The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose +complex conjugate) is replaced by the physically transparent condition of space?time reflection ( ) symmetry. If H has an unbroken symmetry, then the spectrum is real. Examples of -symmetric non-Hermitian quantum-mechanical Hamiltonians are and . Amazingly, the energy levels of these Hamiltonians are all real and positive!Does a -symmetric Hamiltonian H specify a physical quantum theory in which the norms of states are positive and time evolution is unitary? The answer is that if H has an unbroken symmetry, then it has another symmetry represented by a linear operator . In terms of , one can construct a time-independent inner product with a positive-definite norm. Thus, -symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution.The Lee model provides an excellent example of a -symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm 'ghost' state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. The correct interpretation of the ghost is simply that the non-Hermitian Lee-model Hamiltonian is -symmetric. The operator for the Lee model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.

https://iopscience.iop.org/article/10.1088/0034-4885/70/6/R03/pdf

Making sense of non-Hermitian Hamiltonians

A short resume is given about the nature of exceptional points (EPs) followed by discussions about their ubiquitous occurrence in a great variety of physical problems. EPs feature in classical as well as in quantum mechanical problems. They are associated with symmetry breaking for -symmetric Hamiltonians, where a great number of experiments has been performed, in particular in optics, and to an increasing extent in atomic and molecular physics. EPs are involved in quantum phase transition and quantum chaos; they produce dramatic effects in multichannel scattering, specific time dependence and more. In nuclear physics, they are associated with instabilities and continuum problems. Being spectral singularities they also affect approximation schemes.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.

https://iopscience.iop.org/article/10.1088/1751-8113/45/44/444016/pdf

The physics of exceptional points

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 ${\mathcal{P}\mathcal{T}}$, the true meaning and significance of the so-called charge operators $\mathcal{C}$ and the ${\mathcal{C}\mathcal{P}\mathcal{T}}$-inner products,...

Pseudo-Hermitian Representation of Quantum Mechanics

Progress in particle and nuclear physics

This chapter examines the algebraic and geometric properties of the interacting boson model-1, and reviews the interacting boson model-2. Explains that the model was originally introduced with only one kind of collective boson variable with angular momentum J=O and J=2 (the interacting boson model-1), and subsequently, a more elaborate version was introduced with two kinds of collective variables, proton bosons and neutron bosons (the interacting boson model-2). Discusses analytic solutions, transitional classes, extensions of the models, coherent states, transitional classes and shape-phase transitions, energy levels, electromagnetic transition rates, other properties, a microscopic description of interacting bosons, generalized seniority, the single j-shell, several j-shells, and the Ginocchio model. Excludes odd-even nuclei and the corresponding interacting boson-fermion models 1 and 2 from the review.

The interacting boson model

Quasi-Hermitian operators in quantum mechanics and the variational principle

The ubiquitous Lipkin model is investigated for an interaction parameter beyond the traditional critical point. It is argued that a phase transition occurs higher up in the spectrum for such larger interaction, where, using appropriate scaling of the energies, the position of the phase transition becomes independent of the particle number. The phase transition is related to near singularities in the complex interaction plane, the exceptional points. Consideration of finite temperature yields the well-known physical features associated with phase transitions.

The large N behaviour of the Lipkin model and exceptional points

Operator equations and Moyal products–metrics in quasi-Hermitian quantum mechanics

Unveiling s- and d-bosons in even- and odd- A nuclei

It is shown that it is not necessary to resort to an explicit mapping of the fermion basis into the boson space when bifermion excitations are treated in a boson picture. One may use the usual ideal boson basis in the boson space. Under certain circumstances this may lead to the occurrence of spurious states. However, these do not mix with the physical states and they can be identified by means of a simple procedure which is developed within the framework of the Dyson boson method. The results are illustrated by means of three interesting cases in the Sp(4) model.

H.B. Geyer

Papers

Quasi-Hermitian operators in quantum mechanics and the variational principle

The large N behaviour of the Lipkin model and exceptional points

Operator equations and Moyal products–metrics in quasi-Hermitian quantum mechanics

Unveiling s- and d-bosons in even- and odd- A nuclei

Adequacy of the boson basis and the identification of spurious states