Unconstrained Hamiltonian formulation of low energy SU(3) Yang-Mills quantum theory

TL;DR

This paper presents an unconstrained Hamiltonian formulation for low energy SU(3) Yang-Mills quantum mechanics, simplifying the analysis of glueball states by Abelianizing constraints and identifying classical zero-energy valleys.

Contribution

It introduces a novel method using minimal embedding and adapted coordinates to derive an unconstrained Hamiltonian for SU(3) Yang-Mills quantum mechanics, extending previous SU(2) approaches.

Findings

Reduction of gluons to unconstrained glueball fields

Identification of classical zero-energy valleys in the potential

Separation of rotational and scalar degrees of freedom

Abstract

An unconstrained Hamiltonian formulation of the SU(3) Yang-Mills quantum mechanics of spatially constant fields is given using the method of minimal embedding of SU(2) into SU(3) by Kihlberg and Marnelius. Using a canonical transformation of the gluon fields to a new set of adapted coordinates (a non-standard type polar decomposition), which Abelianizes the Non-Abelian Gauss law constraints to be implemented, the corresponding unconstrained Hamiltonian and total angular momentum are derived. This reduces the colored spin-1 gluons to unconstrained colorless spin-0, spin-1, spin-2 and spin-3 glueball fields. The obtained unconstrained Hamiltonian is then rewritten into a form, which separates the rotational from the scalar degrees of freedom. It is shown that the chromomagnetic potential has classical zero-energy valleys for two arbitrarily large classical glueball fields, which are the unconstrained analogs of the well-known "constant Abelian fields". On the quantum level, practically all glueball excitation energy is expected to go into the increase of the strengths of these two fields. Finally, as an outlook, the straightforward generalization to low energy SU(3) Yang-Mills quantum theory in analogy to the SU(2) case is indicated, leading to an expansion in the number of spatial derivatives, equivalent to a strong coupling expansion, with the SU(3) Yang-Mills quantum mechanics constituting the leading order.