Group Theoretical Approach to Conformal Invariant Quantum Field Theory

Abstract

In this set of lectures we will address our-selves to the question of the construction and properties of a nontrivial exactly conformal invariant quantum field theory (QFT). As has been explained in some detail in Symanzik’s lectures (1) such a theory, if it exists, has a good chance of being relevant to the description of the real world in certain high energy limits — or possibly at intermediate energies as was suggested by Wilson (2]. The basic hypothesis and some predictions of the theory can therefore in principle be tested by experiment. Besides, conformal QFT is also interesting as a laboratory, because it can be analysed to a remark-able extent by strictly non-perturbative methods, i.e. without re-course to iterative techniques. It therefore offers welcome insight into the structure of local quantum field theory. To one’s surprise one finds structures much reminiscent of dual resonance models. We hope to illustrate this in what follows. Lastly, it offers an example of how the geometry of spacetime can affect and to some extent determine the structure of a theory of fundamental processes. This aspect is less trivial than might be thought, see the discussion below in Sees.3 (fixed points) and especially Sec.5. There is also another approach to conformal QFT which is based on iterative techniques; it has already been reviewed in the author’s Kaiserslautern lectures (3).