Talk presented by Pinar Emirdag

Parallel Session: Theoretical Developments II

Friday July 2nd, 15.50 - 16.10, Room 8

New Numerical Methods for Quantum Field Theories on the Continuum

Abstract: A new computational method for continuum field theories is being developed. This method, which is called Source Galerkin technique, was first studied on the lattice by Garc\`{\i}a, Guralnik, and Lawson. It is not based on any statistical methods and has controllable errors. It also promises an increase in accuracy and speed of calculations. In this method, we treat field theories in the presence of external sources. The functional relations become a set of coupled differential equations for the generating functional $Z$. Source Galerkin is used to solve these equations for the Green's functions of the theory. According to this technique, the set of the residuals and the test functions are required to be orthogonal with respect to some inner product. The test functions that we are using are polynomials that consists of source terms and preserve the group symmetry of the problem. Symmetries of translation and reflection invariance are used in constructing the solution. A good choice of test functions and ans\"{a}tz solution speeds the convergence to the result. The accuracy of the approximation can be judged by measuring its numerical stability and convergence. The solution can be improved by introducing more structure to the ans\"{a}tz according to the nature of the problem. Techniques of this sort for solving sets of differential equations are known as Galerkin methods. Application of this method to various field theories are investigated. Physical quantities such as mass gap and $\beta$ function are calculated and compared to the results in the literature.

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