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31 pages, 667 KiB

Open AccessArticle

Covariant Integral Quantization of the Semi-Discrete SO(3)-Hypercylinder

by Jean-Pierre Gazeau and Romain Murenzi

Symmetry 2023, 15(11), 2044; https://doi.org/10.3390/sym15112044 - 10 Nov 2023

Cited by 1 | Viewed by 1819

Covariant integral quantization with rotational

SO (3)

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 non-unitary operator that operates on the Hilbert space of square-integrable functions on

SO (3)

. This operator serves as the counterpart to the unitary Weyl or displacement operator used in constructing standard Schrödinger–Glauber–Sudarshan coherent states. We unveil a diverse range of properties associated with the quantizations and their corresponding semi-classical phase-space portraits, which are derived from different weight functions on the considered discrete-continuous hypercylinder. Certain classes of these weight functions lead to families of coherent states. Moreover, our approach allows us to define a Wigner distribution, satisfying the standard marginality conditions, along with its related Wigner transform. Full article

(This article belongs to the Special Issue Symmetry: Feature Papers 2023)

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20 pages, 382 KiB

Open AccessFeature PaperArticle

Quantum Models à la Gabor for the Space-Time Metric

by Gilles Cohen-Tannoudji, Jean-Pierre Gazeau, Célestin Habonimana and Juma Shabani

Entropy 2022, 24(6), 835; https://doi.org/10.3390/e24060835 - 16 Jun 2022

Cited by 3 | Viewed by 2091

Abstract

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 = (x^{μ})

values are space-time variables, and the

k = (k^{μ})

[...] Read more.

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 = (x^{μ})

values are space-time variables, and the

k = (k^{μ})

values are their conjugate frequency-wave vector variables. The procedure is first applied to the variables

(x, k)

and produces essentially canonically conjugate self-adjoint operators. It is next applied to the metric field

g_{μ ν} (x)

of general relativity and yields regularized semi-classical phase space portraits

{\overset{ˇ}{g}}_{μ ν} (x)

. The latter give rise to modified tensor energy density. Examples are given with the uniformly accelerated reference system and the Schwarzschild metric. Interesting probabilistic aspects are discussed. Full article

(This article belongs to the Special Issue Quantum Structures and Logics)

26 pages, 681 KiB

Open AccessArticle

Quantum Mixmaster as a Model of the Primordial Universe

by Hervé Bergeron, Ewa Czuchry, Jean Pierre Gazeau and Przemysław Małkiewicz

Universe 2020, 6(1), 7; https://doi.org/10.3390/universe6010007 - 31 Dec 2019

Cited by 18 | Viewed by 3107

Abstract

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 quantizations and more generally affine and Weyl–Heisenberg covariant integral quantizations. The adiabatic and vibronic approximations widely used in molecular physics can be employed to qualitatively study the dynamics of the model on both quantum and semi-classical levels. Moreover, the semi-classical approach with the exact anisotropy potential can be effective in the numerical integration of some solutions. Some promising physical features such as the singularity resolution, smooth bouncing, the excitation of anisotropic oscillations and a substantial amount of post-bounce inflation as the backreaction to the latter are pointed out. Finally, a realistic cosmological scenario based on the quantum mixmaster model, which includes the formation and evolution of local structures is outlined. Full article

(This article belongs to the Special Issue Quantum Models for Cosmology)

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22 pages, 535 KiB

Open AccessArticle

Geometric Model of Black Hole Quantum N-portrait, Extradimensions and Thermodynamics

by Antonia M. Frassino, Sven Köppel and Piero Nicolini

Entropy 2016, 18(5), 181; https://doi.org/10.3390/e18050181 - 14 May 2016

Cited by 25 | Viewed by 6915

Abstract

Recently a short scale modified black hole metric, known as holographic metric, has been proposed in order to capture the self-complete character of gravity. In this paper we show that such a metric can reproduce some geometric features expected from the quantum N-portrait beyond the semi-classical limit. We show that for a generic N this corresponds to having an effective energy momentum tensor in Einstein equations or, equivalently, non-local terms in the gravity action. We also consider the higher dimensional extension of the metric and the case of an AdS cosmological term. We provide a detailed thermodynamic analysis of both cases, with particular reference to the repercussions on the Hawking-Page phase transition. Full article

(This article belongs to the Special Issue Entropy in Quantum Gravity and Quantum Cosmology)

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