ON THE CONNECTION BETWEEN THE CONFORMAL GROUP AND THE ULTRAVIOLET SINGULARITIES IN QUANTUM FIELD THEORY
H.A. Kastrup · eScholarship (California Digital Library) · 1965
We investigate the hypothesis that the ultraviolet singularities of quantum field theory are connected with conformal invariance at very high energies, and we demonstrate how they can be removed in an elementary way.The main point is that a whole class af new and independent solutions of the field equation is connected with the conformal group.These solutions cannot be described within the framework of the usual test function class S of the distribution calculus.But it is possible to define new test functions which are appropriate for dealing with these new solutions and their essential singularity on the light cone.In the quantum field theory of the Klein-Gordon equation one now obtains two sets of creation and annihilation operators, the second of which belongs to the new solution.The metric of the Hilbert space concerned is semidefinite and degenerate.It also turns out that the vacuum is infinitely degenerate.But the secondary vacua have a vanishing norm.As an immediate consequence of the semidefinite metric the mass~independent singularities on the light cone of the generalized Green's function cancel ea~h other.The results obtained bear a striking resemblance to the ideas already discussed by Heisenberg in his attempt to remove the fundamental difficulties of current quantum field theory.•-;An unknown world of new physical quantities, which is isomorphic to that of the linear momenta but nevertheless quite different, seems to be connected with the additional conformal symmetry at very high energies.,.