Planar Yang-Mills theory: Hamiltonian, regulators and mass gap

Abstract

We carry out the Hamiltonian analysis of non-Abelian gauge theories in (2+1) dimensions in a gauge-invariant matrix parametrization of the fields. A detailed discussion of regularization issues and the construction of the renormalized Laplace operator on the configuration space, which is proportional to the kinetic energy, are given. The origin of the mass gap is analyzed and the lowest eigenstates of the kinetic energy are explicitly obtained; these have zero charge and exhibit a mass gap . The nature of the corrections due to the potential energy, the possibility of an improved perturbation theory and a Schrodinger-like equation for the states are also discussed.Comment: 41 pages, Plain TeX, discussion added, typos correcte

Manifest covariance and the Hamiltonian approach to mass gap in (2+1)-dimensional Yang-Mills theory

“…The volume element on the space C of gauge-invariant configurations was calculated explicitly in [1,2] and found to be…”

“…It is clear that this cannot be done at any finite order in perturbation theory. However, one can define an "improved" perturbation theory where a partial resummation of the perturbative expansion has been carried out [2]. This improvement would be equivalent to keeping the leading term of I(H) as in (6) in the exponent in (2).…”

“…However, one can define an "improved" perturbation theory where a partial resummation of the perturbative expansion has been carried out [2]. This improvement would be equivalent to keeping the leading term of I(H) as in (6) in the exponent in (2). For example, if we write H = e t a ϕ a ≈ 1 + t a ϕ a , as would be appropriate in perturbation theory, we find…”

“…This will be holomorphically invariant with the Green's functions G = ∂ −1 andḠ =∂ −1 transforming in a certain way as discussed in [2]. This way of writing F involves the additional use of Green's functions, over and above the Green's functions which appear in the construction of H (or M, M † ) from the potentials.…”

“…In a series of recent papers we have carried out a Hamiltonian analysis of Yang-Mills theories in (2+1) dimensions, Y M 2+1 [1,2,3]. A matrix parametrization of the gauge potentials A µ was used which facilitated calculations using manifestly gauge-invariant variables.…”