Gherardo Piacitelli

On the Unitarity problem in space-time noncommutative theories

Abstract It is shown that the violation of unitarity observed in space/time noncommutative field theories is due to an improper definition of quantum field theory on noncommutative spacetime.

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−qk 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−qk 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.

Field theory on noncommutative spacetimes: Quasiplanar Wick products

We give a definition of admissible counterterms appropriate for massive quantum field theories on the noncommutative Minkowski space, based on a suitable notion of locality. We then define products of fields of arbitrary order, the so-called quasiplanar Wick products, by subtracting only such admissible counterterms. We derive the analogue of Wicks theorem and comment on the consequences of using quasiplanar Wick products in the perturbative expansion.

Quantum Spacetime: a Disambiguation ?

We review an approach to non-commutative geometry, where models are constructed by quantisation of the coordinates. In particular we focus on the full DFR model and its irreducible components; the (arbitrary) restriction to a particular irreducible component is often referred to as the “canonical quantum spacetime”. The aim is to distinguish and compare the approaches under various points of view, including motivations, prescriptions for quantisation, the choice of mathematical objects and concepts, approaches to dynamics and to covariance. Some incorrect statements as “universality of Planck scale conflicts with Lorentz-Fitzgerald contraction and requires a modification of covariance”, or “stability of the geometric background requires an absolute

Twisted Covariance as a Non Invariant Restriction of the Fully Covariant DFR Model

We discuss twisted covariance over the noncommutative spacetime algebra generated by the relations

The κ‐Minkowski Spacetime: Trace, Classical Limit and Uncertainty Relations

{[q_\theta^\mu,q_\theta^\nu]=i\theta^{\mu\nu}}

Lorentz Covariant k-Minkowski Spacetime

, where the matrix θ is treated as fixed (not a tensor), and we refrain from using the asymptotic Moyal expansion of the twists.We show that the tensor nature of θ is only hidden in the formalism: in particular if θ fulfils the DFR conditions, the twisted Lorentz covariant model of the flat quantum spacetime may be equivalently described in terms of the DFR model, if we agree to discard a huge non-invariant set of localisation states; it is only this last step which, if taken as a basic assumption, severely breaks the relativity principle.We also will show that the above mentioned, relativity breaking, ad hoc rejection of localisation states is an independent, unnecessary assumption, as far as some popular approaches to quantum field theory on the quantum Minkowski spacetime are concerned.The above should raise some concerns about speculations on possible observable consequences of arbitrary choices of θ in arbitrarily selected privileged frames.

Twisted Covariance and Weyl Quantisation

We review the investigations on the quantum structure of spacetime, to be found at the Planck scale if one takes into account the operational limitations to the localization of events which result from the concurrence of Quantum Mechanics and General Relativity. We also discuss the different approaches to (perturbative) Quantum Field Theory on Quantum Spacetime, and some of the possible cosmological consequences.