The conformal invariance in quantum field theory

Abstract

The properties of a conformal invariant quantum field theory are considered. A short discussion of the conformal group in four dimensions and of the topology, introduced into the pseudo-euclidean space by this group is given. With the help of the commutation relations the spectrum of the generators in Hilbert-space is investigated. We find that the only possible discrete eigenvalue ofP2 and of thePv’s is zero and that the generator for scale transformationsS has a continuous spectrum. The eigenfunctions ofS in thex-representation are calculated, they form a complete set. The conservation laws valid in an invariant theory and the commutation relations predict a certain form of the conserved quantities expressed in terms of the energy-momentum tensor and of the co-ordinates. For scalar, spinor and vector fields the generators are derived by the action principle of Schwinger.