Dirac's Observables for the Higgs Model: I) the Abelian Case

TL;DR

This paper analyzes the Dirac observables in the classical Abelian Higgs model with fermions, revealing two distinct phases with different properties of the electromagnetic field and Hamiltonian structure.

Contribution

It provides a canonical basis of Dirac's observables for the Abelian Higgs model, distinguishing electromagnetic and Higgs phases and analyzing their Hamiltonian and Lagrangian formulations.

Findings

Electromagnetic phase has a massless field with nonlocal Hamiltonian and Lagrangian.

Higgs phase features a massive electromagnetic field with a local, nonanalytic Hamiltonian.

The model exhibits two disjoint solution sectors corresponding to different physical phases.

Abstract

We search a canonical basis of Dirac's observables for the classical Abelian Higgs model with fermions in the case of a trivial U(1) principal bundle. The study of the Gauss law first class constraint shows that the model has two disjoint sectors of solutions associated with two physically different phases. In the electromagnetic phase, the electromagnetic field remains massless: after the determination of the Dirac's observables we get that both the reduced physical Hamiltonian and Lagrangian are nonlocal. In the Higgs phase, the electromagnetic field becomes massive and in terms of Dirac's observables we get a local, but nonanalytic in the electric charge (or equivalently in the sum of the electromagnetic mass and of the residual Higgs field), physical Hamiltonian; however the associated Lagrangian is nonlocal. Some comments on the R-gauge-fixing, the possible elimination of the residual Higgs field and on the Nielsen-Olesen vortex solution close the paper.