Delta expansion for local gauge theories. I. A one-dimensional model.

Abstract

The principles of the \ensuremath{\delta} perturbation theory were first proposed in the context of self-interacting scalar quantum field theory. There it was shown how to expand a (${\mathrm{\ensuremath{\varphi}}}^{2}$${)}^{1+\mathrm{\ensuremath{\delta}}}$ theory as a series in powers of \ensuremath{\delta} and how to recover nonperturbative information about a ${\mathrm{\ensuremath{\varphi}}}^{4}$ field theory from the \ensuremath{\delta} expansion at \ensuremath{\delta}=1. The purpose of this series of papers is to extend the notions of \ensuremath{\delta} perturbation theory from boson theories to theories having a local gauge symmetry. In the case of quantum electrodynamics one introduces the parameter \ensuremath{\delta} by generalizing the minimal coupling terms to \ensuremath{\psi}\ifmmode\bar\else\textasciimacron\fi{}(\ensuremath{\partial}-ieA${)}^{\mathrm{\ensuremath{\delta}}}$\ensuremath{\psi} and expanding in powers of \ensuremath{\delta}. This interaction preserves local gauge invariance for all \ensuremath{\delta}. While there are enormous benefits in using the \ensuremath{\delta} expansion (obtaining nonperturbative results), gauge theories present new technical difficulties not encountered in self-interacting boson theories because the expression (\ensuremath{\partial}-ieA${)}^{\mathrm{\ensuremath{\delta}}}$ contains a derivative operator. In the first paper of this series a one-dimensional model whose interaction term has the form \ensuremath{\psi}\ifmmode\bar\else\textasciimacron\fi{}[d/dt-ig\ensuremath{\varphi}(t)${]}^{\mathrm{\ensuremath{\delta}}}$\ensuremath{\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 \ensuremath{\gamma} matrices. In the next paper of this series we consider two-dimensional electrodynamics and show how to calculate the anomaly in the \ensuremath{\delta} expansion.