Conformal Quantum Field Theory

Abstract

In this chapter, a non-perturbative definition of quantum field theory in Minkowski space-time is given starting from the Wightman axioms. The fundamental observables are Wightman correlation functions, admitting both position-space and momentum-space representations. Juggling between the two, the constraints imposed by Poincaré symmetry are analyzed. Time-ordered correlation functions are introduced with the help of the spectral representation. The implications of scale and special conformal symmetry are then presented successively, focusing on 2-point functions for simplicity. This leads to a complete definition of the theory in terms of operators acting on the vacuum state. The chapter closes on a brief discussion of divergences and anomalies in quantum field theory with conformal symmetry.