*Barus & Holley 529, x2623
*

Lattice and Continuum Quantum Field Theory, Particle Physics. [CHEP group page]

**New Numerical Method for Fermion Field Theory**- John W. Lawson and G. S. Guralnik
- hep-th/9507131
**Source Galerkin Calculations in Scalar Field Theory**- John W. Lawson and G. S. Guralnik
- hep-th/9507130
**A New Method for Numerical Quantum Field Theory**- S. García, G. S. Guralnik, and J. W. Lawson
- hep-ph/9312236

A new deterministic, numerical method to solve fermion field theories is presented. This approach is based on finding solutions Z[J] to the lattice functional equations for field theories in the presence of an external source J. Using polynomial expansions for the generating functional Z, we calculate for systems of interacting fermions. These calculations are straightforward to perform and are executed rapidly compared to Monte Carlo. The bulk of the computation involves a single matrix inversion. Because it is not based on a statistical technique, it does not have many of the difficulties often encountered when simulating fermions. Since no determinant is ever calculated, solutions to problems with dynamical fermions are handled more easily. This approach is very flexible, and can be tailored to specific problems based on convenience and computational constraints. We present simple examples to illustrate the method; more general schemes are desirable for more complicated systems.

In this paper, we extend previous work on scalar phi^4 theory using the Source Galerkin method. This approach is based on finding solutions Z[J] to the lattice functional equations for field theories in the presence of an external source J. Using polynomial expansions for the generating functional Z, we calculate propagators and mass-gaps for a number of systems. These calculations are straightforward to perform and are executed rapidly compared to Monte Carlo. The bulk of the computation involves a single matrix inversion. The use of polynomial expansions illustrates in a clear and simple way the ideas of the Source Galerkin method. But at the same time, this choice has serious limitations. Even after exploiting symmetries, the size of calculations become prohibitive except for small systems. The calculations in this paper were made on a workstation of modest power using a fourth order polynomial expansion for lattices of size 8^2, 4^3, 2^4 in 2D, 3D, and 4D. In addition, we present an alternative to the Galerkin procedure that results in sparse matrices to invert.

In this note we present a new numerical method for solving Lattice Quantum Field Theory. This Source Galerkin Method is fundamentally different in concept and application from Monte Carlo based methods which have been the primary mode of numerical solution in Quantum Field Theory. Source Galerkin is not probabilistic and treats fermions and bosons in an equivalent manner.

gerry@het.brown.edu