Fermions in spherical field theory

Spherical field theory is a new non-perturbative method for studying quantum field theory. It was introduced in [1] and was used to describe the interactions of scalar boson fields. In this paper we show how to extend the spherical field method to fermionic systems. The central idea of spherical field theory is to treat a d-dimensional system as a set of coupled one-dimensional systems. This is done by expanding field configurations of the functional integral in terms of spherical partial waves. Regarding each partial wave as a distinct field in a new one-dimensional theory, we interpret the functional integral as a time-evolution equation, where the radial distance in the original theory serves as the time parameter. For a purely bosonic system the time-evolution equation corresponds with a multidimensional partial differential equation. In the case of a purely fermionic system, we find that the time evolution is described by a system of first-order ordinary differential equations. In future work we will study mixed systems with both bosons and fermions which are described by coupled partial differential equations. Unlike lattice methods, spherical field theory yields an expansion which, at any order, corresponds with a continuous system. It is therefore able to avoid problems associated with discrete approximation methods. There is no doubling of fermion states, and we find the correct axial anomaly. Fur-