The source galerkin method for scalar field theory

This is part I of a two-part series on the Source Galerkin method. This approach is based on the differential formulation of quantum field theory. On a finite lattice, the functional differential equations for a theory in the presence of an external source becomes a set of coupled differential equations for the generating functional Z. Systematic approximations to these equations are found using the Galerkin method. Calculations are straightforward to perform, and are executed rapidly compared to Monte Carlo. The bulk of the computation involves a single matrix inversion. In addition, bosons and fermions are treated in a symmetric manner. In this paper, we consider power series solutions for scalar field theory in D = 2, 3,4. Propagators and mass gaps are calculated for a number of systems. The calculations in this paper were made on a work station of modest power using a fourth order polynomial expansion for lattices of size 8², 4³, 2⁴ in 2D 3D, and 4D. In part II we consider the fermionic formulation.