Numerical solutions of lattice quantum fields with a hierarchy of Schroedinger-like equations.

Abstract

Systems of quantized fields can be described by an infinite hierarchy of coupled equations. Such a hierarchy is derived from first principles for a simple interacting field theory to illustrate this type of a representation. The perturbation series for the S matrix is derived from the hierarchy equations in order to show its equivalence to the usual expansion in Feynman amplitudes. An inquiry is then conducted to determine whether this type of representation is useful for solving problems. Truncations of the hierarchy which predict simple bound states are examined in the weak coupling limit, and equations describing a hydrogen-like atom are obtained. Next, the numerical approximation of a truncated hierarchy is studied, and a scattering/particle creation process is modeled in one dimension with a resulting accuracy of 1 to 2 percent. Finally, the mathematical questions of convergence which arise in connection with quantized fields are discussed within the context of the hierarchy equations.