Ultraviolet Finite Quantum Field Theory on Quantum Spacetime

 We discuss a formulation of quantum field theory on quantum space time where the perturbation expansion of the S-matrix is term by term ultraviolet finite. The characteristic feature of our approach is a quantum version of the Wick product at coinciding points: the differences of coordinates q _j −q _k are not set equal to zero, which would violate the commutation relation between their components. We show that the optimal degree of approximate coincidence can be defined by the evaluation of a conditional expectation which replaces each function of q _j −q _k by its expectation value in optimally localized states, while leaving the mean coordinates invariant. The resulting procedure is to a large extent unique, and is invariant under translations and rotations, but violates Lorentz invariance. Indeed, optimal localization refers to a specific Lorentz frame, where the electric and magnetic parts of the commutator of the coordinates have to coincide 11]. Employing an adiabatic switching, we show that the S-matrix is term by term finite. The matrix elements of the transfer matrix are determined, at each order in the perturbative expansion, by kernels with Gaussian decay in the Planck scale. The adiabatic limit and the large scale limit of this theory will be studied elsewhere. Received: 15 January 2003 / Accepted: 20 March 2003 Published online: 5 May 2003 RID="*" ID="*" Research supported by MIUR and GNAMPA-INDAM RID="*" ID="*" Research supported by MIUR and GNAMPA-INDAM Communicated by H. Araki and D. Buchholz