact, most of gauge theory as presented here is a topic of mathematical study in itself.
A gauge transformation is thus a transformation of this degree of freedom which does not modify any physical observable properties.
Gauge theories are usually discussed in the language of differential geometry.
If we have a principle bundle whose base space is space or spacetime and structure group is a Lie group G, then, the space of smooth (although in physics, we often don't deal with smooth functions) sections of this bundle forms a group, called the group of gauge transformations. We can define a connection on this principle bundle, yielding a Lie algebra valued 1-form, A. From this 1-form, we can construct a Lie algebra valued 2-form, F by
where d stands for the exterior derivative and stands for the wedge product.
Infinitesimal gauge transformations form a Lie algebra, which is characterized by a smooth Lie algebra valued scalar, ε. Under such an infinitesimal gauge transformation,
where is the Lie bracket.
One thing nice is if , then where D is the covariant derivative . Also, , which means F transforms covariantly.
One thing to note is that not all gauge transformations can be generated by infinitesimal gauge transformations in general. For example, if the base manifold is a compact boundariless manifold such that the homotopy class of mappings from that manifold to G is nontrivial. See instanton for an example.
The Yang-Mills action is now given by
where * stands for the Hodge dual and the integral is defined as in differential geometry.
A quantity which is invariant under gauge transformations is the Wilson loop, which is defined over any closed path, &gamma, as follows:
where χ is the character of a complex representation ρ and represents the path ordered operator.
Chern-Simons forms
See Chern-Simons.
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