This is the second paper in a series in which we show how to use the principles of the $\delta$ expansion to obtain nonperturbative solutions to gauge theories. The approach consists of replacing the usual minimal-coupling term $\overline{\psi }(\partial -eA)\psi$ by $\overline{\psi }{(i\partial -eA)}^{\delta }\psi$ and then expanding the new theory in powers of $\delta$. For all values of $\delta$ the theory is locally gauge invariant. Thus, local gauge invariance holds order by order in powers of $\delta$. In this paper we show how to calculate the photon propagator and thus the anomaly in the Schwinger model (two-dimensional massless quantum electrodynamics) to first order in $\delta$. At $\delta =1\mathrm{}$ the exact value for the anomaly, $\frac{e^2}{\pi }$, is obtained.

• Received 2 April 1991

DOI:https://doi.org/10.1103/PhysRevD.45.1261