Restoring the gauge invariance in non-Abelian second-class theories

Abstract

In this paper, we propose a generalization of an improved gauge unfixing formalism in order to generate gauge symmetries in the non-Abelian valued systems. This generalization displays a proper and formal reformulation of second-class systems within the phase space itself. We then present our formalism in a manifestly gauge invariant resolution of the SU (N ) massive Yang-Mills and SU (2) Skyrme models where gauge invariant variables are derived allowing then the achievement of Dirac brackets, gauge invariant Hamiltonians and first-class Lagrangians.

Improved gauge-unfixing formalism through a prototypical second-class system

“…showing that we have in fact achieved a parent first-class Abelian strongly involutive description for the original prototypical starting system. Actually, the result (50) was naturally expected and confirms consistency, as we have been able to solve (38) as a strong equality in whole phase space. To see how all this works in practical terms, in the next section we discuss an application to the nonlinear sigma model.…”

“…In fact, (39) demands gauge invariance under T α (q k )generated transformations (35), while (40) assures that the gauge-fixing choice χ α = 0 recovers the original secondclass system. In this way, following [26], we expand (38) in powers of χ α and write qi = q i +b iα χ α +b iαβ χ α χ β +b iαβγ χ α χ β χ γ +. .…”

Improved Gauge-Unfixing Formalism through a Prototypical Second-Class System

“…In fact, (39) demands gauge invariance under T α (q k )-generated transformations (35), while (40) assures that the gauge-fixing choice χ α = 0 recovers the original second-class system. In this way, following [26], we expand (38) in powers of χ α and write qi =…”

“…Then this gauge-invariant form can be used to easily obtain the modified Hamiltonian, constraints and all gauge invariant phase space functions. Modern relevant applications of the GU formalism can be seen in references [33,34,35,36,37,38].…”