Abstract
The quantum mechanics of spatially constant SU(2) Yang–Mills- and Dirac-fields minimally coupled to each other is investigated as the strong coupling limit of 2-color-QCD. Using a canonical transformation of the quark and gluon fields, which Abelianises the Gauss law constraints to be implemented, the corresponding unconstrained Hamiltonian and total angular momentum are derived. In the same way as this reduces the colored spin-1 gluons to unconstrained colorless spin-0 and spin-2 gluons, it reduces the colored spin-12 quarks to unconstrained colorless spin-0 and spin-1 quarks. These however continue to satisfy anti-commutation relations and hence the Pauli-exclusion principle. The obtained unconstrained Hamiltonian is then rewritten into a form, which separates the rotational from the scalar degrees of freedom. In this form the low-energy spectrum can be obtained with high accuracy. As an illustrative example, the spin-0 energy-spectrum of the quark–gluon system is calculated for massless quarks of one flavor. It is found, that only for the case of 4 reduced quarks (half-filling) satisfying the boundary condition of particle–antiparticle C-symmetry, states with energy lower than for the pure-gluon case are obtained. These are the ground state, with an energy about 20% lower than for the pure-gluon case and the formation of a quark condensate, and the sigma–antisigma excitation with an energy about a fifth of that of the first glueball excitation.
Highlights
For a complete description of the physical properties of low-energy QCD, such as color confinement, chiral symmetry breaking, the formation of condensates and flux-tubes, and the spectra and strong interactions of hadrons, it might be advantageous if one could first reformulate QCD in terms of gauge invariant dynamical variables, before applying any approximation schemes
Using a canonical transformation of the dynamical variables, which Abelianises the Non-Abelian Gauss-law constraints, such a reformulation has been achieved for pure SU (2) Yang-Mills theory on the classical [2, 3, 4] and on the quantum level [5]
The leading order term in this expansion constitutes the unconstrained Hamiltonian of SU (2) Yang-Mills quantum mechanics of spatially constant gluon fields [6]-[12], for which the low-energy spectra can be calculated with high accuracy
Summary
For a complete description of the physical properties of low-energy QCD, such as color confinement, chiral symmetry breaking, the formation of condensates and flux-tubes, and the spectra and strong interactions of hadrons, it might be advantageous if one could first reformulate QCD in terms of gauge invariant dynamical variables, before applying any approximation schemes (see e.g.[1]). The leading order term in this expansion constitutes the unconstrained Hamiltonian of SU (2) Yang-Mills quantum mechanics of spatially constant gluon fields [6]-[12], for which the low-energy spectra can be calculated with high accuracy. Subject of the present work is its generalisation to the case of SU (2) Dirac-Yang-Mills quantum mechanics of quark and gluon fields. Subject of the present work is its generalisation to the case of SU (2) Dirac-Yang-Mills quantum mechanics of quark and gluon fields1 First steps in this direction on the classical level have been done in [2]. On the remaining dynamical degrees of freedom Aai, Πai, ψαr and ψα∗r are quantized in the Schrodinger functional approach by imposing the equal time commutation relations. For carrying out quantum mechanical calculations it is desirable to have a corresponding unconstrained Schrodinger equation and to find its eigenstates in an effective way with high accuracy at least for the lowest states
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