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$^\mathrm{c}$ and Spin$^\mathrm{c}$ 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