393 research outputs found
Curved Noncommutative Tori as Leibniz Quantum Compact Metric Spaces
We prove that curved noncommutative tori, introduced by Dabrowski and Sitarz,
are Leibniz quantum compact metric spaces and that they form a continuous
family over the group of invertible matrices with entries in the commutant of
the quantum tori in the regular representation, when this group is endowed with
a natural length function.Comment: 16 Pages, v3: accepted in Journal of Math. Physic
Reply to Norsen's paper "Are there really two different Bell's theorems?"
Yes. That is my polemical reply to the titular question in Travis Norsen's
self-styled "polemical response to Howard Wiseman's recent paper." Less
polemically, I am pleased to see that on two of my positions --- that Bell's
1964 theorem is different from Bell's 1976 theorem, and that the former does
not include Bell's one-paragraph heuristic presentation of the EPR argument ---
Norsen has made significant concessions. In his response, Norsen admits that
"Bell's recapitulation of the EPR argument in [the relevant] paragraph leaves
something to be desired," that it "disappoints" and is "problematic". Moreover,
Norsen makes other statements that imply, on the face of it, that he should
have no objections to the title of my recent paper ("The Two Bell's Theorems of
John Bell"). My principle aim in writing that paper was to try to bridge the
gap between two interpretational camps, whom I call 'operationalists' and
'realists', by pointing out that they use the phrase "Bell's theorem" to mean
different things: his 1964 theorem (assuming locality and determinism) and his
1976 theorem (assuming local causality), respectively. Thus, it is heartening
that at least one person from one side has taken one step on my bridge. That
said, there are several issues of contention with Norsen, which we (the two
authors) address after discussing the extent of our agreement with Norsen. The
most significant issues are: the indefiniteness of the word 'locality' prior to
1964; and the assumptions Einstein made in the paper quoted by Bell in 1964 and
their relation to Bell's theorem.Comment: 13 pages (arXiv version) in http://www.ijqf.org/archives/209
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 beat of a fuzzy drum: fuzzy Bessel functions for the disc
The fuzzy disc is a matrix approximation of the functions on a disc which
preserves rotational symmetry. In this paper we introduce a basis for the
algebra of functions on the fuzzy disc in terms of the eigenfunctions of a
properly defined fuzzy Laplacian. In the commutative limit they tend to the
eigenfunctions of the ordinary Laplacian on the disc, i.e. Bessel functions of
the first kind, thus deserving the name of fuzzy Bessel functions.Comment: 30 pages, 8 figure
An Obstruction to Quantization of the Sphere
In the standard example of strict deformation quantization of the symplectic
sphere , the set of allowed values of the quantization parameter
is not connected; indeed, it is almost discrete. Li recently constructed a
class of examples (including ) in which can take any value in an
interval, but these examples are badly behaved. Here, I identify a natural
additional axiom for strict deformation quantization and prove that it implies
that the parameter set for quantizing is never connected.Comment: 23 page. v2: changed sign conventio
Strict Deformation Quantization for a Particle in a Magnetic Field
Recently, we introduced a mathematical framework for the quantization of a
particle in a variable magnetic field. It consists in a modified form of the
Weyl pseudodifferential calculus and a C*-algebraic setting, these two points
of view being isomorphic in a suitable sense. In the present paper we leave
Planck's constant vary, showing that one gets a strict deformation quantization
in the sense of Rieffel. In the limit h --> 0 one recovers a Poisson algebra
induced by a symplectic form defined in terms of the magnetic field.Comment: 23 page
Unstable solitons on noncommutative tori and D-branes
We describe a class of exact solutions of super Yang-Mills theory on
even-dimensional noncommutative tori. These solutions generalize the solitons
on a noncommutative plane introduced in hep-th/0009142 that are conjectured to
describe unstable D2p-D0 systems. We show that the spectrum of quadratic
fluctuations around our solutions correctly reproduces the string spectrum of
the D2p-D0 system in the Seiberg-Witten decoupling limit. In particular the
fluctuations correctly reproduce the 0-0 string winding modes. For p=1 and p=2
we match the differences between the soliton energy and the energy of an
appropriate SYM BPS state with the binding energies of D2-D0 and D4-D0 systems.
We also give an example of a soliton that we conjecture describes branes of
intermediate dimension on a torus such as a D2-D4 system on a four-torus.Comment: 22 pages, Latex; v.2: references adde
On the role of twisted statistics in the noncommutative degenerate electron gas
We consider the problem of a degenerate electron gas in the background of a
uniformly distributed positive charge, ensuring overall neutrality of the
system, in the presence of non-commutativity. In contrast to previous
calculations that did not include twisted statistics, we find corrections to
the ground state energy already at first order in perturbation theory when the
twisted statistics is taken into account. These corrections arise since the
interaction energy is sensitive to two particle correlations, which are
modified for twisted anti-commutation relations
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