Ultraviolet Finite Quantum Field Theory on Quantum Spacetime

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 qj − 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 qj − q k by its expectation value in optimally localized states, while leaving the mean coordinatesThe 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.

Relativistic corrections to the Moyal-Weyl spacetime

“…If we would have first deformed our conjugate operators and then performed a second quantization we would not have any particle number appearing terms, except for the N 2 -term which would be linear in N . However, one could also define the conjugate operators with an N −1 as was done in [BDFP03] for the mean event coordinate operator. The change will only be visible at the level of the commutator relations since N commutes with all particle number preserving operators.…”

Non Local Theories: New Rules for Old Diagrams

“…The interactions L Q I and L E I both are covariant under translations and space rotations. The lagrangian L E I was found ultraviolet regular in [3]. Notwithstanding that it fails to be covariant even under space rotations, the lagrangian L σ I has also been extensively investigated on its own in the literature, motivated either by [1] or by later results in string theory.…”

Minimal length in quantum space and integration of the line element in noncommutative geometry

“…3 Strictly speaking, one should talk of "state" only for C * -algebras. In case A is not C * , we consider states of its C * -closureĀ with respect to the operator norm coming from the representation on H .…”

“…The latter is a pure state of the tensor product of complex algebras E ⊗ E . However, to guarantee that (that is, the commutators of the coordinates of two independent quantum points are equal), it has been proposed in [3] that the tensor product E ⊗ C 0 (Σ) E over the center C 0 (Σ) of E should be used instead. This has the following importance consequence:…”