Quantization with Action-Angle Coherent States

For a single degree of freedom confined mechanical system with given energy, we know that the motion is always periodic and action-angle variables are convenient choice as conjugate phase-space variables. We construct action-angle coherent states in view to provide a quantization scheme that yields precisely a given observed energy spectrum ${E_n}$ for such a system. This construction is based on a Bayesian approach: each family corresponds to a choice of probability distributions such that the classical energy averaged with respect to this probability distribution is precisely $E_n$ up to a constant shift. The formalism is viewed as a natural extension of the Bohr-Sommerfeld rule and an alternative to the canonical quantization. In particular, it also yields a satisfactory angle operator as a bounded self-adjoint operator

A natural fuzzyness of de Sitter space-time

A non-commutative structure for de Sitter spacetime is naturally introduced by replacing ("fuzzyfication") the classical variables of the bulk in terms of the dS analogs of the Pauli-Lubanski operators. The dimensionality of the fuzzy variables is determined by a Compton length and the commutative limit is recovered for distances much larger than the Compton distance. The choice of the Compton length determines different scenarios. In scenario I the Compton length is determined by the limiting Minkowski spacetime. A fuzzy dS in scenario I implies a lower bound (of the order of the Hubble mass) for the observed masses of all massive particles (including massive neutrinos) of spin s>0. In scenario II the Compton length is fixed in the de Sitter spacetime itself and grossly determines the number of finite elements ("pixels" or "granularity") of a de Sitter spacetime of a given curvature.Comment: 16 page

Nested quasicrystalline discretisations of the line

One-dimensional cut-and-project point sets obtained from the square lattice in the plane are considered from a unifying point of view and in the perspective of aperiodic wavelet constructions. We successively examine their geometrical aspects, combinatorial properties from the point of view of the theory of languages, and self-similarity with algebraic scaling factor $\theta$. We explain the relation of the cut-and-project sets to non-standard numeration systems based on $\theta$. We finally examine the substitutivity, a weakened version of substitution invariance, which provides us with an algorithm for symbolic generation of cut-and-project sequences

Asymptotic behavior of beta-integers

Beta-integers (``$\beta$-integers'') are those numbers which are the counterparts of integers when real numbers are expressed in irrational basis $\beta > 1$. In quasicrystalline studies $\beta$-integers supersede the ``crystallographic'' ordinary integers. When the number $\beta$ is a Parry number, the corresponding $\beta$-integers realize only a finite number of distances between consecutive elements and somewhat appear like ordinary integers, mainly in an asymptotic sense. In this letter we make precise this asymptotic behavior by proving four theorems concerning Parry $\beta$-integers.Comment: 17 page

Semi-classical behavior of P\"oschl-Teller coherent states

We present a construction of semi-classical states for P\"oschl-Teller potentials based on a supersymmetric quantum mechanics approach. The parameters of these "coherent" states are points in the classical phase space of these systems. They minimize a special uncertainty relation. Like standard coherent states they resolve the identity with a uniform measure. They permit to establish the correspondence (quantization) between classical and quantum quantities. Finally, their time evolution is localized on the classical phase space trajectory.Comment: 7 pages, 2 figures, 1 animatio

A discrete nonetheless remarkable brick in de Sitter: the "massless minimally coupled field"

Over the last ten years interest in the physics of de Sitter spacetime has been growing very fast. Besides the supposed existence of a "de sitterian period" in inflation theories, the observational evidence of an acceleration of the universe expansion (interpreted as a positive cosmological constant or a "dark energy" or some form of "quintessence") has triggered a lot of attention in the physics community. A specific de sitterian field called "massless minimally coupled field" (mmc) plays a fundamental role in inflation models and in the construction of the de sitterian gravitational field. A covariant quantization of the mmc field, `a la Krein-Gupta-Bleuler was proposed in [1]. In this talk, we will review this construction and explain the relevance of such a field in the construction of a massless spin 2 field in de Sitter space-time.Comment: Proceedings of the XXVII Colloquium on Group Theoretical Methods in Physics, Yerevan, August 200

The nature of Λ\Lambda and the mass of the graviton: A critical view

The existence of a non-zero cosmological constant $\Lambda$ gives rise to controversial interpretations. Is $\Lambda$ a universal constant fixing the geometry of an empty universe, as fundamental as the Planck constant or the speed of light in the vacuum? Its natural place is then on the left-hand side of the Einstein equation. Is it instead something emerging from a perturbative calculus performed on the metric $g\_{\mu\nu}$ solution of the Einstein equation and to which it might be given a material status of (dark or bright) "energy"? It should then be part of the content of the right-hand side of the Einstein equations. The purpose of this paper is not to elucidate the fundamental nature of $\Lambda$, but instead we aim to present and discuss some of the arguments in favor of both interpretations of the cosmological constant. We conclude that if the fundamental of the geometry of space-time is minkowskian, then the square of the mass of the graviton is proportional to $\Lambda$; otherwise, if the fundamental state is deSitter/AdS, then the graviton is massless in the deSitterian sense.Comment: 39 page