Numerical Quantum Field Theory on the Continuum and a New Look at Perturbation Theory

The Source Galerkin method finds approximate solutions to the functional differential equations of field theories in the presence of external sources. While developing this process, it was recognized that approximations of the spectral representations of the Greens functions by Sinc function expansions are an extremely powerful calculative tool. Specifically, this understanding makes it not only possible to apply the Source Galerkin method to higher dimensional field theories, but also leads to a new approach to perturbation theory calculations in scalar and fermionic field theories. This report summarizes the methodologies for solving quantum field theories with the Source Galerkin method and for performing perturbation theory calculations using Sinc approximations.

New Numerical Methods for Quantum Field Theories on the Continuum

The Source Galerkin Method is a new numerical technique that is being developed to solve Quantum Field Theories on the continuum. It is not based on Monte Carlo techniques and has a measure to evaluate relative errors. It promises to increase the accuracy and speed of calculations, and takes full advantage of symmetries of the theory. The application of this method to the non-linear sigma model is outlined.

New Numerical Methods for Iterative or Perturbative Solution of Quantum Field Theory

A new computational idea for continuum quantum field theories is outlined. This approach is based on the lattice source Galerkin methods developed by Garcia, Guralnik and Lawson. The method has many promising features including treating fermions on a relatively symmetric footing with bosons. As a spinoff of the technology developed for ``exact solutions, the numerical methods used have a special case application to perturbation theory. We are in the process of developing an entirely numerical approach to evaluating graphs to high perturbative order.

Numerical Field Theory on the Continuum

An approach to calculating approximate solutions to the continuum Schwinger-Dyson equations is outlined, with examples for phi^4 in D=1. This approach is based on the source Galerkin methods developed by Garcia, Guralnik and Lawson. Numerical issues and opportunities for future calculations are also discussed briefly.