Quantum canonical transformations of the second kind and the non-Hermitian realizations of the basic canonical commutation relations are investigated with a special interest in the generalization of the conventional ladder operators. The operator ordering problem is shown to be resolved when the non-Hermitian realizations for the canonical variables that cannot be measured simultaneously with the energy are chosen for the canonical quantizations. Another merit of the non-Hermitian representation is that it naturally allows us to introduce the generalized ladder operators with which one can solve eigenvalue problems quite neatly. \textcopyright{} 1996 The American Physical Society.
Quantum canonical transformations of the second kind and the non-Hermitian realizations of the basic canonical commutation relations are investigated with a special interest in the generalization of the conventional ladder operators. The operator ordering problem is shown to be resolved when the non-Hermitian realizations for the canonical variables that cannot be measured simultaneously with the energy are chosen for the canonical quantizations. Another merit of the non-Hermitian representation is that it naturally allows us to introduce the generalized ladder operators with which one can solve eigenvalue problems quite neatly. \textcopyright{} 1996 The American Physical Society.
The quantum-mechanical version of the four kinds of classical canonical transformations is investigated by using non-Hermitian operator techniques. To help understand the usefulness of this approach, the eigenvalue problem of a harmonic oscillator is solved in two different types of canonical transformations. The quantum form of the classical Hamilton-Jacobi theory is also employed to solve time-dependent Schr\"odinger wave equations, showing that when one uses the classical action as a generating function of the quantum canonical transformation of time evolutions of state vectors, the corresponding propagator can easily be obtained.
In a recent series of papers, we have analyzed a certain deformation of the canonical commutation relations producing an interesting functional structure which has been proved to have some connections with physics, and in particular with quasi-Hermitian quantum mechanics. Here we repeat a similar analysis starting with the canonical anticommutation relations. We will show that in this case, most of the assumptions needed in the former situation are automatically satisfied, making our construction rather friendly . We discuss some examples of our construction, again related to quasi-Hermitian quantum mechanics, and the bi-coherent states for the system. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.
The non-Hermitian Schrödinger equation is re-expressed generally in the form of Hamilton's canonical equation without any approximation. Its quantization called non-Hermitian quantum field theory is discussed. By virtue of the canonical equation, the theory of non-Hermitian quantum mechanics is totally reformulated, including the probability amplitudes of states, the expectations of operators, as well as the expressions of interaction terms. The conventional difficulties in non-Hermitian quantum mechanics are totally overcome by the reformulation. Specifically, the imaginary parts the non-Hermitian eigenenergy and adiabatic geometric phase are actually unphysical, although they are mathematically perfect.
Based on (i) the spectral resolution definition of the momentum operator, (ii) the linearity of correspondence between physical observables and quantum Hermitian operators, (iii) the definition of conjugate coordinate-momentum variables in classical mechanics, and (iv) the invariance of the classical Hamiltonian to canonical transformations (transformations that change from one set of conjugate variables to another), we show that the $[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x},\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}]$ coordinate-momentum commutator must have the value ih where h is a real nonzero number (which we can identify with the experimentally determined $\ensuremath{\hbar}$). The results are then extended to include all generalized coordinates and their conjugate momenta as well as the Cartesian special-relativistic case.
The classical system is usually quantized using the canonical formalism, i.e., in the language of coordinates and momenta. In addition to this “purely algebraic” ambiguity, there is also an ambiguity in the choice of realization of the operators p^ and q^: the commutation relations do not determine it uniquely, but up to a unitary equivalence (the well known uniqueness theorem of von Neumann pertains only to the Weyl form of the commutation relations). The realization must be chosen on the basis of additional considerations regarding the nature of the canonical variables p and q. In this book the term “theory” is used in various ways: in a broad sense (quantum field theory, relativistic theory) and in a narrow sense, to refer to a particular system or class of systems with certain dynamics. In quantum theory there are also systems whose operators corresponding to canonical variables satisfy not commutation relations, but anticommutation relations.
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.