γ-algebra algorithms for canonical hypercomplex representations

Abstract

The Dirac γ-matrices which are used in relativistic quantum field theory to describe free and interacting fermions are treated as generating elements of a 16-dimensional Clifford-algebra, which may be viewed construed from the non-interfering product of two quaternion skew fields. A canonical form for the elements of this algebra is defined; the corresponding arithmetic is implemented in ALDES using the SAC-2 system at Karlsruhe.

Bounds for the maximal computing times of the algorithms are given. A remarkable fact is that in this representation the algorithm for getting the trace of a γ-expression essentially consists in a single step. The computing times are polynomial functions of the computing times for multivariate polynomial arithmetic.