Quantum harmonic analysis. An introduction
type: Item
instance of: scholarly article
title: Quantum harmonic analysis. An introduction
zbMATH Open document ID: 1476.81001
DOI: 10.1515/9783110722772
published in: Advances in Analysis and Geometry
publication date: 2021-06-01T00:00:00Z
review text: The book provides a compact and interesting presentation of both standard and advanced mathematical tools in harmonic analysis relevant for quantum mechanical systems described in \(L^2(\mathbb{R}^n)\) spaces. The book is structured in 17 relatively short chapters. Chapter 1 fixes notation and relevant background, while chapter 2 introduces the Heisenberg-Weyl or displacements operators, together with the parity or Grossman-Royer operator and the symplectic Fourier transform, to be used in the following chapter to introduce the Wigner and socalled cross-Wigner transform. Properties and physical relevance of the latter are briefly discussed, putting into evidence the Moyal identity, suggested as a bridge between orthogonality in \(L^2(\mathbb{R}^n)\) and \(L^2(\mathbb{R}^{2n})\). After considering the Wigner transform of functions of special interest, the connection between phase-space functions and operators, the socalled quantization, is introduced by means of the Weyl transform in chapter 5, further considering alternative quantization schemes in the following chapters. Chapter 8 and 9 are devoted to introduce the metaplectic group and its use in studying covariance properties of the Weyl quantization. Chapter 10 introduces the Feichtinger algebra, providing an alternative to the Schwartz space in considering a Gelfand triplet and characterized by invariance properties with respect to the action of the symplectic group. The remaining chapters are devoted to the definition and study of relevant classes of operators appearing in quantum mechanics treatment, such as trace-class operators, and therefore in particular density operators, and Hilbert-Schmidt operators. The formulation is in terms of Wigner transform and Weyl symbols, but connection is made to the standard Hilbert space treatment. Crucial properties for the quantum description, such as compliance with the uncertainty principle, positivity and separability of states are considered in detail. These properti…
reviewed by: Bassano Vacchini
Mathematics Subject Classification ID: 81-01
Mathematics Subject Classification ID: 42-01
Mathematics Subject Classification ID: 81S30
Mathematics Subject Classification ID: 81P16
Mathematics Subject Classification ID: 81P40
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 47B90
zbMATH DE Number: 7353463
zbMATH Keywords: Wigner function
zbMATH Keywords: Weyl transform
zbMATH Keywords: symplectic group
zbMATH Keywords: metaplectic group
zbMATH Keywords: Heisenberg group
zbMATH Keywords: quantization
zbMATH Keywords: positivity
zbMATH Keywords: Feichtinger algebra
zbMATH Keywords: Gromov's theorem
author: Maurice de Gosson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1515/9783110722772
OpenAlex ID: W3175115915
description: scientific article; zbMATH DE number 7353463