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Is quantum gravity a non-commutative geometry?

Final Activity Report Summary - QGNC (Is quantum gravity a noncommutative geometry?)

There is a strong theoretical evidence that the usual continuum model of spacetime is not correct at very short distance (i.e. the Planck scale of order 10^-33 cm). It is rather straightforward to show that if the principles of quantum theory hold right down to very small length scales where gravitational forces come into play then the familiar picture of space-time as a manifold must be abandoned. Non-commutative geometry provides a mathematical framework to study this new picture of space-time. In this project, we studied various approaches of non-commutative geometry applied to quantum gravity.

On the one side, together with the Rome group of Amelino-Camelia and his students, we establish the first steps of a Noether analysis on non-commutative spaces. Here, non-commutative spaces should be intended as spaces whose coordinates no longer commute. We worked on two particular cases, where the commuter is either a constant (we talk about a twist of the commutative space) or another coordinate (Lie algebraic deformation). In three publications we established the first steps towards a Noether analysis on these non-commutative spaces. Namely, we put into evidence conserved currents and charges associated to the symmetry of those non-commutative spaces, specifically rotations, boosts and translations.

The symmetries appear as a deformation of classical Minkovsky space, best expressed in terms of Hopf algebras. As a consequence, we found conserved currents which looks like a deformation of classical currents. Besides writing the explicit forms of these currents, the main result is the observation that these symmetries cannot be considered independently: for instance in the Lie-deformed Minkovski space, one cannot consider a transformation that does not include a translation part. Any symmetry transformation, in order to be mathematically coherent with the deformation, must include a certain amount of translation. This property, that has no classical counterpart, should now be expressed in a more systematic way, for instance in the framework of differential calculus on quantum group.
% On the other side, I continued my work on Connes' approach to non-commutative geometry. Especially I generalized to arbitrary dimensions the result obtained in a precedent work on the metric interpretation of gauge fields within a non-commutative geometry framework. This paper, rather technical, has been submitted to a mathematical review. The computations are long and difficult, illustrating that Connes's distance formula is far from trivial and certainly deserves more attention that it has nowadays. Together with a former student of Rome's group, we undertake the computation of Connes's distance formula on some examples of non-commutative spaces given by deformation of coordinates, namely the Moyal plane. The set of pure states of the corresponding algebra is unknown, so we tackle the case of non-pure states. Very interestingly it turned out that already in the classical case (i.e. the plane R^2) the result was interesting since the spectral distance provided by non-commutative geometry is closely linked to the so-called Monge-Kantorovich metric between probability distribution.

As a conclusion, these two years in Rome led to new and innovating results on noncommutative spaces, and a deeper comprehension of Connes' non-commutative geometry. Several contacts have been established, including R. Longo from the math department of Roma 2 University, Tor Vergata.