Nonabelian gauge symmetries

10 Relativistic Lagrangian theories 10.3.3 Nonabelian gauge symmetries 

Electro weak theory (EWT) [73,340,217] makes use of the gauge group (7( 1) x SU(2). The representation matrices are of the form 

For the abelian phase factor of the (7(1) group, U = exp ieyix) and ly, = —(ie/Tic)A/x, which produces the usual gauge transformation As, + 

The 5(7(2) gauge field W), has three components, corresponding to the isospin vector of matrices r, with no relationship to the coordinate space ct, x. By implication, the fermion field i// is a set of spinors, one for each value of the isospin index. The covariant derivative 

Considering only the SU(2) field interaction, the Euler-Lagrange equation for the fermion field is 

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Because field quantization falls outside the scope of the present text, the discussion here has been limited to properties of classical fields that follow from Lorentz and general nonabelian gauge invariance of the Lagrangian densities. Treating the interacting fermion field as a classical field allows derivation of symmetry properties and of conservation laws, but is necessarily restricted to a theory of an isolated single particle. When this is extended by field quantization, so that the field amplitude rjr becomes a sum of fermion annihilation operators, the theory becomes applicable to the real world of many fermions and of physical antiparticles, while many qualitative implications of classical gauge field theory remain valid. 

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