POVM Quantization

We present a general formalism for giving a measure space paired with a separable Hilbert space a quantum version based on normalized positive operator-valued measure. The latter are built from families of density operators labelled by points of the measure space. We specially focus on various probabilistic aspects of these constructions. Simple or more elaborate examples illustrate the procedure: circle, 2-sphere, plane, half-plane. Links with POVM quantum measurement and quantum statistical inference are sketched

On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon

The spectrum of a weighted Dirac comb on the Thue-Morse quasicrystal is investigated, and characterized up to a measure zero set, by means of the Bombieri-Taylor conjecture, for Bragg peaks, and of another conjecture that we call Aubry-Godr\`eche-Luck conjecture, for the singular continuous component. The decomposition of the Fourier transform of the weighted Dirac comb is obtained in terms of tempered distributions. We show that the asymptotic arithmetics of the $p$-rarefied sums of the Thue-Morse sequence (Dumont; Goldstein, Kelly and Speer; Grabner; Drmota and Skalba,...), namely the fractality of sum-of-digits functions, play a fundamental role in the description of the singular continous part of the spectrum, combined with some classical results on Riesz products of Peyri\`ere and M. Queff\'elec. The dominant scaling of the sequences of approximant measures on a part of the singular component is controlled by certain inequalities in which are involved the class number and the regulator of real quadratic fields.Comment: 35 pages In honor of the 60-th birthday of Henri Cohe

Three paths toward the quantum angle operator

We examine mathematical questions around angle (or phase) operator associated with a number operator through a short list of basic requirements. We implement three methods of construction of quantum angle. The first one is based on operator theory and parallels the definition of angle for the upper half-circle through its cosine and completed by a sign inversion. The two other methods are integral quantization generalizing in a certain sense the Berezin-Klauder approaches. One method pertains to Weyl-Heisenberg integral quantization of the plane viewed as the phase space of the motion on the line. It depends on a family of "weight" functions on the plane. The third method rests upon coherent state quantization of the cylinder viewed as the phase space of the motion on the circle. The construction of these coherent states depends on a family of probability distributions on the line.Comment: 20 page

Complex and real Hermite polynomials and related quantizations

It is known that the anti-Wick (or standard coherent state) quantization of the complex plane produces both canonical commutation rule and quantum spectrum of the harmonic oscillator (up to the addition of a constant). In the present work, we show that these two issues are not necessarily coupled: there exists a family of separable Hilbert spaces, including the usual Fock-Bargmann space, and in each element in this family there exists an overcomplete set of unit-norm states resolving the unity. With the exception of the Fock-Bargmann case, they all produce non-canonical commutation relation whereas the quantum spectrum of the harmonic oscillator remains the same up to the addition of a constant. The statistical aspects of these non-equivalent coherent states quantizations are investigated. We also explore the localization aspects in the real line yielded by similar quantizations based on real Hermite polynomials.Comment: 15 pages, 6 figure

Krein Spaces in de Sitter Quantum Theories

Experimental evidences and theoretical motivations lead to consider the curved space-time relativity based on the de Sitter group $SO_0(1,4)$ or $Sp(2,2)$ as an appealing substitute to the flat space-time Poincare relativity. Quantum elementary systems are then associated to unitary irreducible representations of that simple Lie group. At the lowest limit of the discrete series lies a remarkable family of scalar representations involving Krein structures and related undecomposable representation cohomology which deserves to be thoroughly studied in view of quantization of the corresponding carrier fields. The purpose of this note is to present the mathematical material needed to examine the problem and to indicate possible extensions of an exemplary case, namely the so-called de Sitterian massless minimally coupled field, i.e. a scalar field in de Sitter space-time which does not couple to the Ricci curvature

Quantum states of the bouncing universe

In this paper we study quantum dynamics of the bouncing cosmological model. We focus on the model of the flat Friedman-Robertson-Walker universe with a free scalar field. The bouncing behavior, which replaces classical singularity, appears due to the modification of general relativity along the methods of loop quantum cosmology. We show that there exist a unitary transformation that enables to describe the system as a free particle with Hamiltonian equal to canonical momentum. We examine properties of the various quantum states of the Universe: boxcar state, standard coherent state, and soliton-like state, as well as Schr{\"o}dinger's cat states constructed from these states. Characteristics of the states such as quantum moments and Wigner functions are investigated. We show that each of these states have, for some range of parameters, a proper semiclassical limit fulfilling the correspondence principle. Decoherence of the superposition of two universes is described and possible interpretations in terms of triad orientation and Belinsky-Khalatnikov-Lifshitz conjecture are given. Some interesting features regarding the area of the negative part of the Wigner function have emerged.Comment: 18 pages, 19 figure