DELTA EXPANSION FOR LOCAL GAUGE-THEORIES .1 A ONE-DIMENSIONAL MODEL

The principles of the delta-perturbation theory were first proposed in the context of self-interacting scalar quantum field theory. There it was shown how to expand a (phi(2))1 + delta theory as a series in powers of delta and how to recover nonperturbative information about a phi(4) field theory from the delta expansion at delta = 1. The purpose of this series of papers is to extend the notions of delta-perturbation theory from boson theories to theories having a local gauge symmetry. In the case of quantum electrodynamics one introduces the parameter delta by generalizing the minimal coupling terms to psiBAR(not partial derivative-ie A)delta psiBAR and expanding in powers of delta. This interaction preserves local gauge invariance for all delta. While there are enormous benefits in using the delta-expansion (obtaining nonperturbative results), gauge theories present new technical difficulties not encountered in self-interacting boson theories because the expression (not partial derivative-ie A)delta contains a derivative operator. In the first paper of this series a one-dimensional model whose interaction term has the form psiBAR[d/dt-ig-phi(t)](delta)psi is considered. The virtue of this model is that it provides a laboratory in which to study fractional powers of derivative operators without the added complexity of gamma-matrices. In the next paper of this series we consider two-dimensional electrodynamics and show how to calculate the anomaly in the delta-expansion.