0 v 1 2 1 A ug 1 99 6 hep - ph / 9608390 FAU - TP 3 - 96 / 12 August 1996 Defects in Modified Axial Gauge QCD 3 + 1 Harald

Magnetically charged vortex defects are shown to arise in canonically quantised modified axial gauge QCD3+1 on a torus. This gauge – being an Abelian projection – keeps only the eigenphases as dynamical variables of the Wilson line in x3-direction. A mismatch between the identification of large gauge transformations before and after gauge fixing is indicated. Talk presented at the “Workshop on QCD” at the American University of Paris, June 3rd – 8th, 1996, and the conference “Quark Confinement and the Hadron Spectrum II” at Villa Olmo, Como, Italy, June 26th – 29th, 1996. Email: [email protected] One of the main issues in non-Abelian gauge theories is the presence of redundant variables. Eliminating them by “gauge fixing”, one hopes to identify the relevant degrees of freedom, the non-perturbative part of which may solve the outstanding questions in the low energy régime of these theories. Monopole field configurations in the Abelian projection gauges [1, 2] seem to be a useful device to explain confinement by the dual Meissner effect [1, 3, 4]. In this context, a Hamiltonian formulation of QCD is especially useful since it allows one to bear in mind all intuition and techniques of ordinary quantum mechanics; formulating the theory in terms of unconstrained, “physical” variables is the easiest way to render gauge invariant results in approximations. The choice of a compact base manifold softens the infrared problem and allows the definition of zero modes. Here, the formulation by Lenz et al. [5] of Hamiltonian QCD in the modified axial gauge [6] on a torus T 3 as spatial manifold is used, in which – in contradistinction to the naive axial gauge A3 = 0 – the eigenphases of the Wilson line/Polyakov loop in x3direction are kept as dynamical variables. This gauge belongs to the Abelian projection gauges, but the magnetically charged configurations found do not have particle character. On the contrary, their appearance indicates a failure of gauge fixing. The outline of these results is the goal of the present paper; a more rigorous treatment may be found in references [7, 8] and a forthcoming publication. As is well known, the Hamiltonian of pure QCD in the Weyl gauge A0 = 0, quantised by imposing the canonical commutation relations between fields ~ A and momenta ~ Π, allows for time independent gauge transformations whose infinitesimal generator is Gauß’ law, G(~x) = ~∂ · ~ Π(~x) + gf ~ A(~x) · ~ Π(~x). It cannot be derived as an equation of motion in the Hamiltonian formalism but has to be imposed on states, defining the physical Hilbert space Hphys as ∀ | phys〉 ∈ Hphys : G (~x) | phys〉 = 0 ∀a, ~x. (1) On a torus T 3 with length of the edge L, one imposes periodic boundary conditions for all fields and derivatives as well as for the gauge transformations 1 ~ A(~x) = ~ A(~x+ L~ei) , ~ Π(~x) = ~ Π(~x+ L~ei) , V (~x) = V (~x+ L~ei) . (2) The functional space H whose subspace Hphys is consists hence of the space of periodic functionals. Eq. (2) is closely related to the vanishing of all total colour charges in the box and to translation invariance [8, 10]. “Gauge fixing” corresponds to a coordinate transformation in field space ~ A(~x) = Ũ(~x)[ ~ A(~x) + i g ~∂]Ũ (~x) (3) to a basis splitting explicitly in unconstrained ( ~ A, ~ Π) and constrained (Ũ , ~̃p ) variables and respective conjugate momenta. Any operator O commuting with Gauß’ law does not contain redundant degrees of freedom and hence can be written as (possibly complicated) Fermions will not affect the arguments given here. Their sole trace is not to allow for twisted boundary conditions [9].