A few recent innovations of the applicability of standard textbook Quantum Theory are reviewed. The three-Hilbert-space formulation of the theory (known from the interacting boson models in nuclear physics) is discussed in its slightly broadened four-Hilbert-space update. Among applications involving several new scattering and bound-state problems the central role is played by models using apparently non-Hermitian (often called “crypto-Hermitian”) Hamiltonians with real spectra. The formalism (originally inspired by the topical need for a mathematically consistent description of tobogganic quantum models) is shown to admit even certain unusual nonlocal and/or “moving-frame” representations H(S) of the standard physical Hilbert space of wave functions.
A few recent innovations of the applicability of standard textbook Quantum Theory are reviewed. The three-Hilbert-space formulation of the theory (known from the interacting boson models in nuclear physics) is discussed in its slightly broadened four-Hilbert-space update. Among applications involving several new scattering and bound-state problems the central role is played by models using apparently non-Hermitian (often called “crypto-Hermitian”) Hamiltonians with real spectra. The formalism (originally inspired by the topical need for a mathematically consistent description of tobogganic quantum models) is shown to admit even certain unusual nonlocal and/or “moving-frame” representations H(S) of the standard physical Hilbert space of wave functions.
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as [Formula: see text], the true meaning and significance of the so-called charge operators [Formula: see text] and the [Formula: see text]-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications and manifestations of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos and biophysics.
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as PT, the true meaning and significance of the charge operators C and the CPT-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos, and biophysics.
We investigate the quantum-mechanical interpretation of models with non-Hermitian Hamiltonians and real spectra. After describing a general framework to reformulate such models in terms of Hermitian Hamiltonians defined on the Hilbert space L2(−infty,infty), we discuss the significance of the algebra of physical observables.
Theories with indefinite metric have a long and very rich history. Such theories are being used time to time in searching the solutions for many outstanding problems in physics over the several decades. However much attentions were not paid for these as they were not mathematically consistent with usual quantum theories. Towards the end of last century this field of research received huge boost, when attempts had been made to accommodate certain class of non-Hermitian theories in rigorous mathematical formulation. Subsequently it has been shown that certain specific types of non-Hermitian theories can lead to fully consistent quantum theories with complete real spectrum, unitary time evolution and probabilistic interpretation in a modified Hilbert space. Due to this important realisation, theories with complex Hamiltonian have become the topic of frontier research over last two decades and enormous applicationswere found in various branches of physics. In this brief note we try to review certain aspects of such theories where we have contributed. In particular, we discuss PT phase transition in non-relativistic, relativistic and quantum field theoretic systems and some new features of scattering in complex systems.
The aim of this thesis is to present a reasonable account of some of the applications that functional analysis, or more specifically operator theory, has in the theory of quantum scattering. We begin by introducing the physical scattering experiment as well as establishing the standard framework for discussing quantum theory, that of a Hilbert space. This leads to a discussion of the properties of operators acting in a Hilbert space and the role played by unbounded selfadjoint operators in quantum physics. We also spend some time discussing the position, momentum and energy operators which are of most relevance here. This in turn leads logically to the problem of formulating the scattering problem mathematically using these operators. The stationary equations of scattering theory are also rigorously derived and their suitability for two-body scattering problems is proven. We show that this analysis does not extend to three-body scattering problems, and introduce the Faddeev formulation for three-body systems to supplement the theory. This thesis concludes with a brief overview of how our scattering equations can be modified to take into account effects such as particle spin and identical particles, before discussing an example of a three-body scattering problem (incorporating these features) and deriving a mathematically satisfactory set of equations which are amenable to numerical solution.