Unconstrained Hamiltonian Formulation of SU(2) Gluodynamics

One of the main issues in the Hamiltonian formulation of Yang-Mills theories is to find the projection from the phase space of canonical variables constrained by the non-Abelian Gauss law to the “smaller” phase space of unconstrained gauge invariant coordinates only. Dealing with the problem of the elimination of the pure gauge degrees of freedom two approaches exist, the perturbative and the nonperturbative one, with complementary features. The conventional perturbative gauge fixing method works successfully for the description of high energy phenomena, but fails in applications in the infrared region. The correct nonperturbative reduction of gauge theories [1][15], on the other hand, leads to representations for the unconstrained Yang-Mills systems, which are valid also in the low energy region, but unfortunately up to now have been rather complicated for practical calculations. The guideline of these investigations is the search for a representation of the gauge invariant variables which is suitable for the description the infrared limit of Yang-Mills theory. To get such a representation for the unconstrained system we are following the Dirac generalized Hamiltonian formalism [16][18] using the method of Hamiltonian reduction (see [19][21] and references therein) instead of the conventional gauge fixing approach [22]. In previous work [23] it was demonstrated that for the case of the mechanics of spatially constant SU(2) Dirac-Yang-Mills fields an unconstrained Hamiltonian can be derived which has a simple practical form. The elimination of the gauge degrees of freedom has been achieved by performing a canonical transformation to a new adapted coordinates, in terms of which both the Abelianization of the Gauss law constraints and the projection onto the constraint shell is simplified. The obtained unconstrained system then describes the dynamics of a symmetric second rank tensor under spatial rotations. The main-axis-transformation of this symmetric tensor allowed us to separate the gauge invariant variables into scalars under ordinary space rotations and into “rotational” degrees of freedom. In this final form the physical Hamiltonian can be quantized without operator ordering ambiguities. In this work we shall generalize our approach from non-Abelian Dirac-Yang-Mills mechanics [23] to field theory. We shall give a Hamiltonian formulation of classical SU(2) Yang-Mills field theory entirely in terms of gauge invariant variables, and separate these into scalars under ordinary space rotations and into “rotational” degrees of freedom. It will be shown that this naturally leads to their identification as fields with “nonrelativistic spin-2 and spin-0”.