Conceptual Aspects of PT-Symmetry and Pseudo-Hermiticity: A status report

Abstract

We survey some of the main conceptual developments in the study of PT-symmetric and pseudo-Hermitian Hamiltonian operators that have taken place during the past ten years or so. We offer a precise mathematical description of a quantum system and its representations that allows us to describe the idea of unitarization of a quantum system by modifying the inner product of the Hilbert space. We discuss the role and importance of the quantum-to-classical correspondence principle that provides the physical interpretation of the observables in quantum mechanics. Finally, we address the problem of constructing an underlying classical Hamiltonian for a unitary quantum system defined by an a priori non-Hermitian Hamiltonian.

11 pages

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• Making Sense of Non-Hermitian Hamiltonians
• Physical realization of $\mathcal{P}\mathcal{T}$-symmetric potential scattering in a planar slab waveguide
• Delta-Function Potential with a Complex Coupling
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Cited by in corpus (33)

• Invisibility in PT-symmetric complex crystals
• Non-Hermitian physics and master equations
• Impact of loss on the wave dynamics in photonic waveguide lattices
• $PT$-symmetric invisible defects and confluent Darboux-Crum transformations
• Wave packet evolution in non-Hermitian quantum systems
• Spectral Singularities of a General Point Interaction
• Interpretation of quantum mechanics with indefinite norm
• Classical and quantum dynamics in the (non-Hermitian) Swanson oscillator
• Exactly solvable time-dependent pseudo-Hermitian $su$(1,1) Hamiltonian models
• Quantum Mechanics of a Photon
• Confluent Crum-Darboux transformations in Dirac Hamiltonians with $PT$-symmetric Bragg gratings
• Imaginary-Scaling versus Indefinite-Metric Quantization of the Pais-Uhlenbeck Oscillator
• Two-qubit entanglement generation through non-Hermitian Hamiltonians induced by repeated measurements on an ancilla
• Quantum phase transitions in nonhermitian harmonic oscillator
• Analytically solvable $2\times 2$ $PT$-symmetry dynamics from su(1,1)-symmetry problems
• A unidirectional invisible PT-symmetric complex crystal with arbitrary thickness
• PT-symmetry breaking with divergent potentials: lattice and continuum cases
• Observing a Changing Hilbert-Space Inner Product
• Spin-chain-star systems: entangling multiple chains of spin qubits
• Multitude of exceptional points in van der Waals magnets
• Non-Hermitian Oscillator and R-deformed Heisenberg Algebra
• Weak Measurements in Non-Hermitian Systems
• Metric operators for non-Hermitian quadratic su(2) Hamiltonians
• Theory Including Future Not Excluded -- Formulation of Complex Action Theory II --
• CPT-symmetric discrete square well
• Mixedness timescale in non-Hermitian quantum systems
• Deformations of Yang-Mills theory
• Momentum relation and classical limit in the future-not-included complex action theory
• Consistent treatment of quantum systems with a time-dependent Hilbert space
• Field operators in topological quantum theories
• $\mathcal{P}\mathcal{T}$-Symmetric Generalized Extended Momentum Operator
• Complementarity versus coordinate transformations: mapping between pseudo-Hermiticity and weak pseudo-Hermiticity
• The operational foundations of PT-symmetric and quasi-Hermitian quantum theory