Operator Regularization

In this thesis, a computational method for perturbative quantum field theory, known as operator regularization, is presented. It is used in conjunction with the Feynman path integral and background field quantization. Green\u27s functions are evaluated using a perturbative expansion due to Schwinger, which employs an expansion of the generating functional in powers of the background field. In contrast to the usual Feynman diagramatic technique, this expansion is made after the evaluation of the path integral. By regulating only the operators occurring in the generating functional, operator regularization does not explicitly break the symmetries of the original lagrangian. No ambiguities arise in the evaluation of anomalies. Surprisingly, it also avoids the occurrence of ultraviolet divergences. At one-loop order, operator regularization is equivalent to dimensional regularization in conjunction with {dollar}M\overline S{dollar} renormalization for purely bosonic theories. Several computations serve to illustrate this technique: the two- and three-point functions in spinor QED are evaluated to one-loop order and are shown to satisfy the Ward identities; the U(1) axial anomaly is computed, and scalar theories are also examined to two-loop order.;Operator regularization is well suited to computations in supersymmetric theories. Explicit calculations to two-loop order in the component Wess-Zumino (WZ) model show that the two- and three-point WTST identities are satisfied. No infinities arise at any stage of these computations. The anomalies in the supercurrent supermultiplet are also evaluated by considering specific one-loop processes