Jean-Pierre Gazeau

Université Paris Cité · AstroParticle and Cosmology Laboratory AstroParticle and Cosmology Laboratory

Covariant integral quantization with rotational SO(3) symmetry is established for quantum motion on this group manifold. It can also be applied to Gabor signal analysis on this group. The corresponding phase space takes the form of a discrete-continuous hypercylinder. The central tool for implementing this procedure is the Weyl–Gabor operator, a no...

We have discovered a class of dynamically stable coherent states for motion on the half-line. The regularization of the half-line boundary and the consequent quantum motion are expounded within the framework of covariant affine quantization, although alternative approaches are also feasible. The former approach is rooted in affine coherent states a...

We present a covariant quantization scheme for the so-called “partially massless” graviton field in de Sitter spacetime. Our approach is founded on the principles of the de Sitter group representation theory (in the sense given by Wigner), the Wightman-Gärding axioms for gauge-invariant fields (Gupta-Bleuler scheme), and the essential analyticity p...

This letter introduces a unitarity-preserving holographic correspondence within a $d$-dimensional de Sitter (dS$_d$) spacetime, distinctly challenging the prevailing notion that the holographic framework of dS$_d$ falls short in maintaining unitarity. The proposed approach is rooted in the geometry of the complex dS$_d$ spacetime and leverages the...

We study a semiclassical model of the mixmaster universe. We first derive the quantum model and then introduce its semiclassical approximation. We employ a general integral quantization method that respects the symmetries of the model given by the affine and the Weyl-Heisenberg groups, and can produce a wide class of quantum models. The semiclassic...

Within the de Sitter ambient space framework, the two different bases of the one-particle Hilbert space of the de Sitter group algebra are presented for the scalar case. Using field operator algebra and its Fock space construction in this formalism, we discuss the existence of asymptotic states in de Sitter QFT under an extension of the adiabatic h...

Covariant integral quantization with rotational symmetry SO(3) is established for the quantum motion on this group manifold, or, alternatively, for Gabor signal analysis on this group. We revisit the action of the related (non-unitary) Weyl-Gabor operator on the Hilbert space of square integrable functions on SO(3) and disclose a set of various pro...

We elaborate the definition and properties of ''massive" elementary systems in the (1+3)-dimensional Anti-de Sitter (AdS$_4$) spacetime, on both classical and quantum levels. We fully exploit the symmetry group Sp$(4,\mathbb R)$, that is, the two-fold covering of SO$_0(2,3)$ (Sp$(4,\mathbb R) \sim$ SO$_0(2,3)\times \mathbb Z_2$), recognized as the...

Majorana stellar representation, which visualizes a quantum spin as points on the Bloch sphere, allows quantum mechanics to accommodate the concept of trajectory, the hallmark of classical physics. We extend this notion to the discrete cylinder, which is the phase space of the canonical pair angle and orbital angular momentum. We demonstrate that t...

We present a covariant quantization scheme for the so-called ``partially massless" graviton field in de Sitter spacetime. Our approach is founded on the principles of the de Sitter group representation theory (in the sense given by Wigner), the Wightman-G\"{a}rding axioms for gauge-invariant fields (Gupta-Bleuler scheme), and the essential analytic...

We revisit the Perelomov SU(1,1) displaced coherent states states as possible quantum states of light. We disclose interesting statistical aspects of these states in relation with photon counting and squeezing. In the non-displaced case we discuss the efficiency of the photodetector as inversely proportional to the parameter k of the discrete serie...

Within the de Sitter ambient space framework, the two different bases of the one-particle Hilbert space of the de Sitter group algebra are presented for the scalar case. Using field operator algebra and its Fock space construction in this formalism, we discuss the existence of asymptotic states in de Sitter QFT under an extension of the adiabatic h...

We study a semi-classical model of the mixmaster universe. We first derive the quantum model and then introduce its semi-classical approximation. We employ a general integral quantization method that respects the symmetries of the model given by the affine and the Weyl-Heisenberg groups, and can produce a wide class of quantum models. The semi-clas...

We revisit the Perelomov SU(1,1)-displaced coherent states as possible quantum states of light. We disclose interesting statistical aspects of these states in relation to photon counting and squeezing. In the non-displaced case, we discuss the efficiency of the photodetector as inversely proportional to the parameter ϰ of the discrete series of uni...

Course given at the Krakow Interdisciplinary Doctoral School under the PROM Project PPI/PRO/2019/1/00016 of the Polish National Agency for Academic Exchange, 9-20 January 2023

In this chapter, we proceed our attempts to present a consistent formulation for elementary systems living in \(1+3\)-dimensional de Sitter (dS\(_4\)) on the quantum field theory (QFT) level.

In this chapter, we review a consistent and univocal definition of mass in \(1+3\)-dimensional de Sitter (dS\(_4\)) relativity.

In this chapter, we study the construction of (projective) unitary irreducible representations (UIR’s) of the \(1+1\)-dimensional de Sitter (dS\(_2\)) relativity group. We point out that (projective) Hilbert spaces carrying the UIR’s (in some restricted sense) identify quantum (“one-particle”) states spaces of dS\(_2\) elementary systems.

This chapter is devoted to a comprehensive review of various mathematical aspects of the \(1+1\)-dimensional de Sitter (dS\(_2\)) relativity group (SO\(_0(1,2)\) or its double-covering group SU(1, 1)), its Lie manifold, its Lie algebra, its (co-)adjoint orbits (as possible classical elementary systems living in dS\(_2\) spacetime).

This chapter is devoted to the construction of (projective) unitary irreducible representations (UIR’s) of the \(1+3\)-dimensional de Sitter (dS\(_4\)) relativity group. [(Projective) Hilbert spaces carrying the UIR’s (in some restricted sense) identify quantum (“one-particle”) states spaces of dS\(_4\) elementary systems.] We also study the physic...

In this chapter, we present a comprehensive review of various mathematical aspects of the 1+3-dimensional de Sitter (dS4) relativity group (SO0(1,4) or its universal covering group Sp(2, 2)), its Lie manifold, its Lie algebra, its (co-)adjoint orbitsOrbitCo-adjoint (as possible classical elementary systems living in dS4 spacetime). Involving the un...

We investigate two aspects of the elementary example of POVMs on the Euclidean plane, namely their status as quantum observables and their role as quantizers in the integral quantization procedure. The compatibility of POVMs in the ensuing quantum formalism is discussed, and a Naimark dilation is found for the quantum operators. The relation with T...

This Special Issue presents a collection of review and original papers on various aspects and applications of quantum cosmological models [...]

Covariant integral quantizations are based on the resolution of the identity by continuous or discrete families of normalized positive operator valued measures (POVM), which have appealing probabilistic content and which transform in a covariant way. One of their advantages is their ability to circumvent problems due to the presence of singularitie...

Covariant integral quantization is implemented 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 thi...

As an extension of Gabor signal processing, the covariant Weyl-Heisenberg integral quantization is implemented to transform functions on the eight-dimensional phase space x,k into Hilbertian operators. The x=xμ values are space-time variables, and the k=kμ values are their conjugate frequency-wave vector variables. The procedure is first applied to...

Quantisation with Gaussian type states offers certain advantages over other quantisation schemes, in particular, they can serve to regularise formally discontinuous classical functions leading to well defined quantum operators. In this work we define a squeezed state quantisation in two dimensions using several families of squeezed states for one-...

As an extension of Gabor signal processing, the covariant Weyl-Heisenberg integral quantization is implemented to transform functions on the eight-dimensional phase space $\left(x,k\right)$ into Hilbertian operators. The $x=\left(x^{\mu}\right)$'s are space-time variables and the $k=\left(k^{\mu}\right)$'s are their conjugate wave vector-frequency...

TITLE: Dark matter as a QCD effect in an Anti de Sitter background (Cosmogonic implications of de Sitter, Anti de Sitter and Poincaré symmetries) AUTHORS: Jean-Pierre Gazeau (U. Paris Cité) (in collaboration with Gilles Cohen-Tannoudji, CEA Saclay) ABSTRACT: The ΛCDM standard model of cosmology involves two dark components of the universe, dark...

A few errors have been identified in the article [Adv. Oper. Theory (2020) 5:901-935] and they are corrected in this Addendum. Furthermore, some notations have been modified in order to avoid any confusion.

We present the quantum Yang-Mills theory in the four-dimensional de Sitter ambient space formalism. In accordance with the SU(3) gauge symmetry the interaction Lagrangian is formulated in terms of interacting color charged fields in curved space-time. The gauge-invariant field equations are obtained in an independent coordinate description, and the...

The real plane with its set of orientations or angles in [0, π) is the simplest non trivial example of a (projective) Hilbert space and provides nice illustrations of quantum formalism. We present some of them, namely covariant integral quantization, linear polarisation of light as a quantum measurement, interpretation of entanglement leading to th...

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 u...

We review the construction of ("free") elementary systems in de Sitter (dS) spacetime, in the Wigner sense, as associated with unitary irreducible representations (UIR's) of the dS (relativity) group. This study emphasizes the conceptual issues arising in the formulation of such systems and discusses known results in a mathematically rigorous way....

Quantisation with Gaussian type states offers certain advantages over other quantisation schemes, in particular, they can serve to regularise formally discontinuous classical functions leading to well defined quantum operators. In this work we define a squeezed state quantisation in two dimensions using several families of squeezed states for one-...

The most recent progresses of observational astrophysics lead to a new cosmological standard model, the so-called CDM model involving two dark components of the universe, dark energy, and dark matter. In a recent publication [Universe 2021, 7(11), 402], we have interpreted dark matter as a gluonic Bose-Einstein condensate in anti-de Sitter spaceti...

The quantum Yang-Mills theory is studied in the four-dimensional de Sitter ambient space formalism. The interaction Lagrangian has reformulated in terms of the SU$(3)$ local gauge symmetry as interacting color charged fields in curved space-time. The invariant field equations are obtained in a coordinate independent description and their correspond...

The real plane with its set of orientations or angles in $[0,\pi)$ is the simplest non trivial example of a (projective) Hilbert space and provides nice illustrations of quantum formalism. We present some of them, namely covariant integral quantization, linear polarisation of light as a quantum measurement, interpretation of entanglement leading to...

In the same way as the realization of some of the famous gedanken experiments imagined by the founding fathers of quantum mechanics has recently led to the current renewal of the interpretation of quantum physics, it seems that the most recent progress of observational astrophysics can be interpreted as the realization of some cosmological gedanken...

Lectures given at "Groups and Lie Algebras, Representation Theory, and their Applications" CIMPA-Math, Bujumbura, Burundi 19-30 July 2021

In quantum information processing, using a receiver device to differentiate between two nonorthogonal states leads to a quantum error probability. The minimum possible error is known as the Helstrom bound. In this work we study statistical aspects and quantum limits for states which generalize the Glauber-Sudarshan coherent states, like non-linear,...

In the same way as the realization of some of the famous gedanken experiments imagined by the founding fathers of quantum mechanics has recently led to the current renewal of the interpretation of quantum physics, it seems that the most recent progresses of observational astrophysics can be interpreted as the realization of some cosmological gedank...

We investigate two aspects of the elementary example of real POVMs on the Euclidean plane, namely their status as quantum observables and their role as quantizers in the integral quantization procedure. The compatibility of POVMs in the ensuing quantum formalism is discussed, and a Naimark dilation is found for the quantum operators. A physical sit...

In the same way as the realization of some of the famous gedanken experiments imagined by the founding fathers of quantum mechanics has recently led to the current renewal of the interpretation of quantum physics, it seems that the most recent progresses of observational astrophysics can be interpreted as the realization of some cosmological gedank...

Susskind–Glogower coherent states, whose Fock expansion coefficients include Bessel functions, have recently attracted considerable attention for their optical properties. Nevertheless, identity resolution is still an open question, which is an essential mathematical property that defines an overcomplete basis in the Fock space and allows a coheren...

In the same way as the realization of some of the famous gedanken experiments imagined by the founding fathers of quantum mechanics has recently led to the current renewal of the interpretation of quantum physics, it seems that the most recent progresses of observational astrophysics can be interpreted as the realization of some cosmological gedank...

In the same way as the realization of some of the famous gedanken experiments imagined by the founding fathers of quantum mechanics has recently led to the current renewal of the interpretation of quantum physics, it seems that the most recent progresses of observational astrophysics can be interpreted as the realization of some cosmological gedank...

Signal analysis is built upon various resolutions of the identity in signal vector spaces, e.g. Fourier, Gabor, wavelets. 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 electromagnet...

Susskind-Glogower coherent states, whose Fock expansion coefficients include Bessel functions, have recently attracted considerable attention for their optical properties. Nevertheless, identity resolution is still an open question, which is an essential mathematical property that defines an overcomplete basis in the Fock space and allows a coheren...

In quantum information processing, {using a receiver device to differentiate between two nonorthogonal states leads to a quantum error probability. The minimum possible error is} known as the Helstrom bound. In this work we study and compare quantum limits for states which generalize the Glauber-Sudarshan coherent states, like non-linear, Perelomov...

In this essay, we present an overview of some important mathematical works of Professor Franciszek Hugon Szafraniec and a survey of his achievements and influence.

We explore the quantization of classical models with position-dependent mass (PDM) terms constrained to a bounded interval in the canonical position. This is achieved through the Weyl-Heisenberg covariant integral quantization by properly choosing a regularizing function $\Pi(q,p)$ on the phase space that smooths the discontinuities present in the...

An explanation of the origin of dark matter is suggested in this work. The argument is based on symmetry considerations about the concept of mass. In Wigner’s view, the rest mass and the spin of a free elementary particle in flat space-time are the two invariants that characterize the associated unitary irreducible representation of the Poincaré gr...

An explanation of the origin of dark matter is suggested in this work. The argument is based on symmetry considerations about the concept of mass. In the Wigner's view, the rest mass and the spin of a free elementary particle in flat space-time are the two invariants that characterize the associated unitary irreducible representation of the Poincar...

Seminar talk at the University of Bialystok, June 2019

We revisit the problem of the uncertainty relation for angle by using quantum hydrodynamics formulated in terms of the stochastic variational method (SVM), where we need not define the angle operator. We derive both the Kennard and Robertson-Schrödinger inequalities for canonical variables in polar coordinates. The inequalities have state-dependent...

We implement a SU(1, 1) covariant integral quantization of functions on the unit disk. The latter can be viewed as the phase space for the motion of a “massive” test 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 disc...

Covariant affine integral quantization is studied and applied to the motion of a particle in a punctured plane \({\mathbb {R}}_{*}^2{:}{=}{\mathbb {R}}^2{\setminus }\{0\}\), for which the phase space is \({\mathbb {R}}_{*}^2 \times {\mathbb {R}}^2\). We examine the consequences of different quantizer operators built from weight functions on \({\mat...

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 electrom...

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 timescale quantum formalism and revealing interesting or unexpected features. Extensions to classical electroma...

The Mixmaster solution to Einstein field equations was examined by C. Misner in an effort to better understand the dynamics of the early universe. We highlight the importance of the quantum version of this model for the early universe. This quantum version and its semi-classical portraits are yielded through affine and standard coherent state quant...

We present a covariant quantization of the “massive” spin-32 Rarita-Schwinger field in de Sitter (dS) spacetime. The dS group representation theory and its Wigner interpretation combined with the Wightman-Gärding axiomatic and analyticity requirements in the complexified pseudo-Riemanian manifold constitute the basis of the quantization scheme, whi...

We revisit the problem of the uncertainty relation for angle by using quantum hydrodynamics formulated in the stochastic variational method (SVM), where we need not define the angle operator. We derive both the Kennard and Robertson-Schroedinger inequalities for canonical variables in polar coordinates. The inequalities have state-dependent minimum...

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 -...

The Mixmaster solution to Einstein field equations was examined by C. Misner in an effort to better understand the dynamics of the early universe. We highlight the importance of the quantum version of this model for early universe. This quantum version and its semi-classical portraits are yielded through affine and standard coherent state quantizat...

Covariant affine integral quantization is studied and applied to the motion of a particle in a punctured plane Pp, for which the phase space is Pp X plane. 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 corre...

In the realm of Boltzmann-Gibbs (BG) statistical mechanics and its q-generalisation for complex systems, we analyse observed sequences of q-triplets, or q-doublets if one of them is the unity, in terms of cycles of successive M\"obius transforms of the line preserving unity ( q=1 corresponds to the BG theory). Such transforms have the form q --> (a...

We present a covariant quantization of the "massive" spin-${\frac{3}{2}}$ Rarita-Schwinger field in de Sitter (dS) spacetime. The dS group representation theory and its Wigner interpretation combined with the Wightman-G$\mbox{\"{a}}$rding axiomatic and analyticity requirements in the complexified pseudo-Riemanian manifold constitute the basis of th...

We implement the so-called Weyl–Heisenberg covariant integral quantization in the case of a classical system constrained by a bounded or semi-bounded geometry. The procedure, which is free of the ordering problem of operators, is illustrated with the basic example of the one-dimensional motion of a free particle in an interval, and yields a fuzzy b...

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

In a recent Letter, we have pointed out that the linearized Einstein gravity in de Sitter (dS) spacetime besides the spacetime symmetries generated by the Killing vectors and the evident gauge symmetry also possesses a hitherto `hidden' local (gauge-like) symmetry which becomes anomalous on the quantum level. This gauge-like anomaly makes the theor...

In a recent Letter, we have pointed out that the linearized Einstein gravity in de Sitter (dS) spacetime besides the spacetime symmetries generated by the Killing vectors and the evident gauge symmetry also possesses a hitherto 'hidden' local (gauge-like) symmetry which becomes anomalous on the quantum level. This gauge-like anomaly makes the theor...

An explanation of the origin of Dark Matter is suggested in this (speculative) work. The argument is based on symmetry considerations about the concept of mass, in Minkowski, in de Sitter and in Anti de Sitter spacetimes

Talk in Rende (Univ Calabria) 19 October 2017 Signal analysis is built on various resolutions of the identity in signal vector spaces, e.g. Fourier, Gabor, wavelets … The same 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...

The three approaches to relativistic generalization of coherent states are discussed in the simplest case of a spinless particle: the standard, canonical coherent states, the Lorentzian states and the coherent states introduced by Kaiser and independently by Twareque Ali, Antoine and Gazeau. All treatments utilize the Newton-Wigner localization and...

The aim of the present article is to introduce and to discuss some of the most basic fundamental concepts of quantum physics by using orientations or angles in the plane, illustrated through linear polarizations. We start with the Euclidean plane, which is certainly a paradigmatic example of a Hilbert space. The orientations in the plane are identi...

For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern development of Fourier analysis during XXth century has e...

We implement the so-called Weyl-Heisenberg covariant integral quantization in the case of a classical system constrained by a bounded or semi-bounded geometry. The procedure, which is free of the ordering problem of operators, is illustrated with the basic example of the one-dimensional motion of a free particle in an interval and yields a fuzzy bo...

We implement the so-called Weyl-Heisenberg covariant integral quantization in the case of a classical system constrained by a bounded or semi-bounded geometry. The procedure, which is free of the ordering problem of operators, is illustrated with the basic example of the one-dimensional motion of a free particle in an interval and yields a fuzzy bo...

In physics, one is often misled in thinking that the mathematical model of a system is part of or is that system itself. Think of expressions commonly used in physics like "point" particle, motion "on the line", "mooth" observables, wave function, and even "going to infinity", without forgetting perplexing phrases like "classical world" versus "qua...

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 foun...

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.

We present a new approach to the study of the dynamics of the Mixmaster universe through the approximation to the anisotropy potential based on the explicitly integrable periodic 3-particle Toda system. We show that the covariant Weyl-Heisenberg integral quantization naturally amplifies the dynamical role of the underlying Toda system. Since the re...

Any quantization maps linearly function on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with the symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integ...

Based on the submitted contribution http ://arxiv.org/abs/1810.06473 to Integrability, Supersymmetry and Coherent States. A volume in honor of V. Hussin CRM series in Mathematical Physics-Springer

The three approaches to relativistic generalization of coherent states are discussed in the simplest case of a spinless particle: the standard, canonical coherent states, the Lorentzian states and the coherent states introduced by Kaiser and independently by Twareque Ali, Antoine and Gazeau. All treatments utilize the Newton–Wigner localization and...

Any quantization maps linearly functions on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral...

A baby Majorana quantum formalism ; arXiv :1701.04026 [quant-ph] Orientations in the plane as quantum states, submitted

Seminar: Mathematical models for the "World" Integral quantization of the motion on the line: Weyl Heisenberg covariant integral quantization

An example of E(2) covariant integral quantization for quantum localisation on the circle

Talk at SEMINARIUM ZAK\L ADU FIZYKI TEORETYCZNEJ 11.04.18

We present a new approach to the study of the dynamics of the Mixmaster universe through the approximation to the anisotropy potential based on the explicitly integrable periodic 3-particle Toda system. We show that the covariant Weyl-Heisenberg integral quantization naturally amplifies the dynamical role of the underlying Toda system. Since the re...

We propose a deformed version of the generalized Heisenberg algebra by using techniques borrowed from the theory of pseudo-bosons. In particular, this analysis is relevant when non self-adjoint Hamiltonians are needed to describe a given physical system. We also discuss relations with nonlinear pseudo-bosons. Several examples are discussed.