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arXiv:0902.4411

Introductive Backgrounds of Modern Quantum Mathematics with Application to Nonlinear Dynamical Systems

Anatoliy K. Prykarpatsky, Nikolai N. Bogolubov, Jolanta Golenia, Ufuk Taneri

Published Feb 25, 2009• 1 citation
arXivHTMLPDF

Abstract

Introductive backgrounds of a new mathematical physics discipline - Quantum Mathematics - are discussed and analyzed both from historical and analytical points of view. The magic properties of the second quantization method, invented by V. Fock in 1932, are demonstrated, and an impressive application to the nonlinear dynamical systems theory is considered.

DOI
10.48550/arxiv.0902.4411

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Preface

This Foundations of Quantum Theory special issue of Journal of Physics A: Mathematical and General contains contributions on basic aspects of modern quantum and mathematical physics. The contribution topics relate to two conferences, `Quantum Theory and Symmetries' held in Krakow (18-21 July 2001, organized by E Kapuscik and A Horzela) and `Symmetry in Sciences' held in Bregenz (23-27 July 2001, organized by B Gruber and M Ramek). Both conferences had special sessions dedicated to the memory of Professor L O'Raifeartaigh. In modern quantum physics there are several important aspects including Bell inequalities, quantum groups, nonlinear dynamics, dissipation and the time-irreversibility problem, alternative approaches and structures in quantum mechanics, quantum control, star-product quantization, etc, as well as different applications in nuclear and elementary particle physics and purely quantum effects analogous to the nonstationary Casimir effect, and different sorts of interference effects for matter waves. All of these problems were discussed at the conferences in Krakow and Bregenz and are presented in the contributions to the special issue. For example, quantum groups were found to be appropriate for the description of nonlinear dynamics and, in particular, nonlinear vibrations for which the phase of the oscillations depends on the energy. It was also found that biHamiltonian systems may correspond to different quantum pictures for the same classical equations of motion including nonlinear vibrations. Thus, it has transpired that there is an interesting link between several aspects of quantum and classical dynamics and the formalism of quantum groups. Another new quantum effect is the nonclassical behaviour of slow packets interacting with macroscopic surfaces. Quantum physics progresses rapidly and we hope that the contributions to this special issue will help readers learn more about the fundamental new aspects of quantum phenomena and the development of a mathematical formalism describing quantum and classical dynamics. Andrzej Horzela, Edward Kapuscik and Vladimir I Man'ko Guest Editors

Journal of Physics A Mathematical and General•Sep 24, 2002•0 citations
Introductive backgrounds of a new mathematical physics discipline - Quantum Mathematics - are discussed and analyzed both from historical and analytical points of view. The magic properties of the second quantization method, invented by V. Fock in 1932, are demonstrated, and an impressive application to the nonlinear dynamical systems theory is considered.
Expository notes which combine a historical survey of the development of quantum physics with a review of selected mathematical topics in quantization theory (addressed to students that are not complete novices in quantum mechanics). After recalling in the introduction the early stages of the quantum revolution, and recapitulating in Sect. 2.1 some basic notions of symplectic geometry, we survey in Sect. 2.2 the so called prequantization thus preparing the ground for an outline of geometric quantization (Sect. 2.3). In Sect. 3 we apply the general theory to the study of basic examples of quantization of Kaehler manifolds. In Sect. 4 we review the Weyl and Wigner maps and the work of Groenewold and Moyal that laid the foundations of quantum mechanics in phase space, ending with a brief survey of the modern development of deformation quantization. Sect. 5 provides a review of second quantization and its mathematical interpretation. We point out that the treatment of (nonrelativistic) bound states requires going beyond the neat mathematical formalization of the concept of second quantization. An appendix is devoted to Pascual Jordan, the least known among the creators of quantum mechanics and the chief architect of the "theory of quantized matter waves".
Nonlinear dynamics (``chaos theory'') and quantum mechanics are two of the scientific triumphs of the 20th century. The former lies at the heart of the modern interdisciplinary approach to science, whereas the latter has revolutionized physics. Both chaos theory and quantum mechanics have achieved a fairly large level of glamour in the eyes of the general public. The study of quantum chaos encompasses the application of dynamical systems theory in the quantum regime. In the present article, we give a brief review of the origin and fundamentals of both quantum mechanics and nonlinear dynamics. We recount the birth of dynamical systems theory and contrast chaotic motion with integrable motion. We similarly recall the transition from classical to quantum mechanics and discuss the origin of the latter. We then consider the interplay between nonlinear dynamics and quantum mechanics via a classification and explanation of the three types of quantum chaos. We include several recent results in this discussion.
Written for a two-semester graduate course in Quantum Mechanics, this comprehensive text helps develop the tools and formalism of Quantum Mechanics and its applications to physical systems. It suits students who have taken some introductory Quantum Mechanics and Modern Physics courses at undergraduate level, but it is self-contained and does not assume any specific background knowledge beyond appropriate fluency in mathematics. The text takes a modern logical approach rather than a historical one and it covers standard material, such as the hydrogen atom and the harmonic oscillator, the WKB approximations and Bohr-Sommerfeld quantization. Important modern topics and examples are also described, including Berry phase, quantum information, complexity and chaos, decoherence and thermalization, nonstandard statistics, as well as more advanced material such as path integrals, scattering theory, multiparticles and Fock space. Readers will gain a broad overview of Quantum Mechanics, as solid preparation for further study or research.
Written for a two-semester graduate course in Quantum Mechanics, this comprehensive text helps develop the tools and formalism of Quantum Mechanics and its applications to physical systems. It suits students who have taken some introductory Quantum Mechanics and Modern Physics courses at undergraduate level, but it is self-contained and does not assume any specific background knowledge beyond appropriate fluency in mathematics. The text takes a modern logical approach rather than a historical one and it covers standard material, such as the hydrogen atom and the harmonic oscillator, the WKB approximations and Bohr-Sommerfeld quantization. Important modern topics and examples are also described, including Berry phase, quantum information, complexity and chaos, decoherence and thermalization, nonstandard statistics, as well as more advanced material such as path integrals, scattering theory, multiparticles and Fock space. Readers will gain a broad overview of Quantum Mechanics, as solid preparation for further study or research.
This Foundations of Quantum Theory special issue of Journal of Physics A: Mathematical and General contains contributions on basic aspects of modern quantum and mathematical physics. The contribution topics relate to two conferences, `Quantum Theory and Symmetries' held in Krakow (18-21 July 2001, organized by E Kapuscik and A Horzela) and `Symmetry in Sciences' held in Bregenz (23-27 July 2001, organized by B Gruber and M Ramek). Both conferences had special sessions dedicated to the memory of Professor L O'Raifeartaigh. In modern quantum physics there are several important aspects including Bell inequalities, quantum groups, nonlinear dynamics, dissipation and the time-irreversibility problem, alternative approaches and structures in quantum mechanics, quantum control, star-product quantization, etc, as well as different applications in nuclear and elementary particle physics and purely quantum effects analogous to the nonstationary Casimir effect, and different sorts of interference effects for matter waves. All of these problems were discussed at the conferences in Krakow and Bregenz and are presented in the contributions to the special issue. For example, quantum groups were found to be appropriate for the description of nonlinear dynamics and, in particular, nonlinear vibrations for which the phase of the oscillations depends on the energy. It was also found that biHamiltonian systems may correspond to different quantum pictures for the same classical equations of motion including nonlinear vibrations. Thus, it has transpired that there is an interesting link between several aspects of quantum and classical dynamics and the formalism of quantum groups. Another new quantum effect is the nonclassical behaviour of slow packets interacting with macroscopic surfaces. Quantum physics progresses rapidly and we hope that the contributions to this special issue will help readers learn more about the fundamental new aspects of quantum phenomena and the development of a mathematical formalism describing quantum and classical dynamics. Andrzej Horzela, Edward Kapuscik and Vladimir I Man'ko Guest Editors