Unconstrained Hamiltonian Formulation of SU(2) Gluodynamics

TL;DR

This paper develops an unconstrained Hamiltonian formulation for SU(2) Yang-Mills theory, simplifying the gauge constraints and deriving an effective low-energy nonlinear sigma model, with potential links to Faddeev-Niemi models.

Contribution

It introduces a canonical transformation that Abelianizes Gauss law constraints and derives a reduced Hamiltonian and effective Lagrangian for SU(2) gluodynamics.

Findings

Reduced Hamiltonian with gauge degrees of freedom eliminated

Derived effective nonlinear sigma model Lagrangian

Analyzed the ground state wave functional in strong coupling limit

Abstract

SU(2) Yang-Mills field theory is considered in the framework of the generalized Hamiltonian approach and the equivalent unconstrained system is obtained using the method of Hamiltonian reduction. A canonical transformation to a set of adapted coordinates is performed in terms of which the Abelianization of the Gauss law constraints reduces to an algebraic operation and the pure gauge degrees of freedom drop out from the Hamiltonian after projection onto the constraint shell. For the remaining gauge invariant fields two representations are introduced where the three fields which transform as scalars under spatial rotations are separated from the three rotational fields. An effective low energy nonlinear sigma model type Lagrangian is derived which out of the six physical fields involves only one of the three scalar fields and two rotational fields summarized in a unit vector. Its possible relation to the effective Lagrangian proposed recently by Faddeev and Niemi is discussed. Finally the unconstrained analog of the well-known nonnormalizable groundstate wave functional which solves the Schr\"odinger equation with zero energy is given and analysed in the strong coupling limit.