Finite Quantum-Field Theory and the Bosonic String Formalism: A Critical Point of View

Abstract

Basics of scalar and vector Finite Quantum Field Theories are recalled, stressing the importance of the quantization of classical physical elds as Operator-Valued- Distributions with specic fast decreasing test functions of the coordinates. The procedure respects full Lorentz and symmetry invariances and, due to the presence of test functions, leads to nite Feynman diagrams directly at the physical dimension D = 2..4. In dimension 2 it is only with such test function that the canonical quantization of the massless scalar eld is found to be fully consistent with the most successfull Conformal Field Theoretic approach, pioneered by Belavin, Polyakov and Zamolodchikov in the early 1980's. The question is then raised how Poliakov's wordline path integral representation of the relativistic string could possibly lead to nite Feynmann diagrams. The natural way of inquiries is through the extension of the string formalism with classical convoluted coordinates leading then to Operator- Valued-Distributions and thereby to Finite Quantum Field Theories. It is shown that in the process some age-old certitudes about quantized strings are somewhat jostled.