Renormalization of Fermionic Theories in Stochastic Quantization

Abstract

>~ 4 > Various aspects, ranging from its usefulness as a scheme for numerical computations in quantum field theory to its viability· as an alternative and perhaps a more flexible scheme for quantizing field theories, have been studied. In order to be able to consider SQM as a complete scheme for quantizing field theories, it becomes necessary to examine the renormalization programme within the framework of SQM. Here by renormalizability of a given theory we mean that the higher order corrections do not alter the structure of the stochastic equations one started out with. This question assumes particular importance in the context of physically relevant theories such as Yang-Mills field theory, quantum electrodynamics and unified models of weak and electromagnetic interactions, where SQM not only permits but also requires introduc­ tion of certain features, such as stochastic gauge fixing, 5 > not present in the conven­ tional field theories and thereby precludes the possibility of any conclusions to be drawn regarding the renormalizability or otherwise of such theories within the framework of SQM on the basis of the results established in the conventional frame­ work. It is, therefore, necessary to study renormalization entirely within the frame­ work of SQM and to develop the necessary techniques so as to have a better intuition about the behaviour of ultra-violet (UV) divergences in SQM which could be useful in discussing quantization of non-holonomic systems which fall outside the realm of ·conventional quantization methods. So far, the renormalizability, within SQM, of only the ¢ 4 theory 6 > and gauge theories with stochastic gauge fixing 7 > has been inves­ tigated. The present investigation is devoted to the study of renormalization of fermionic theories in SQM. It is now known that the fermionic theories in SQM without a non-trivial kernel require new counterterms not present in the original unrenormal­ ized formulation. It is, therefore, important that the renorma_Iizability properties of fermionic theories in SQM be studied and understood. The formalism of SQM offers