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
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
This special issue of Journal of Physics A: Mathematical and Theoretical appears on the occasion of the 5th International Symposium on Quantum Theory and Symmetries (QTS5), held in Valladolid, Spain, from 22–28 July 2007. This is the fith in a series of conferences previously held in Goslar (Germany) 1999, QTS1; Cracow (Poland) 2001, QTS2; Cincinnati (USA) 2003, QTS3; and Varna (Bulgaria) 2005, QTS4. The QTS5 symposium gathered 181 participants from 39 countries working in different fields of theoretical physics. The spirit of the QTS conference series is to join researchers in a wide variety of topics in theoretical physics, as a way of making accessible recent results and the new lines of different fields. This is based on the feeling that it is good for a physicist to have a general overview as well as expertise in his/her own field. There are many other conferences devoted to specific topics, which are of interest to gain deeper insight in many technical aspects and that are quite suitable for discussions due to their small size. However, we believe that general conferences like this are interesting and worth keeping. We like the talks, in both plenary and parallel sessions, which are devoted to specific topics, to be prepared so as to be accessible to any researcher in any branch of theoretical physics. We think that this objective is compatible with rigour and high standards. As is well known, similar methods and techniques can be useful for many problems in different fields. We hope that this has been appreciated during the sessions of the QTS5 conference. The QTS5 conference offered the following list of topics: 1. Symmetries in string theory, quantum gravity and related topics 2. Symmetries in quantum field theories, conformal and related field theories, lattice and noncommutative theories, gauge theories 3.Quantum computing, information and control 4. Foundations of quantum theory 5. Quantum optics, coherent states, Wigner functions 6. Dynamical and integrable systems 7. Symmetries in condensed matter and statistical physics 8. Symmetries in particle physics, nuclear, atomic and molecular physics 9. Nonlinear quantum mechanics 10. Time asymmetric quantum mechanics 11. SUSY quantum mechanics, PT symmetries and pseudo-Hamiltonians 12. Mathematical methods for symmetries and quantum theories 13. Symmetries in chemistry, biology and other sciences Papers accepted for publication in this issue aim to provide a survey of the state of the art in different fields and contain contributions from plenary speakers. In addition, the issue contains contributions from other participants and it has also been open to other authors whose work fits into the topics of the conference. In any case, all the contributions have been refereed according to the high standards of Journal of Physics A: Mathematical and Theoretical . We are much indebted to several institutions; without their support the organization of the QTS5 symposium would not have been possible. In this respect we acknowledge the Ministerio de Educación of Spain and Junta de Castilla y León for general financial support; to Fundación Universidades de Castilla y León for a number of grants to young researchers who otherwise would not have attended the conference; also to the European Physical Society that provided a number of grants for eastern countries, and to the University of Valladolid where the event took place. We thank IOP Publishing and the staff of Journal of Physics A: Mathematical and Theoretical for the publication of this special issue. In addition, we want to express our gratitude to other members of the Local Organizing Committe of QTS5, who are not Editors of this special issue: Oscar Arratia, Juan A Calzada and Fernando Gómez. Finally, we would like to thank all the participants in the QTS5 conference for their presence, contributions, and for the good atmosphere achieved during their stay. We hope that the experience of spending these days in Valladolid has been most fruitful for all of them.
In recent decades, substantial theoretical and experimental progress was achieved in understanding the quantum nature of physical phenomena that serves as the foundation of present and future quantum technologies. Quantum correlations like the entanglement of the states of composite systems, the phenomenon of quantum discord, which captures other aspects of quantum correlations, quantum contextuality and, connected with these phenomena, uncertainty relations for conjugate variables and entropies, like Shannon and Rényi entropies, and the inequalities for spin states, like Bell inequalities, reflect the recently understood quantum properties of micro and macro systems. The mathematical methods needed to describe all quantum phenomena mentioned above were also the subject of intense studies in the end of the last, and beginning of the new, century. In this section of CAMOP 'Mathematical Methods of Studying Physical Phenomena' new results and new trends in the rapidly developing domain of quantum (and classical) physics are presented. Among the particular topics under discussion there are some reviews on the problems of dynamical invariants and their relations with symmetries of the physical systems. In fact, this is a very old problem of both classical and quantum systems, e.g. the systems of parametric oscillators with time-dependent parameters, like Ermakov systems, which have specific constants of motion depending linearly or quadratically on the oscillator positions and momenta. Such dynamical invariants play an important role in studying the dynamical Casimir effect, the essence of the effect being the creation of photons from the vacuum in a cavity with moving boundaries due to the presence of purely quantum fluctuations of the electromagnetic field in the vacuum. It is remarkable that this effect was recently observed experimentally. The other new direction in developing the mathematical approach in physics is quantum tomography that provides a new vision of quantum states. In the tomographic picture of quantum mechanics, the states are identified with fair conditional probability distributions, which contain the same information on the states as the wave function or the density matrix. The mathematical methods of the tomographic approach are based on studying the star-product (associative product) quantization scheme. The tomographic star-product technique provides an additional understanding of the associative product, which is connected with the existence of specific pairs of operators called quantizers and dequantizers. These operators code information on the kernels of all the star-product schemes, including the traditional phase-space Weyl–Wigner–Moyal picture describing the quantum-system evolution. The new equation to find quantizers, if the kernel of the star product of functions is given, is presented in this CAMOP section. For studying classical systems, the mathematical methods developed in quantum mechanics can also be used. The case of paraxial-radiation beams propagating in waveguides is a known example of describing a purely classical phenomenon by means of quantum-like equations. Thus, some quantum phenomenon like the entanglement can be mimicked by the properties of classical beams, for example, Gaussian modes. The mathematical structures and relations to the symplectic symmetry group are analogous for both classical and quantum phenomena. Such analogies of the mathematical classical and quantum methods used in research on quantum-like communication channels provide new tools for constructing a theoretical basis of the new information–transmission technologies. The conventional quantum mechanics and its relation to classical mechanics contain mathematical recipes of the correspondence principle and quantization rules. Attempts to find rules for deriving the quantum-mechanical formalism starting from the classical field theory, taking into account the influence of classical fluctuations of the field, is considered in these papers. The methods to solve quantum equations and formulate the boundary conditions in the problems with singular potentials are connected with the mathematical problems of self-adjointness of the Hamiltonians. The progress and some new results in this direction are reflected in this CAMOP section. The Gaussian states of the photons play an important role in quantum optics. The multimode electromagnetic field and quantum correlations in the Gaussian states are considered in this section. The new results in the statistical properties of the laser radiation discussed here are based on applications of mathematical methods in this traditional domain of physics. It is worth stressing that the universality of the mathematical procedures permitted to consider the physical phenomena in the ocean is on the same footing as the phenomena in the microworld. In this CAMOP section, there are also papers devoted to traditional problems of solving the Schrödinger equation for interesting quantum systems. Recently obtained results related to different domains of theoretical physics are united by applying mathematical methods and tools, that provide new possibilities to better understand the theoretical foundations needed to develop new quantum technologies like quantum computing and quantum communications. The papers are arranged alphabetically by the name of the first author. We are grateful to all authors who accepted our invitation to contribute to this CAMOP section.
The Wigner Symposia series emerged from an International Symposium On SpaceTime Symmetries, held in College Park, Maryland, 1988 organized by Y.S. Kim and W. Zachary and commemorating the 50'th anniversary of Wigner's fundamental paper On the Unitary Representation of the Inhomogenous Lorentz Group. That symposium covered various areas of physics, of theoretical and mathematical physics to which Eugene Paul Wigner has contributed. It has shown how new and fruitful ideas connected with symmetries for a mathematical modeling of complex physical systems developed under the influence of Wigners work. Following the idea of the symposia to discuss new developments in physics centered around fundamental questions topics with emphasis on theoretical and mathematical physics were chosen and new frontiers like neuronal networks and quantum holography, whose origins can be traced back to Wigner's work, where included. The aim was to show the unity of different parts in physics and to demonstrate the power of arguments which are directly or indirectly related with mathematical realizations of symmetries. (1) Foundations of Quantum Mechanics; (2) General group Theoretical and Quantization Methods; (3) Coherent States; (4) Berry Phases; (5) Phases Space and Wigner Distributions; (6) Applications of Quantum Mechanics From Channel Space to Wigner Crystal; (7) Quantum Fields and Particles; (8) C-Algebraic and Methods; (9) Differential Geometric Methods: and (10) Nonlinear Partial Differential Equations, Dynamical Systems and Neural Networks.
The main motivation behind the call for this special issue was to gather recent results, developments and open problems in quantum physics with non-Hermitian operators. There have been previous special issues in this journal [1, 2] and elsewhere on this subject. The intention of this issue is to reflect the current state of this rapidly-developing field. It has therefore been open to all contributions containing new results on non-Hermitian theories that are explicitly -symmetric and/or pseudo-Hermitian or quasi-Hermitian. In the last decade these types of systems have proved to be viable self-consistent physical theories with well defined unitary time-evolution and real spectra. As the large number of responses demonstrates, this is a rapidly evolving field of research. A consensus has been reached regarding most of the fundamental problems, and the general ideas and techniques are now readily being employed in many areas of physics. Nonetheless, this issue still contains some treatments of a more general nature regarding the spectral analysis of these models, in particular, the physics of the exceptional points, the breaking of the -symmetry, an interpretation of negative energies and the consistent implementation of the WKB analysis. This issue also contains a treatment of a scattering theory associated with these types of systems, weak measurements, coherent states, decoherence, unbounded metric operators and the inclusion of domain issues to obtain well defined self-adjoint theories. Contributions in the form of applications of the general ideas include: studies of classical shock-waves and tunnelling, supersymmetric models, spin chain models, models with ring structure, random matrix models, the Pauli equation, the nonlinear Schrödinger equation, quasi-exactly solvable models, integrable models such as the Calogero model, Bose–Einstein condensates, thermodynamics, nonlinear oligomers, quantum catastrophes, the Landau–Zener problem and pseudo-Fermions. Applications close to experimental realization are proposed in optics, including short light pulse models, waveguides and laser systems, and also in electronics. We hope that this issue will become a valuable reference and inspiration for the broader scientific community working in mathematical and theoretical physics. References [1] Fring A, Jones H F and Znojil M (ed) 2008 J. Phys. A: Math. Theor. 41 240301 [2] Geyer H, Heiss D and Znojil M (ed) 2006 J. Phys. A: Math. Gen. 39 9963
Quantum mechanics has brought a gigantic breakthrough to physics since its introduction more than a century ago. It has led to many discoveries at the microscopic level, initiated new pathways to unsuspected properties of matter, and produced new and amazing usable applications. This Special Issue of Symmetry is devoted to recent advances in theoretical developments in quantum mechanics. It is part of the global effort to provide a continuous supply of information to the research community. This volume gathers contributions in various directions and their contents are best described chronologically by their abstracts, which are hereafter listed for the convenience of the reader.
Quantum mechanics has brought a gigantic breakthrough to physics since its introduction more than a century ago. It has led to many discoveries at the microscopic level, initiated new pathways to unsuspected properties of matter, and produced new and amazing usable applications. This Special Issue of Symmetry is devoted to recent advances in theoretical developments in quantum mechanics. It is part of the global effort to provide a continuous supply of information to the research community. This volume gathers contributions in various directions and their contents are best described chronologically by their abstracts, which are hereafter listed for the convenience of the reader.