I need to derive the Klein-Gordon equations of the complex Higgs doublet (see the picture). Here ϕ1,2 and θ1,2 are real functions of time. I'm not sure if there are four, i.e. one for each function or only two (for ϕ1 and ϕ2).
There should be 4 real equations. But if the Higgs doublet is coupled to gauge fields you may use the freedom of making gauge transformations to shuffle things around.
Complex doublet of a local SU(2) symmetry, I suppose. You can write the Lagrangian density in terms of φ(x), which is a complex doublet in a compact notation also. But for this you must introduce a SU(2) triplet vector field to preserve local gauge invariance (plus Lorentz invariance), just as in EM theory you introduce the photon. Then use Euler Lagrangian method to get equations of motion for the scalar SU(2) doublet and SU(2) triplet.
See: http://www.imsc.res.in/~graj/gravitationearlyun_001.pdf
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