Coherent states (CS) of the harmonic oscillator (also called canonical CS) were introduced in 1926 by Schrödinger in answer to a remark by Lorentz on the classical interpretation of the wave function. They were rediscovered in the early... more
Coherent states (CS) of the harmonic oscillator (also called canonical CS) were introduced in 1926 by Schrödinger in answer to a remark by Lorentz on the classical interpretation of the wave function. They were rediscovered in the early 1960s, first (somewhat implicitly) by Klauder in the context of a novel representation of quantum states, then by Glauber and Sudarshan for
Research Interests: Group Representation, Quantum Physics, Public Opinion, Quantum Gravity, Quantum Optics, and 23 moreSignal Processing, Particle Physics, Quantum Information, Quantum Information Processing, Group Theory, Loop Quantum Gravity, Magnetic field, Time-Frequency Analysis, Wavelet Analysis, Quantum Information Theory, Mathematical Sciences, Physical sciences, Phase transition, Oscillations, Quantum Computer, Spectrum, Functional integration, Cross Section, Coherent States, Quantum Harmonic Oscillator, Reproducing Kernel Hilbert Space, Physical Properties, and Orthogonal Polynomial
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The paper is devoted to integral quantization, a procedure based on operator-valued measure and resolution of the identity. We insist on covariance properties in the important case where group representation theory is involved. We also... more
The paper is devoted to integral quantization, a procedure based on operator-valued measure and resolution of the identity. We insist on covariance properties in the important case where group representation theory is involved. We also insist on the inherent probabilistic aspects of this classical-quantum map. The approach includes and generalizes coherent state quantization. Two applications based on group representation are carried out. The first one concerns the Weyl-Heisenberg group and the euclidean plane viewed as the corresponding phase space. We show that there exists a world of quantizations which yield the canonical commutation rule and the usual quantum spectrum of the harmonic oscillator. The second one concerns the affine group of the real line and gives rise to an interesting regularization of the dilation origin in the half-plane viewed as the corresponding phase space.
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We present the quantization of the vacuum Bianchi IX model. Using a compound quantization procedure (based on affine coherent states and Weyl quantization) and the Born-Oppenheimer approximation, we develop a complete analytical treatment... more
We present the quantization of the vacuum Bianchi IX model. Using a compound quantization procedure (based on affine coherent states and Weyl quantization) and the Born-Oppenheimer approximation, we develop a complete analytical treatment on the semi-classical level. The resolution of the classical singularity occurs due to a repulsive potential generated by the affine quantization. This procedure shows that during contraction the quantum energy of anisotropic degrees of freedom grows much slower than the classical one. Our treatment is put in the general context of methods of molecular physics which include both adiabatic (Born-Oppenheimer) and non-adiabatic (vibronic) approximations.
We present a quantum version of the vacuum Bianchi IX model by implementing affine coherent state quantization combined with a Born-Oppenheimer-like adiabatic approximation. The analytical treatment is carried out on both quantum and... more
We present a quantum version of the vacuum Bianchi IX model by implementing affine coherent state quantization combined with a Born-Oppenheimer-like adiabatic approximation. The analytical treatment is carried out on both quantum and semiclassical levels. The resolution of the classical singularity occurs by means of a repulsive potential generated by our quantization procedure. The quantization of the oscillatory degrees of freedom produces a radiation energy density term in the semiclassical constraint equation. The Friedmann-like lowest energy eigenstates of the system are found to be dynamically stable.
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We study the orbital magnetism of a two dimensional electron gas confined by an isotropic harmonic potential by using a coherent state approach. A rigorous derivation of the magnetic moment gives us a full description of the phase diagram... more
We study the orbital magnetism of a two dimensional electron gas confined by an isotropic harmonic potential by using a coherent state approach. A rigorous derivation of the magnetic moment gives us a full description of the phase diagram of the magnetization. We show a paramagnetic behavior in the thermodynamical limit as well as in the quasiclassical limit under a weak field.
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We show that quantization through standard (Gaussian) coherent states (CS) enables us to construct fairly reasonable quantum versions of irregular observables living on the classical phase space, such as the argument function or even a... more
We show that quantization through standard (Gaussian) coherent states (CS) enables us to construct fairly reasonable quantum versions of irregular observables living on the classical phase space, such as the argument function or even a large set of distributions comprising the tempered distributions. Enlarging in this way the set of quantizable classical observables allows to obtain any finite dimensional projector in the Hilbert space of quantum states.
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We investigate in a geometrical way the sieving process of fl for obtaining the Delone set fl fl of fl - integers where fl is a Perron number in the context of linear asymp- totic invariants associated with a canonical inductive system... more
We investigate in a geometrical way the sieving process of fl for obtaining the Delone set fl fl of fl - integers where fl is a Perron number in the context of linear asymp- totic invariants associated with a canonical inductive system constructed from fl . When fl is a Pisot number, we exhibit a canonical cut-and-project scheme, a model
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Page 1. Syed Twareque Ali Jean-Pierre Antoine Jean-Pierre Gazeau Coherent States> Wavelets and Their Generalizations Springer Page 2. Page 3. Page 4. Page 5. Graduate Texts in Contemporary Physics Series Editors: R ...
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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... more
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 $...
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ABSTRACT One more coherent state quantization of a complex plane is presented. Although the complex plane is equipped with a non-rotationally invariant measure, we still obtain a canonical commutation rule (up to a simple rescaling). We... more
ABSTRACT One more coherent state quantization of a complex plane is presented. Although the complex plane is equipped with a non-rotationally invariant measure, we still obtain a canonical commutation rule (up to a simple rescaling). We explain how the involved coherent states, built from holomorphic continuations of Hermite polynomials, are related to the non-commutative plane.
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Research Interests: Functional Analysis, Quantum Gravity, Quantum Optics, Signal Processing, Quantum Information, and 10 moreQuantum Mechanics, Non-commutative Geometry, Representation Theory, Mathematical Sciences, Scientific Communication, Physical sciences, Quantum Measurement, Coherent States, Reproducing Kernel Hilbert Space, and Supersymmetric Quantum Mechanics
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We present the construction of a symmetry plane-group for a quasiperiodic point-set in the plane named τ-lattice, τ being the golden-ratio. The τ-lattice generalizes the notion of periodic lattice to quasiperiodicity. The algebraic... more
We present the construction of a symmetry plane-group for a quasiperiodic point-set in the plane named τ-lattice, τ being the golden-ratio. The τ-lattice generalizes the notion of periodic lattice to quasiperiodicity. The algebraic framework is issued from the counting system of τ-integers. The set of τ-integers can be equipped with an abelian group structure and an internal multiplicative law. These arithmetic structures lead to a freely generated symmetry plane-group for the τ-lattice, based on repetitions of discrete “τ-rotations” and “τ-translations” in the plane. Hence the τ-lattice, endowed with these adapted rotations and translations, can be viewed as a lattice with “rotational” symmetries.