arXiv subjects

Marco Matassa

Publications and source records attributed to Marco Matassa.

25 records · Page 2Linked to original sources

arXiv2014

Quantum Dimension and Quantum Projective Spaces

We show that the family of spectral triples for quantum projective spaces introduced by D'Andrea and Dabrowski, which have spectral dimension equal to zero, can be reconsidered as modular spectral triples by taking into account the action of the element $K_{2ρ}$ or its inverse. The spectral dimension computed in this sense coincides with the dimension of the classical projective spaces. The connection with the well known notion of quantum dimension of quantum group theory is pointed out.

math.QA↗

arXiv2014

Non-commutative integration, zeta functions and the Haar state for $SU_{q}(2)$

We study a notion of non-commutative integration, in the spirit of modular spectral triples, for the quantum group $SU_{q}(2)$. In particular we define the non-commutative integral as the residue at the spectral dimension of a zeta function, which is constructed using a Dirac operator and a weight. We consider the Dirac operator introduced by Kaad and Senior and a family of weights depending on two parameters, which are related to the diagonal automorphisms of $SU_{q}(2)$. We show that, after fixing one of the parameters, the non-commutative integral coincides with the Haar state of $SU_{q}(2)$. Moreover we can impose an additional condition on the zeta function, which also fixes the second parameter. For this unique choice the spectral dimension coincides with the classical dimension.

math-ph↗

arXiv2013

On the spectral and homological dimension of k-Minkowski space

We extend the construction of a spectral triple for k-Minkowski space, previously given for the two-dimensional case, to the general n-dimensional case. This takes into account the modular group naturally arising from the symmetries of the geometry, and requires the use of notions that have been recently developed in the frameworks of twisted and modular spectral triples. First we compute the spectral dimension, using an appropriate weight, and show that in general it coincides with the classical one. We also study the classical limit and the analytic continuation of the associated zeta function. Then we compare this notion of dimension with the one coming from homology. To this end, we compute the twisted Hochschild dimension of the universal enveloping algebra underlying k-Minkowski space. The result is that twisting avoids the dimension drop, similarly to other examples coming from quantum groups. In particular, the simplest such twist is given by the inverse of the modular group mentioned above.

math-ph↗

arXiv2013

A modular spectral triple for $κ$-Minkowski space

We present a spectral triple for $κ$-Minkowski space in two dimensions. Starting from an algebra naturally associated to this space, a Hilbert space is built using a weight which is invariant under the $κ$-Poincaré algebra. The weight satisfies a KMS condition and its associated modular operator plays an important role in the construction. This forces us to introduce two ingredients which have a modular flavor: the first is a twisted commutator, used to obtain a boundedness condition for the Dirac operator, the second is a weight replacing the usual operator trace, used to measure the growth of the resolvent of the Dirac operator. We show that, under some assumptions related to the symmetries and the classical limit, there is a unique Dirac operator and automorphism such that the twisted commutator is bounded. Then, using the weight mentioned above, we compute the spectral dimension associated to the spectral triple and find that is equal to the classical dimension. Finally we briefly discuss the introduction of a real structure.

math-ph↗

arXiv2012

Deformed Lorentz symmetry and relative locality in a curved/expanding spacetime

The interest of part of the quantum-gravity community in the possibility of Planck-scale-deformed Lorentz symmetry is also fueled by the opportunities for testing the relevant scenarios with analyses, from a signal-propagation perspective, of observations of bursts of particles from cosmological distances. In this respect the fact that so far the implications of deformed Lorentz symmetry have been investigated only for flat (Minkowskian) spacetimes represents a very significant limitation, since for propagation over cosmological distances the curvature/expansion of spacetime is evidently tangible. We here provide a significant step toward filling this gap by exhibiting an explicit example of Planck-scale-deformed relativistic symmetries of a spacetime with constant rate of expansion (deSitterian). Technically we obtain the first ever example of a relativistic theory of worldlines of particles with 3 nontrivial relativistic invariants: a large speed scale ("speed-of-light scale"), a large distance scale (inverse of the "expansion-rate scale"), and a large momentum scale ("Planck scale"). We address some of the challenges that had obstructed success for previous attempts by exploiting the recent understanding of the connection between deformed Lorentz symmetry and relativity of spacetime locality. We also offer a preliminary analysis of the differences between the scenario we here propose and the most studied scenario for broken (rather than deformed) Lorentz symmetry in expanding spacetimes.

hep-th↗

arXiv2012

Taming nonlocality in theories with Planck-scale-deformed Lorentz symmetry

We report a general analysis of worldlines for theories with deformed relativistic symmetries and momentum dependence of the speed of photons. Our formalization is faithful to Einstein's program, with spacetime points viewed as an abstraction of physical events. The emerging picture imposes the renunciation of the idealization of absolutely coincident events, but is free from some pathologies which had been previously conjectured.

gr-qc↗

arXiv2010

Testing quantum-spacetime relativity with gamma-ray telescopes

Observations of gamma-ray bursts are being used to test for a momentum dependence of the speed of photons, partly motivated by preliminary results reported in analyses of some quantum-spacetime scenarios. The relationship between time of arrival, momentum of the photon and redshift of the source which is used for these purposes assumes a "breakdown" of relativistic symmetries, meaning that it is a preferred-frame scenario which does not satisfy the Relativity Principle. The alternative hypothesis of a "deformation" of relativistic symmetries, which preserves the Relativity Principle by adopting deformed laws of relativistic transformation between observers, could not so far be tested in gamma-ray-burst observations because it was not known how to formulate it in expanding spacetimes. We here provide such a formulation, and we find that also for the symmetry-deformation scenario the analysis of gamma-ray-burst data take us very close to the desired "Planck-scale sensitivity".

astro-ph.HE↗