Papers by Jean Pierre Gazeau
L'accès aux archives de la revue « Annales de l'I. H. P., section A » implique l'accord avec les ... more L'accès aux archives de la revue « Annales de l'I. H. P., section A » implique l'accord avec les conditions générales d'utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
Theoretical and Mathematical Physics, 2014
Journal of Mathematical Physics, 2013
We consider some modifications of the two dimensional canonical commutation relations, leading to... more We consider some modifications of the two dimensional canonical commutation relations, leading to non commutative bosons and we show how biorthogonal bases of the Hilbert space of the system can be obtained out of them. Our construction extends those recently introduced by one of us (F.B.), modifying the canonical anticommutation relations. We also briefly discuss how bicoherent states, producing a resolution of the identity, can be defined.
Physical Review D, 2017
We study the spectral properties of the anisotropic part of Hamiltonian entering the quantum dyna... more We study the spectral properties of the anisotropic part of Hamiltonian entering the quantum dynamics of the Mixmaster universe. We derive the explicit asymptotic expressions for the energy spectrum in the limit of large and small volumes of the universe. Then we study the threshold condition between both regimes. Finally we prove that the spectrum is purely discrete for any volume of the universe. Our results validate and improve the known approximations to the anisotropy potential. They should be useful for any approach to the quantization of the Mixmaster universe.
Entropy, 2022
In quantum information processing, using a receiver device to differentiate between two non-ortho... more In quantum information processing, using a receiver device to differentiate between two non-orthogonal states leads to a quantum error probability. The minimum possible error is known as the Helstrom bound. In this work, we study the conditions for state discrimination using an alphabet of squeezed coherent states and compare them with conditions using the Glauber-Sudarshan, i.e., standard, coherent states.
Quantum versions of cylindric phase space, like for the motion of a particle on the circle, are o... more Quantum versions of cylindric phase space, like for the motion of a particle on the circle, are obtained through different families of coherent states. The latter are built from various probability distributions of the action variable. The method is illustrated with Gaussian distributions and uniform distributions on intervals, and resulting quantizations are explored.
Acta Polytechnica, 2010
Berezin-Klauder-Toeplitz (“anti-Wick”) or “coherent state” quantization of the complex plane, vie... more Berezin-Klauder-Toeplitz (“anti-Wick”) or “coherent state” quantization of the complex plane, viewed as the phase space of a particle moving on the line, is derived from the resolution of the unity provided by the standard (or gaussian) coherent states. The construction of these states and their attractive properties are essentially based on the energy spectrum of the harmonic oscillator, that is on natural numbers. We follow in this work the same path by considering sequences of non-negative numbers and their associated “non-linear” coherent states. We illustrate our approach with the 2-d motion of a charged particle in a uniform magnetic field. By solving the involved Stieltjes moment problem we construct a family of coherent states for this model. We then proceed with the corresponding coherent state quantization and we show that this procedure takes into account the circle topology of the classical motion.
Landscapes of Time-Frequency Analysis, 2020
arXiv: General Relativity and Quantum Cosmology, 2016
The massless minimally coupled scalar field in de Sitter ambient space formalism might play a sim... more The massless minimally coupled scalar field in de Sitter ambient space formalism might play a similar role to what the Higgs scalar field accomplishes within the electroweak standard model. With the introduction of a "local transformation" for this field, the interaction Lagrangian between the scalar field and the spinor field can be made similar to a gauge theory. In the null curvature limit, the Yukawa potential can be constructed from that Lagrangian. Finally the one-loop correction of the scalar-spinor interaction is presented, which is free of any infrared divergence.
Advances in Operator Theory, 2020
Covariant affine integral quantization is studied and applied to the motion of a particle in a pu... more Covariant affine integral quantization is studied and applied to the motion of a particle in a punctured plane R 2 , for which the phase space is R 2ˆR2. We examine the consequences of different quantizer operators built from weight functions on R 2ˆR2. To illustrate the procedure, we examine two examples of weights. The first one corresponds to 2-D coherent state families, while the second one corresponds to the affine inversion in the punctured plane. The later yields the usual canonical quantization and a quasi-probability distribution (2-D affine Wigner function) which is real, marginal in both position q and momentum p.
Entropy, 2019
In the realm of Boltzmann-Gibbs (BG) statistical mechanics and its q-generalisation for complex s... more In the realm of Boltzmann-Gibbs (BG) statistical mechanics and its q-generalisation for complex systems, we analysed sequences of q-triplets, or q-doublets if one of them was the unity, in terms of cycles of successive Möbius transforms of the line preserving unity ( q = 1 corresponds to the BG theory). Such transforms have the form q ↦ ( a q + 1 - a ) / [ ( 1 + a ) q - a ] , where a is a real number; the particular cases a = - 1 and a = 0 yield, respectively, q ↦ ( 2 - q ) and q ↦ 1 / q , currently known as additive and multiplicative dualities. This approach seemingly enables the organisation of various complex phenomena into different classes, named N-complete or incomplete. The classification that we propose here hopefully constitutes a useful guideline in the search, for non-BG systems whenever well described through q-indices, of new possibly observable physical properties.
Physical Review D, 2018
We present a regularisation approach to the study of the quantum dynamics of the Mixmaster univer... more We present a regularisation approach to the study of the quantum dynamics of the Mixmaster universe which allows to approximate the anisotropy potential with the explicitly integrable periodic 3-particle Toda system. This approach is based on a covariant Weyl-Heisenberg integral quantization. Such a procedure naturally amplifies the dynamical role of the underlying Toda system by smoothing out the three canyons of the anisotropy potential. Since the respective eigenfunctions can be explicitly constructed, our finding paves the way to a novel perturbative approach to the quantum Mixmaster dynamics.
Annals of Physics, 2016
We examine mathematical questions around angle (or phase) operator associated with a number opera... more 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. Contents 1. Introduction 2. Requirements for angle operator 3. Angle operator from the quantum harmonic oscillator 3 3.1. Spatial extension of the oscillator; heuristic pattern 3.2. The abstract, operator theoretic, setup 3.3. Building the full angle operator 3.4. Covariance of the half-circle angle operator 6 4. Quantum angle or phase from Weyl-Heisenberg integral quantization 5. Quantum angle for cylindric phase space 6. Conclusion Acknowledgments Appendix A. From Taylor to Fourier for angle functions and vice-versa A.1. Trigonometric argument function(s) A.2. Canonical angle operator Appendix B. Unitary Weyl-Heisenberg group representation Appendix C. Inequalities Appendix D. Normal law coherent states for the motion on the circle References
Physical Review D, 2015
We present a quantum version of the vacuum Bianchi IX model by implementing affine coherent state... 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.
Physics Letters A, 2008
A new family of 2-component vector-valued coherent states for the quantum particle motion in an i... more A new family of 2-component vector-valued coherent states for the quantum particle motion in an infinite square well potential is presented. They allow a consistent quantization of the classical phase space and observables for a particle in this potential. We then study the resulting position and (well-defined) momentum operators. We also consider their mean values in coherent states and their quantum dispersions.
Coherent States in Quantum Physics, 2009
Part One Coherent States 1 1 Introduction 3 1.1 The Motivations 3 2 The Standard Coherent States:... more Part One Coherent States 1 1 Introduction 3 1.1 The Motivations 3 2 The Standard Coherent States: the Basics 13 2.1 Schrödinger Definition 13 2.2 Four Representations of Quantum States 13 2.2.1 Position Representation 14 2.2.2 Momentum Representation 14 2.2.3 Number or Fock Representation 15 2.2.4 A Little (Lie) Algebraic Observation 16 2.2.5 Analytical or Fock-Bargmann Representation 16 2.2.6 Operators in Fock-Bargmann Representation 17 2.3 Schrödinger Coherent States 18 2.3.1 Bergman Kerne' as a Coherent State 18 2.3.2 A First Fundamental Property 19 2.3.3 Schrödinger Coherent States in the Two Other Representations 19 2.4 Glauber-Klauder-Sudarshan or Standard Coherent States 20 2.5 Why the Adjective Coherent? 20 3 The Standard Coherent States: the (Elementary) Mathematics 25 3.1 Introduction 25 3.2 Properties in the Hilbertian Framework 26 3.2.1 A "Continuity" from the Classical Complex Plane to Quantum States 26 3.2.2 "Coherent" Resolution of the Unity 26 3.2.3 The Interplay Between the Circle (as a Set of Parameters) and the Plane (as a Euclidean Space) 27 3.2.4 Analytical Bridge 28 3.2.5 Overcompleteness and Reproducing Properties 29 3.3 Coherent States in the Quantum Mechanical Context 30 3.3.1 Symbols 30 3.3.2 Lower Symbols 30
Physical Review D, 2013
In this paper we study quantum dynamics of the bouncing cosmological model. We focus on the model... more 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 Schrödinger'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.
Journal of Physics A: Mathematical and Theoretical, 2012
Quantum versions of cylindric phase space, like for the motion of a particle on the circle, are o... more Quantum versions of cylindric phase space, like for the motion of a particle on the circle, are obtained through different families of coherent states. The latter are built from various probability distributions of the action variable. The method is illustrated with Gaussian distributions and uniform distributions on intervals, and resulting quantizations are explored.
Journal of Physics A: Mathematical and Theoretical, 2012
We build coherent states (CS) for unbounded motions along two different procedures. In the first ... more We build coherent states (CS) for unbounded motions along two different procedures. In the first one we adapt the Malkin-Manko construction for quadratic Hamiltonians to the motion of a particle in a linear potential. A generalization to arbitrary potentials is discussed. The second one extends to continuous spectrum previous constructions of action-angle coherent states in view of a consistent energy quantization.
Journal of Physics A: Mathematical and Theoretical, 2010
By using a coherent state quantization of paragrassmann variables, operators are constructed in f... more By using a coherent state quantization of paragrassmann variables, operators are constructed in finite Hilbert spaces. We thus obtain in a straightforward way a matrix representation of the paragrassmann algebra. This algebra of finite matrices realizes a deformed Weyl-Heisenberg algebra. The study of mean values in coherent states of some of these operators leads to interesting conclusions.
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Papers by Jean Pierre Gazeau