Dirac's Observables for the Higgs Model: II) the non-Abelian SU(2) Case

TL;DR

This paper analyzes the classical non-Abelian Higgs model with SU(2) symmetry, identifying Dirac observables and revealing a nonanalyticity in the coupling constant within the Higgs phase.

Contribution

It provides a canonical basis of Dirac's observables for the non-Abelian Higgs model and characterizes the physical Hamiltonian and Lagrangian, highlighting nonanalytic behavior in the coupling.

Findings

Identified two phases: massless SU(2) and Higgs phase with massive SU(2).

Derived the Dirac observables and reduced Hamiltonian for the Higgs phase.

Discovered nonanalyticity in the SU(2) coupling constant.

Abstract

We search a canonical basis of Dirac's observables for the classical non-Abelian Higgs model with fermions in the case of a trivial SU(2) principal bundle with a complex doublet of Higgs fields and with the fermions in a given representation of SU(2). Since each one of the three Gauss law first class constraints can be solved either in the corresponding longitudinal electric field or in the corresponding Higgs momentum, we get a priori eight disjoint phases of solutions of the model. The only two phases with SU(2) covariance are the SU(2) phase with massless SU(2) fields and the Higgs phase with massive SU(2) fields. The Dirac observables and the reduced physical (local) Hamiltonian and (nonlocal) Lagrangian of the Higgs phase are evaluated: the main result is the nonanalyticity in the SU(2) coupling constant, or equivalently in the sum of the residual Higgs field and of the mass of the SU(2) fields. Some comments on the function spaces needed for the gauge fields are made.