15 research outputs found
Connes distance by examples: Homothetic spectral metric spaces
We study metric properties stemming from the Connes spectral distance on
three types of non compact noncommutative spaces which have received attention
recently from various viewpoints in the physics literature. These are the
noncommutative Moyal plane, a family of harmonic Moyal spectral triples for
which the Dirac operator squares to the harmonic oscillator Hamiltonian and a
family of spectral triples with Dirac operator related to the Landau operator.
We show that these triples are homothetic spectral metric spaces, having an
infinite number of distinct pathwise connected components. The homothetic
factors linking the distances are related to determinants of effective Clifford
metrics. We obtain as a by product new examples of explicit spectral distance
formulas. The results are discussed.Comment: 23 pages. Misprints corrected, references updated, one remark added
at the end of the section 3. To appear in Review in Mathematical Physic
The Lorentzian distance formula in noncommutative geometry
For almost twenty years, a search for a Lorentzian version of the well-known
Connes' distance formula has been undertaken. Several authors have contributed
to this search, providing important milestones, and the time has now come to
put those elements together in order to get a valid and functional formula.
This paper presents a historical review of the construction and the proof of a
Lorentzian distance formula suitable for noncommutative geometry.Comment: 16 pages, final form, few references adde
Symmetries of noncommutative scalar field theory
We investigate symmetries of the scalar field theory with harmonic term on
the Moyal space with euclidean scalar product and general symplectic form. The
classical action is invariant under the orthogonal group if this group acts
also on the symplectic structure. We find that the invariance under the
orthogonal group can be restored also at the quantum level by restricting the
symplectic structures to a particular orbit.Comment: 12 pages, revised versio
Renormalization of the commutative scalar theory with harmonic term to all orders
The noncommutative scalar theory with harmonic term (on the Moyal space) has
a vanishing beta function. In this paper, we prove the renormalizability of the
commutative scalar field theory with harmonic term to all orders by using
multiscale analysis in the momentum space. Then, we consider and compute its
one-loop beta function, as well as the one on the degenerate Moyal space. We
can finally compare both to the vanishing beta function of the theory with
harmonic term on the Moyal space.Comment: 16 page
Metric Properties of the Fuzzy Sphere
The fuzzy sphere, as a quantum metric space, carries a sequence of metrics
which we describe in detail. We show that the Bloch coherent states, with these
spectral distances, form a sequence of metric spaces that converge to the round
sphere in the high-spin limit.Comment: Slightly shortened version, no major changes, two new references,
version to appear on Letters in Mathematical Physic