In physics, 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 principal 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 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
A quantity which is invariant under gauge transformations is the Wilson loop, which is defined over any closed path, &gamma, as follows:
In the theories of the electroweak interaction and quantum chromodynamics of the Standard Model of particle physics, the Lagrangians of bosons, which mediate interactions between fermions, are invariant under gauge transformations. This is why these bosons are called gauge bosons.
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