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Czechoslovak Journal of Physics

Proceedings of the 1st International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics, Prague, 2003

Published Jan 1, 2004• 13 citations
Publisher
DOI
10.1023/b:cjop.0000014619.62079.77

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Attempts to understand the quantum mechanics of non-Hermitian Hamiltonian systems can be traced back to the early days, one example being Heisenberg's endeavour to formulate a consistent model involving an indefinite metric. Over the years non-Hermitian Hamiltonians whose spectra were believed to be real have appeared from time to time in the literature, for instance in the study of strong interactions at high energies via Regge models, in condensed matter physics in the context of the XXZ-spin chain, in interacting boson models in nuclear physics, in integrable quantum field theories as Toda field theories with complex coupling constants, and also very recently in a field theoretical scenario in the quantization procedure of strings on an AdS 5 x S 5 background. Concrete experimental realizations of these types of systems in the form of optical lattices have been proposed in 2007. In the area of mathematical physics similar non-systematic results appeared sporadically over the years. However, intensive and more systematic investigation of these types of non- Hermitian Hamiltonians with real eigenvalue spectra only began about ten years ago, when the surprising discovery was made that a large class of one-particle systems perturbed by a simple non-Hermitian potential term possesses a real energy spectrum. Since then regular international workshops devoted to this theme have taken place. This special issue is centred around the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics held in July 2007 at City University London. All the contributions contain significant new results or alternatively provide a survey of the state of the art of the subject or a critical assessment of the present understanding of the topic and a discussion of open problems. Original contributions from non-participants were also invited. Meanwhile many interesting results have been obtained and consensus has been reached on various central conceptual issues in the growing community of this subject. It is, for instance, well understood that the reality of the spectrum can be attributed either to the unbroken PT-symmetry of the entire system, that is, invariance of the Hamiltonian and the corresponding wavefunctions under a simultaneous parity transformation and time reversal, or more generally to its pseudo-Hermiticity . When the spectrum is real and discrete the Hamiltonian is actually quasi-Hermitian, with a positive-definite metric operator, and can in principle be related by a similarity transformation to an isospectral Hermitian counterpart. For all approaches well-defined procedures have been developed, which allow one to construct metric operators and therefore a consistent description of the underlying quantum mechanical observables. Even though the general principles have been laid out, it remains a challenge in most concrete cases to implement the entire procedure. Solvable models in this sense, some of which may be found in this issue, remain a rare exception. Nonetheless, despite this progress some important questions are still unanswered. For instance, according to the current understanding the non-Hermitian Hamiltonian does not uniquely define the physics of the system since a meaningful metric can no longer be associated with the system in a non-trivial and unambiguous manner. A fully consistent scattering theory has also not yet been formulated. Other issues remain controversial, such as the quantum brachistochrone problem, the problem of forming a mixture between a Hermitian and non-Hermitian system, the new phenomenological possibilities of forming a kind of worm-hole effect, etc. We would like to acknowledge the financial support of the London Mathematical Society, the Institute of Physics, the Doppler Institute in Prague and the School of Engineering and Mathematical Science of City University London. We hope this special issue will be useful to the newcomer as well as to the expert in the subject. Participants of the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics.

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This special issue contains contributions presented at the workshop on The Physics of Non-Hermitian Operators held at Stellenbosch, South Africa in November 2005. A few additional articles are included from researchers who could not attend the meeting. The workshop was the fourth meeting in a series on Pseudo-Hermitian Hamiltonians in Quantum Physics which commenced in Prague in 2003 as a result of growing interest in non-Hermitian Hamiltonians in general, and PT-symmetric quantum mechanics and quantum field theory in particular. Naturally, Hamiltonians of open systems are an obvious part of this topic, but the focus here is rather on systems where unitary time evolution as such is not undermined. This rather new, and perhaps ambitious, field thus explores a possible generalization of quantum mechanics which incorporates non-Hermitian Hamiltonians into a consistent quantum mechanical framework. New results of a physical and mathematical nature have brought about appreciable progress; earlier work has been uncovered or even rediscovered and links between sub-fields have been established. This issue should provide a stimulus for initiating further advances in the field, re-focus on earlier perspectives and explore open and controversial questions. While non-Hermitian operators have been encountered in theoretical physics since the advent of quantum mechanics, their systematic mathematical investigation and their possible physical implications is a relatively new subject, further explored in this issue. We would like to thank the University of Stellenbosch and the South African National Research Foundation for their financial support, without which it would not have been possible to plan and organize the workshop. Organizational aspects and technical support were very ably taken care of by Christine Ruperti and Tinus Botha whom we would like to thank again. Finally, we also appreciate and acknowledge the invitation from the Editorial Board of Journal of Physics A: Mathematical and General for the opportunity to publish this special issue and trust that this will indeed widen the interest and stimulate progress in the field.

Journal of Physics A Mathematical and General•Jul 26, 2006•27 citations
This article contains a short summary of an oral presentation in the 2nd International Workshop on "Pseudo-Hermitian Hamiltonians in Quantum Physics" (14.-16.6.2004, Villa Lanna, Prague, Czech Republic). The purpose of the presentation has been to introduce a non-Hermitian generalization of pseudo-Hermitian Quantum Theory allowing to reconcile the orthogonal concepts of causality, Poincare invariance, analyticity, and locality. We conclude by considering interesting applications like non-Hermitian supersymmetry.
Attempts to understand the quantum mechanics of non-Hermitian Hamiltonian systems can be traced back to the early days, one example being Heisenberg's endeavour to formulate a consistent model involving an indefinite metric. Over the years non-Hermitian Hamiltonians whose spectra were believed to be real have appeared from time to time in the literature, for instance in the study of strong interactions at high energies via Regge models, in condensed matter physics in the context of the XXZ-spin chain, in interacting boson models in nuclear physics, in integrable quantum field theories as Toda field theories with complex coupling constants, and also very recently in a field theoretical scenario in the quantization procedure of strings on an AdS 5 x S 5 background. Concrete experimental realizations of these types of systems in the form of optical lattices have been proposed in 2007. In the area of mathematical physics similar non-systematic results appeared sporadically over the years. However, intensive and more systematic investigation of these types of non- Hermitian Hamiltonians with real eigenvalue spectra only began about ten years ago, when the surprising discovery was made that a large class of one-particle systems perturbed by a simple non-Hermitian potential term possesses a real energy spectrum. Since then regular international workshops devoted to this theme have taken place. This special issue is centred around the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics held in July 2007 at City University London. All the contributions contain significant new results or alternatively provide a survey of the state of the art of the subject or a critical assessment of the present understanding of the topic and a discussion of open problems. Original contributions from non-participants were also invited. Meanwhile many interesting results have been obtained and consensus has been reached on various central conceptual issues in the growing community of this subject. It is, for instance, well understood that the reality of the spectrum can be attributed either to the unbroken PT-symmetry of the entire system, that is, invariance of the Hamiltonian and the corresponding wavefunctions under a simultaneous parity transformation and time reversal, or more generally to its pseudo-Hermiticity . When the spectrum is real and discrete the Hamiltonian is actually quasi-Hermitian, with a positive-definite metric operator, and can in principle be related by a similarity transformation to an isospectral Hermitian counterpart. For all approaches well-defined procedures have been developed, which allow one to construct metric operators and therefore a consistent description of the underlying quantum mechanical observables. Even though the general principles have been laid out, it remains a challenge in most concrete cases to implement the entire procedure. Solvable models in this sense, some of which may be found in this issue, remain a rare exception. Nonetheless, despite this progress some important questions are still unanswered. For instance, according to the current understanding the non-Hermitian Hamiltonian does not uniquely define the physics of the system since a meaningful metric can no longer be associated with the system in a non-trivial and unambiguous manner. A fully consistent scattering theory has also not yet been formulated. Other issues remain controversial, such as the quantum brachistochrone problem, the problem of forming a mixture between a Hermitian and non-Hermitian system, the new phenomenological possibilities of forming a kind of worm-hole effect, etc. We would like to acknowledge the financial support of the London Mathematical Society, the Institute of Physics, the Doppler Institute in Prague and the School of Engineering and Mathematical Science of City University London. We hope this special issue will be useful to the newcomer as well as to the expert in the subject. Participants of the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics.
This special issue contains contributions presented at the workshop on The Physics of Non-Hermitian Operators held at Stellenbosch, South Africa in November 2005. A few additional articles are included from researchers who could not attend the meeting. The workshop was the fourth meeting in a series on Pseudo-Hermitian Hamiltonians in Quantum Physics which commenced in Prague in 2003 as a result of growing interest in non-Hermitian Hamiltonians in general, and PT-symmetric quantum mechanics and quantum field theory in particular. Naturally, Hamiltonians of open systems are an obvious part of this topic, but the focus here is rather on systems where unitary time evolution as such is not undermined. This rather new, and perhaps ambitious, field thus explores a possible generalization of quantum mechanics which incorporates non-Hermitian Hamiltonians into a consistent quantum mechanical framework. New results of a physical and mathematical nature have brought about appreciable progress; earlier work has been uncovered or even rediscovered and links between sub-fields have been established. This issue should provide a stimulus for initiating further advances in the field, re-focus on earlier perspectives and explore open and controversial questions. While non-Hermitian operators have been encountered in theoretical physics since the advent of quantum mechanics, their systematic mathematical investigation and their possible physical implications is a relatively new subject, further explored in this issue. We would like to thank the University of Stellenbosch and the South African National Research Foundation for their financial support, without which it would not have been possible to plan and organize the workshop. Organizational aspects and technical support were very ably taken care of by Christine Ruperti and Tinus Botha whom we would like to thank again. Finally, we also appreciate and acknowledge the invitation from the Editorial Board of Journal of Physics A: Mathematical and General for the opportunity to publish this special issue and trust that this will indeed widen the interest and stimulate progress in the field.