Pseudo-Hermitian Representation of Quantum Mechanics (0810.5643v4)

Abstract: 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.

• The paper develops a pseudo-Hermitian framework by modifying the Hilbert space inner product to render non-Hermitian Hamiltonians effectively Hermitian with a real spectrum.
• The paper builds on PT-symmetry concepts by generalizing to broader antilinear symmetries, thus expanding the class of operators that ensure unitary quantum evolution.
• The paper explores applications such as the quantum brachistochrone problem, suggesting potential for faster quantum evolution and wider implications in theoretical physics.

Overview of "Pseudo-Hermitian Representation of Quantum Mechanics"

The paper presented in "Pseudo-Hermitian Representation of Quantum Mechanics" by Ali Mostafazadeh explores the extension of conventional quantum mechanics to systems governed by non-Hermitian Hamiltonians, which can nonetheless possess a real spectrum, enabling a formulation of consistent quantum theories. The central focus is the adaptation of the Hilbert space's inner product to render these non-Hermitian Hamiltonians effectively Hermitian, achieving unitarity in the modified framework.

Key Concepts and Contributions

1. Pseudo-Hermiticity and its Implications: The paper develops a rigorous framework for pseudo-Hermitian quantum mechanics, defining a Hamiltonian operator $H$ as pseudo-Hermitian if there exists a metric operator $\eta$ such that $H^\dagger = \eta H \eta^{-1}$. This condition ensures the realness of the Hamiltonian's spectrum, provided it is diagonalizable. The authors highlight that the pseudo-metric operator $\eta$ alters the inner product in the Hilbert space, allowing the construction of a real and positive-definite spectral decomposition akin to conventional Hermitian systems.
2. Relation to $\mathcal{PT}$-Symmetry: The concept extends Bender's work on $\mathcal{PT}$-symmetric quantum mechanics, where a pseudo-Hermitian framework relaxes the necessity for explicit $\mathcal{PT}$-symmetry while generalizing to broader classes of antilinear symmetries. Mostafazadeh shows that a pseudo-Hermitian framework accommodates such symmetries as a special case, broadening the spectrum of operators that maintain quantum mechanical consistency beyond the $\mathcal{PT}$-invariance constraint.
3. Mathematical Formulation and Structural Results: The paper meticulously sets forth mathematical tools needed for dealing with pseudo-Hermitian operators, including spectral properties, biorthonormal systems, and the implications of various symmetry operations (like charge and antilinear symmetries). These enable a comprehensive understanding of the underlying structure of quantum systems represented in this generalized framework.
4. Implications for Fundamental Physics: The pseudo-Hermitian formulation holds significant implications for theoretical physics, offering a potential pathway to reconcile aspects like the reality of certain potentials and providing an effective method for handling open quantum systems, quantum cosmology, and even aspects of quantum field theory where conventional Hermitian approaches face obstacles.
5. Quantum Brachistochrone Problem: Investigating the quantum Brachistochrone problem within pseudo-Hermitian systems reveals unique insights into the speed of quantum evolution. By adjusting the metric operator, it demonstrates the theoretical possibility of faster-than-standard evolution, showing how appropriate manipulation within this framework might yield practical computational advantages.
6. Applications Across Various Domains: Concrete applications of the pseudo-Hermitian framework cited in the paper include electromagnetic wave propagation, quantum chaos, and biophysics, highlighting the versatility and potential utility of the approach across different physical systems.

Outlook and Future Directions

The work effectively expands the boundaries of quantum mechanics by incorporating non-Hermitian operators into the mainstream fold without sacrificing physical interpretability or mathematical rigor. Moving forward, the paper suggests exploring the application of pseudo-Hermitian systems in solving real-world quantum problems, particularly those involving systems where non-Hermiticity naturally arises. Moreover, further research could examine how pseudo-Hermitian treatments influence quantum entanglement, decoherence, and other fundamental quantum phenomena.

The ongoing development of this framework aims to enrich our understanding of quantum mechanics by providing a toolset capable of addressing the peculiarities and challenges posed by non-Hermitian but physically significant Hamiltonians. This is particularly salient in fields ranging from quantum field theory to quantum cosmology where complex potentials present unique challenges and opportunities.