55 research outputs found
Quasi-Dirac Operators and Quasi-Fermions
We investigate examples of quasi-spectral triples over two-dimensional
commutative sphere, which are obtained by modifying the order-one condition. We
find equivariant quasi-Dirac operators and prove that they are in a
topologically distinct sector than the standard Dirac operator.Comment: 11 page
Metaplectic and spin representations: a parallel treatment
The analogies between symplectic and orthogonal groups, regarded as
symmetries of real bilinear forms, are manifest in their (metaplectic and spin)
projective representations. In finite dimensions, those are true
representations of doubly covering groups; but one may also use group
extensions by a circle. Here we lay out a parallel treatment of of the
Mp and Spin covering groups, acting on the respective
Fock spaces by permuting certain Gaussian vectors. The cocycles of these
extensions exhibit interesting similarities.Comment: Latex, 38 page
On the ultraviolet behaviour of quantum fields over noncommutative manifolds
By exploiting the relation between Fredholm modules and the
Segal-Shale-Stinespring version of canonical quantization, and taking as
starting point the first-quantized fields described by Connes' axioms for
noncommutative spin geometries, a Hamiltonian framework for fermion quantum
fields over noncommutative manifolds is introduced. We analyze the ultraviolet
behaviour of second-quantized fields over noncommutative 3-tori, and discuss
what behaviour should be expected on other noncommutative spin manifolds.Comment: 10 pages, RevTeX version, a few references adde
On summability of distributions and spectral geometry
Modulo the moment asymptotic expansion, the Cesàro and parametric behaviours of distributions at infinity are equivalent. On the strength of this result, we construct the asymptotic analysis for spectral densities arising from elliptic pseudodifferential operators. We show how Cesàro developments lead to efficient calculations of the expansion coefficients of counting number functionals and Green functions. The bosonic action functional proposed by Chamseddine and Connes can more generally be validated as a Cesàro asymptotic development
Contracting the Wigner kernel of a spin to the Wigner kernel of a particle
A general relation between the Moyal formalisms for a spin and a particle is established. Once the formalism has been set up for a spin, the phase-space description of a particle is obtained from contracting the group of rotations to the oscillator group. In this process, turn into a spin Wigner kernel turns into the Wigner kernel of a particle. In fact, only one out of 22s different possible kernels for a spin shows this behavior
Fermion Hilbert Space and Fermion Doubling in the Noncommutative Geometry Approach to Gauge Theories
In this paper we study the structure of the Hilbert space for the recent
noncommutative geometry models of gauge theories. We point out the presence of
unphysical degrees of freedom similar to the ones appearing in lattice gauge
theories (fermion doubling). We investigate the possibility of projecting out
these states at the various levels in the construction, but we find that the
results of these attempts are either physically unacceptable or geometrically
unappealing.Comment: plain LaTeX, pp. 1
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
Inverted spectroscopy and interferometry for quantum-state reconstruction of systems with SU(2) symmetry
We consider how the conventional spectroscopic and interferometric schemes
can be rearranged to serve for reconstructing quantum states of physical
systems possessing SU(2) symmetry. The discussed systems include a collection
of two-level atoms, a two-mode quantized radiation field with a fixed total
number of photons, and a single laser-cooled ion in a two-dimensional harmonic
trap with a fixed total number of vibrational quanta. In the proposed
rearrangement, the standard spectroscopic and interferometric experiments are
inverted. Usually one measures an unknown frequency or phase shift using a
system prepared in a known quantum state. Our aim is just the inverse one,
i.e., to use a well-calibrated apparatus with known transformation parameters
to measure unknown quantum states.Comment: 8 pages, REVTeX. More info on
http://www.ligo.caltech.edu/~cbrif/science.htm
Almost-Commutative Geometries Beyond the Standard Model II: New Colours
We will present an extension of the standard model of particle physics in its
almost-commutative formulation. This extension is guided by the minimal
approach to almost-commutative geometries employed in [13], although the model
presented here is not minimal itself.
The corresponding almost-commutative geometry leads to a Yang-Mills-Higgs
model which consists of the standard model and two new fermions of opposite
electro-magnetic charge which may possess a new colour like gauge group. As a
new phenomenon, grand unification is no longer required by the spectral action.Comment: Revised version for publication in J.Phys.A with corrected Higgs
masse
Spin-Hall effect with quantum group symmetry
We construct a model of spin-Hall effect on a noncommutative 4 sphere with
isospin degrees of freedom (coming from a noncommutative instanton) and
invariance under a quantum orthogonal group. The corresponding representation
theory allows to explicitly diagonalize the Hamiltonian and construct the
ground state; there are both integer and fractional excitations. Similar models
exist on higher dimensional noncommutative spheres and noncommutative
projective spaces.Comment: v2: 14 pages, latex. Several changes and additional material; two
extra sections added. To appear in LMP. Dedicated to Rafael Sorkin with
friendship and respec
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