37 research outputs found
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
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
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
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
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
Almost-Commutative Geometries Beyond the Standard Model III: Vector Doublets
We will present a new extension of the standard model of particle physics in
its almostcommutative formulation. This extension has as its basis the algebra
of the standard model with four summands [11], and enlarges only the particle
content by an arbitrary number of generations of left-right symmetric doublets
which couple vectorially to the U(1)_YxSU(2)_w subgroup of the standard model.
As in the model presented in [8], which introduced particles with a new colour,
grand unification is no longer required by the spectral action. The new model
may also possess a candidate for dark matter in the hundred TeV mass range with
neutrino-like cross section
The lowest excited configuration of harmonium
The harmonium model has long been regarded as an exactly solvable laboratory
bench for quantum chemistry [Heisenberg, 1926]. For studying correlation
energy, only the ground state of the system has received consideration
heretofore. This is a spin singlet state. In this work we exhaustively study
the lowest excited (spin triplet) harmonium state, with the main purpose of
revisiting the relation between entanglement measures and correlation energy
for this quite different species. The task is made easier by working with
Wigner quasiprobabilities on phase space.Comment: Latex, 21 pages. Minor changes, 4 improved figures, references added.
To appear in Physical Review
Weyl-Underhill-Emmrich quantization and the Stratonovich-Weyl quantizer
Weyl-Underhill-Emmrich (WUE) quantization and its generalization are
considered. It is shown that an axiomatic definition of the Stratonovich-Weyl
(SW) quantizer leads to severe difficulties. Quantization on the cylinder
within the WUE formalism is discussed.Comment: 15+1 pages, no figure