Abelian gauge symmetries symmetry

The formalism generalizes immediately to higher global non-Abelian gauge symmetries. Let Tj j = be the generators... 

From the foregoing, it becomes clear that fields and potentials are freely intermingled in the symmetry-broken Lagrangians of the Higgs mechanism. To close this section, we address the question of whether potentials are physical (Faraday and Maxwell) or mathematical (Heaviside) using the non-Abelian Stokes theorem for any gauge symmetry ... 

The non-Abelian Stokes theorem is a relation between covariant derivatives for any gauge group symmetry ... 

In order to form a self-consistent description [44] of interferometry and the Aharonov-Bohm effect, the non-Abelian Stokes theorem is required. It is necessary, therefore, to provide a brief description of the non-Abelian Stokes theorem because it generalizes the ordinary Stokes theorem, and is based on the following relation between covariant derivatives for any internal gauge group symmetry ... 

This non-Abelian gauge theory satisfies the usual transformation properties. If Jl is the base manifold in four dimensions, then the gauge theory is determined by an internal set of symmetries described by a principal bundle. Let Ua, where a = 1,2,be an atlas of charts on the Ji. The transitions from one chart to another is given by gap f/p —> Ua, where these determine the transition functions between sections on the principal bundle. The transform between one section to another is given by... 

T. Kibble, Symmetry breaking in non-abelian gauge theories, Phys. Rev. 155 (1966) 1554-1561. 

Non-Abelian electrodynamics has been presented in considerable detail in a nonrelativistic setting. However, all gauge fields exist in spacetime and thus exhibits Poincare transformation. In flat spacetime these transformations are global symmetries that act to transform the electric and magnetic components of a gauge field into each other. The same is the case for non-Abelian electrodynamics. Further, the electromagnetic vector potential is written according to absorption and emission operators that act on element of a Fock space of states. It is then reasonable to require that the theory be treated in a manifestly Lorentz covariant manner. 

The gauge transformations form a group. It is Abelian, i.e. diflferent transformations of the group commute with each other, and it is onedimensional, i.e. the transformations are specified by one parameter 0. This group is C/(l), the group of unitary transformations in one dimension. We say that 17(1) is a symmetry group of , and that the functions form a one-dimensional representation of U 1). 

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