Ultraviolet Finite Quantum Field Theory on Quantum Spacetime

TL;DR

This paper presents a quantum spacetime formulation of quantum field theory where the perturbation expansion is ultraviolet finite at each order, utilizing a quantum Wick product that respects coordinate commutation relations but breaks Lorentz invariance.

Contribution

It introduces a novel quantum Wick product at coinciding points that ensures ultraviolet finiteness in perturbative quantum field theory on quantum spacetime.

Findings

Perturbation series are ultraviolet finite at each order.

Kernel functions exhibit Gaussian decay at the Planck scale.

The approach is invariant under translations and rotations, but not Lorentz transformations.

Abstract

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 (q_1 + ... + q_n)/n 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 *). 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. -- *) S. Doplicher, K. Fredenhagen, and J.E.Roberts, Commun. Math. Phys. 172, 187 (1995) [arXiv:hep-th/0303037]