arXiv subjects

Jean Pierre Gazeau

Publications and source records attributed to Jean Pierre Gazeau.

At least 19 recordsLinked to original sources

arXiv2024

Weyl-Heisenberg covariant quantization for the discrete torus

Covariant integral quantization is implemented for systems whose phase space is $Z_{d} \times Z_{d}$, i.e., for systems moving on the discrete periodic set $Z_d= \{0,1,\dotsc d-1$ mod$ d\}$. The symmetry group of this phase space is the periodic discrete version of the Weyl-Heisenberg group, namely the central extension of the abelian group $Z_d \times Z_d$. In this regard, the phase space is viewed as the left coset of the group with its center. The non-trivial unitary irreducible representation of this group, as acting on $L^2(Z_{N})$, is square integrable on the phase phase. We derive the corresponding covariant integral quantizations from (weight) functions on the phase space, and display their phase space portrait.

quant-ph↗

arXiv2022

Integral Quantization for the Discrete Cylinder

Covariant integral quantizations are based on the resolution of the identity by continuous or discrete families of normalised positive operator valued measures (POVM), which have appealing probabilistic content and which transform in a covariant way. One of their advantages is to allow to circumvent problems due to the presence of singularities in the classical models. In this paper we implement covariant integral quantizations for systems whose phase space is $\mathbb{Z}\times\,\mathbb{S}^1$, i.e., for systems moving on the circle. The symmetry group of this phase space is the discrete \& compact version of the Weyl-Heisenberg group, namely the central extension of the abelian group $\mathbb{Z}\times\,\mathrm{SO}(2)$. In this regard, the phase space is viewed as the right coset of the group with its center. The non-trivial unitary irreducible representation of this group, as acting on $L^2(\mathbb{S}^1)$, is square integrable on the phase space. We show how to derive corresponding covariant integral quantizations from (weight) functions on the phase space {and resulting resolution of the identity}. {As particular cases of the latter} we recover quantizations with de Bièvre-del Olmo-Gonzales and Kowalski-Rembielevski-Papaloucas coherent states on the circle. Another straightforward outcome of our approach is the Mukunda Wigner transform. We also look at the specific cases of coherent states built from shifted gaussians, Von Mises, Poisson, and Fejér kernels. Applications to stellar representations are in progress.

quant-ph↗

arXiv2021

2-D Covariant Affine Integral Quantization(s)

Covariant affine integral quantization is studied and applied to the motion of a particle in a punctured plane R^2_\ast=R^2\{0}, for which the phase space is R^2_\ast=R^2\{0}X R^2. We examine the consequences of different quantizer operators built from weight functions on this phase space. 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 and momentum.

math-ph↗

arXiv2020

Signal analysis and quantum formalism: Quantizations with no Planck constant

Signal analysis is built upon various resolutions of the identity in signal vector spaces, e.g. Fourier, Gabor, wavelets, etc. Similar resolutions are used as quantizers of functions or distributions, paving the way to a time-frequency or time-scale quantum formalism and revealing interesting or unexpected features. Extensions to classical electromagnetism viewed as a quantum theory for waves and not for photons are mentioned.

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arXiv2018

Coherent states in Quantum Optics: An oriented overview

In this survey, various generalisations of Glauber-Sudarshan coherent states are described in a unified way, with their statistical properties and their possible role in non-standard quantisations of the classical electromagnetic field. Some statistical photon-counting aspects of Perelomov SU(2) and SU(1,1) coherent states are emphasized.

quant-ph↗

arXiv2018

Covariant integral quantization of the unit disk

We implement a SU(1,1) covariant integral quantization of functions or distributions on the unit disk. The latter can be viewed as the phase space for the motion of a test "massive" particle on 1+1 Anti de Sitter space-time, and the relevant unitary irreducible representations of SU(1,1) corresponding to the quantum version of such motions are found in the discrete series and its lower limits. Our quantization method depends on a weight function on the phase space, and it includes Perelomov coherent states quantization. Semi-classical portraits or lower symbols of main physically relevant operators are determined.

math-ph↗

arXiv2018

Primordial gravitational waves in a quantum model of big bounce

We quantise and solve the dynamics of gravitational waves in a quantum Friedmann-Lemaitre-Robertson-Walker spacetime filled with perfect fluid. The classical model is formulated canonically. The Hamiltonian constraint is de-parametrised by setting a fluid variable as the internal clock. The obtained reduced (i.e. physical) phase space is then quantised. Our quantisation procedure is implemented in accordance with two different phase space symmetries, namely, the Weyl-Heisenberg symmetry for the perturbation variables, and the affine symmetry for the background variables. As an appealing outcome, the initial singularity is removed and replaced with a quantum bounce. The quantum model depends on a free parameter that is naturally induced from quantisation and determines the scale of the bounce. We study the dynamics of the quantised gravitational waves across the bounce through three different methods ("thin-horizon", analytical and numerical) which give consistent results and we determine the primordial power spectrum for the case of radiation-dominated universe. Next, we use the instantaneous radiation-matter transition transfer function to make approximate predictions for late universe and constrain our model with LIGO and Planck data. We also give an estimate of the quantum uncertainties in the present-day universe.

gr-qc↗

arXiv2017

Higgs Field and the Massless Minimally Coupled Scalar Field in de Sitter Universe

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.

gr-qc↗

arXiv2017

Three examples of quantum dynamics on the half-line with smooth bouncing

This article is an introductory presentation of the quantization of the half-plane based on affine coherent states (ACS). The half-plane is viewed as the phase space for the dynamics of a positive physical quantity evolving with time, and its affine symmetry is preserved due to the covariance of this type of quantization. We promote the interest of such a procedure for transforming a classical model into a quantum one, since the singularity at the origin is systematically removed, and the arbitrariness of boundary conditions can be easily overcome. We explain some important mathematical aspects of the method. Three elementary examples of applications are presented, the quantum breathing of a massive sphere, the quantum smooth bouncing of a charged sphere, and a smooth bouncing "dust" sphere as a simple model of quantum Newtonian cosmology.

quant-ph↗

arXiv2017

Quantum localisation on the circle

Covariant integral quantisation using coherent states for semidirect product groups is studied and applied to the motion of a particle on the circle. In the present case the group is the Euclidean group E$(2)$. We implement the quantisation of the basic classical observables, particularly the $2π$-periodic discontinuous angle function and the angular momentum, and compute their corresponding lower symbols. An important part of our study is devoted to the angle operator given by our procedure, its spectrum and lower symbol, its commutator with the quantum angular momentum, and the resulting Heisenberg inequality. Comparison with other approaches to the long-standing question of the quantum angle is discussed.

math-ph↗

arXiv2017

A baby Majorana quantum formalism

The aim of the present paper is to introduce and to discuss the most basic fundamental concepts of quantum physics by means of a simple and pedagogical example. An appreciable part of its content presents original results. We start with the Euclidean plane which is certainly a paradigmatic example of a Hilbert space. The pure states form the unit circle (actually a half of it), the mixed states form the unit disk (actually a half of it), and rotations in the plane rule time evolution through Majorana-like equations involving only real quantities for closed and open systems. The set of pure states or a set of mixed states solve the identity and they are used for understanding the concept of integral quantization of functions on the unit circle and to give a semi-classical portrait of quantum observables. Interesting probabilistic aspects are developed. Since the tensor product of two planes, their direct sum, their cartesian product, are isomorphic (2 is the unique solution to x^x= x\times x = x+x), and they are also isomorphic to C^2, and to the quaternion field H (as a vector space), we describe an interesting relation between entanglement of real states, one-half spin cat states, and unit-norm quaternions which form the group SU$(2)$. We explain the most general form of the Hamiltonian in the real plane by considering the integral quantization of a magnetic-like interaction potential viewed as a classical observable on the unit $2$-sphere. Finally, we present an example of quantum measurement with pointer states lying also in the Euclidean plane.

quant-ph↗

arXiv2016

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.

quant-ph↗

arXiv2016

Covariant Affine Integral Quantization(s)

Covariant affine integral quantization of the half-plane is studied and applied to the motion of a particle on the half-line. We examine the consequences of different quantizer operators built from weight functions on the half-plane. To illustrate the procedure, we examine two particular choices of the weight function, yielding thermal density operators and affine inversion respectively. The former gives rise to a temperature-dependent probability distribution on the half-plane whereas the later yields the usual canonical quantization and a quasi-probability distribution (affine Wigner function) which is real, marginal in both momentum p and position q.

quant-ph↗

arXiv2015

${\mathcal D}$-Pseudo-Bosons, Complex Hermite Polynomials, and Integral Quantization

The ${\mathcal D}$-pseudo-boson formalism is illustrated with two examples. The first one involves deformed complex Hermite polynomials built using finite-dimensional irreducible representations of the group ${\rm GL}(2,{\mathbb C})$ of invertible $2 \times 2$ matrices with complex entries. It reveals interesting aspects of these representations. The second example is based on a pseudo-bosonic generalization of operator-valued functions of a complex variable which resolves the identity. We show that such a generalization allows one to obtain a quantum pseudo-bosonic version of the complex plane viewed as the canonical phase space and to understand functions of the pseudo-bosonic operators as the quantized versions of functions of a complex variable.

math-ph↗

arXiv2015

Smooth Quantum Dynamics of Mixmaster Universe

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.

gr-qc↗

arXiv2014

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.

quant-ph↗

arXiv2014

Smooth Big Bounce from Affine Quantization

We examine the possibility of dealing with gravitational singularities on a quantum level through the use of coherent state or wavelet quantization instead of canonical quantization. We consider the Robertson-Walker metric coupled to a perfect fluid. It is the simplest model of a gravitational collapse and the results obtained here may serve as a useful starting point for more complex investigations in future. We follow a quantization procedure based on affine coherent states or wavelets built from the unitary irreducible representation of the affine group of the real line with positive dilation. The main issue of our approach is the appearance of a quantum centrifugal potential allowing for regularization of the singularity, essential self-adjointness of the Hamiltonian, and unambiguous quantum dynamical evolution.

gr-qc↗

arXiv2013

Integral quantizations with two basic examples

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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