Unconstrained SU(2) Yang-Mills Theory with Topological Term in the Long-Wavelength Approximation

TL;DR

This paper reduces SU(2) Yang-Mills theory with a topological term to an unconstrained form using Hamiltonian methods, analyzing topological invariance and deriving an approximate local Lagrangian with topological insights.

Contribution

It provides a Hamiltonian reduction of SU(2) Yang-Mills theory with a heta term to an unconstrained nonlocal form and develops a consistent local approximation preserving topological properties.

Findings

Proves the heta independence of the reduced theory.

Derives an approximate local Lagrangian up to second order in derivatives.

Identifies a nonlinear sigma model with a Hopf invariant for degenerate configurations.

Abstract

The Hamiltonian reduction of SU(2) Yang-Mills theory for an arbitrary \theta angle to an unconstrained nonlocal theory of a self-interacting positive definite symmetric 3 \times 3 matrix field S(x) is performed. It is shown that, after exact projection to a reduced phase space, the density of the Pontryagin index remains a pure divergence, proving the \theta independence of the unconstrained theory obtained. An expansion of the nonlocal kinetic part of the Hamiltonian in powers of the inverse coupling constant and truncation to lowest order, however, lead to violation of the \theta independence of the theory. In order to maintain this property on the level of the local approximate theory, a modified expansion in the inverse coupling constant is suggested, which for a vanishing \theta angle coincides with the original expansion. The corresponding approximate Lagrangian up to second order in derivatives is obtained, and the explicit form of the unconstrained analogue of the Chern-Simons current linear in derivatives is given. Finally, for the case of degenerate field configurations S(x) with rank|S| = 1, a nonlinear \sigma-type model is obtained, with the Pontryagin topological term reducing to the Hopf invariant of the mapping from the three-sphere S^3 to the unit two-sphere S^2 in the Whitehead form.