Interactions and State Constraints via Induced Nonlinear Realizations of Lie Groups

Abstract: This is a study of induced nonlinear realizations of a Lie group G in which the presence of one field induces nonlinear transformations on another field. The covariant derivative structure is similar in form to that for local gauge theory. For an arbitrary Lie group, basic equations and non standard invariant Lagrangian forms are described. Covariant constraint equations that place restrictions on field components are presented. With G = SU(2), a detail application to the electroweak model is discussed. We first show that the standard Lagrangian for the gauge $SU(2) \times U(1)$ electroweak model is invariant. We then show that an alternate invariant Lagrangian is also possible. In it, the intermediate boson masses arise from the adjoint field rather than from the Higgs doublet. An alternate invariant lepton Lagrangian is presented. Covariant constraints on the right-handed lepton field lead to two right-handed neutrino fields. One vanishes at the point where we obtain a massless (photon) field and the second one has no interaction. This provides a clear explanation why weak interactions do not involve right-handed neutrinos, while permitting neutrinos with mass. This model also indicates a different region of matter involving coupled leptons that are "blind" to the massless electromagnetic field but "see" four massive potentials that are themselves blind to the electromagnetic field. We argue that these more difficult to detect "dark" fields provide a possible contribution to the missing mass.