Spatial Geometry of Hamiltonian Gauge Theories

The Hamiltonians of SU(2) and SU(3) gauge theories in 3+1 dimensions can be expressed in terms of gauge invariant spatial geometric variables, i.e., metrics, connections and curvature tensors which are simple local functions of the non-Abelian electric field. The transformed Hamiltonians are local. New results from the same procedure applied to the SU(2) gauge theory in 2+1 dimensions are also given. Talk presented at the Conference QCD ’94 Montpellier, France, 7-13 July 1994 CERN-TH.7391/94 August 1994 We outline a formalism which contains a rather new approach to non-perturbative dynamics of the gluon sector of QCD. What is achieved is a formally exact transformation of the Hamiltonian on the physical subspace of states obeying the Gauss law constraint. The new Hamiltonian is local and is expressed in terms of gauge invariant spatial geometric variables, i.e., a dynamical metric Gij(x) which is a simple function of the non-Abelian electric field E (x) and the Christoffel connection Γjk and curvature-tensor R i jkl computed from Gij by the standard formulas of Riemannian geometry. For gauge group SU(2) the underlying geometry is purely Riemannian, and the six gauge-invariant variables contained in Gij are essentially all that are required. For gauge group SU(3) there is a more complicated metric-preserving geometry with torsion, and the torsion tensors are expressed in terms of a set of 16 gauge-invariant variables. The Hamiltonian we find is admittedly complicated and has some strange features. But it also has some physical features, and I am moderately optimistic that physical and geometric insight can be combined so that results of physical interest can be drawn from the formalism. We start with an observation about the basic equations of canonical Hamiltonian dynamics in A0 = 0 gauge with conjugate variables A a i (x), the non-Abelian vector potential, and E (x). The equal-time commutation relations, the Gauss law constraint, the non-Abelian magnetic field, and the Hamiltonian are [Ai (x), E (x)] = iδδ i δ (x− x) (1) G(x)ψ = 1 g (∂iE ai + gfAiE )ψ = 0 (2) B(x) = ǫ[∂jA a k + 1 2 gfAjA c k] (3)