Reformulating Yang-Mills theory in terms of local gauge invariant variables

Abstract

An explicit canonical transformation is constructed to relate the physical subspace of Yang-Mills theory to the phase space of the ADM variables of general relativity. This maps 3+1 dimensional Yang-Mills theory to local evolution of metrics on 3 manifolds.A long standing problem in Yang-Mills theories is whether its dynamics can be written in terms of gauge invariant variables. This is very important for our understanding of confinement in QCD. One option is to write the theory as dynamics of Wilson loops. However these are nonlocal variables and their evolution equations are difficult to handle. Another approach has been to use composite variables constructed out of the non-Abelian electric [1][2][3] or magnetic fields [4,5].In three dimensions, gauge invariant formulation of Yang-Mills theory was carried out in close analogy to gravity in [6]. A similar approach was tried out for 3+1 dimensions in [7], but the theory turned out to be quite complicated. Here we do a manifestly gauge invariant formulation of 3+1 dimensional Yang-Mills theory using Ashtekar like variables. This formulation is also closely related to ADM formulation of gravity. We identify the physical variables and the Gauss law and then rewrite the Hamiltonian in terms of the new variables.Starting with the usual Euclidean partition function, we introduce an auxiliary field E ia and integrate over A a 0 to obtain(1) * Present address:

UnconstrainedSU(2)Yang-Mills theory with a topological term in the long-wavelength approximation

“…For a complete understanding of the low-energy quantum phenomena of Yang-Mills theory, it is necessary to have a nonperturbative, gauge invariant description of the underlying classical theory including the θ-dependent Pontryagin term [1]- [4]. Several representations of Yang-Mills theory in terms of local gauge invariant fields have been proposed [5]- [24] in recent decades, implementing the Gauss law as a generator of small gauge transformations. However, in dealing with such local gauge invariant fields special consideration is needed when the topological term is included, since it is the four-divergence of a current changing under large gauge transformations.…”

On unconstrained SU(2) gluodynamics with theta angle

“…We therefore pose at this place the question whether it is possible to express the topological term in the classical action as a total divergence of a gauge invariant current using the unconstrained formulation of gauge theories [5]- [20]. In the hope to obtain such a representation of the topological term we would like to generalize in the present notes the Hamiltonian reduction of classical SU(2) Yang-Mills field theory given in [18] to arbitrary θ-angle by including the CP odd part (1.2) of the action.…”