Quasi-Hermitian formulation of quantum mechanics using two conjugate Schrödinger equations

Abstract

In an amended version of non-Hermitian interaction picture we propose to work with the states $\psi (t)$ in a dyadic representation. The control of evolution via two conjugate Schrödiner equations then renders the usual necessity of the construction of the time-dependent inner-product-metric operator $\Theta (t)$ redundant. The primary information about dynamics is assumed carried by a non-Hamiltonian observable (say, $R(t)$). A specific realization of phase transitions is then rendered possible via the Kato's exceptional-point (EP) degeneracy of the eigenvalues of $R(t)$ at the EP time $t=t^{(EP)}$. For illustration a cosmological model is proposed mimicking the unitary-evolution birth of the Universe from an initial quantum Big Bang singularity.

arXiv comment27 pp., 1 figure

References in corpus (41)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Making Sense of Non-Hermitian Hamiltonians
• Background Independent Quantum Gravity: A Status Report
• Pseudo-Hermitian Representation of Quantum Mechanics
• Absence of Singularity in Loop Quantum Cosmology
• Biorthogonal Quantum Mechanics
• Geometric Quantum Mechanics
• Geometry of quantum phase transitions
• Time-dependent version of cryptohermitian quantum theory
• Time-Dependent Pseudo-Hermitian Hamiltonians Defining a Unitary Quantum System and Uniqueness of the Metric Operator
• Unitary quantum evolution for time-dependent quasi-Hermitian systems with non-observable Hamiltonians
• Time evolution of non-Hermitian Hamiltonian systems
• Hilbert Space Structures on the Solution Space of Klein-Gordon Type Evolution Equations
• Time Dependent $\mathcal{P}\mathcal{T}$-Symmetric Quantum Mechanics
• Three-Hilbert-Space Formulation of Quantum Mechanics
• Quantum control and long-range quantum correlations in dynamical Casimir arrays
• Conceptual Problems in Scattering from Localized non-Hermitian Potentials
• Quantum Mechanics of Klein-Gordon-Type Fields and Quantum Cosmology
• Interface between Hermitian and non-Hermitian Hamiltonians in a model calculation
• Geometric Phase in PT-Symmetric Quantum Mechanics
• Quantum Mechanics of Klein-Gordon Fields I: Hilbert Space, Localized States, and Chiral Symmetry
• Exact analytical solutions for time-dependent Hermitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians
• Flattening the Curve with Einstein's Quantum Elevator: Hermitization of Non-Hermitian Hamiltonians via a Generalized Vielbein Formalism
• Non-Hermitian interaction representation and its use in relativistic quantum mechanics
• A return to observability near exceptional points in a schematic PT-symmetric model
• Eternal life of entropy in non-Hermitian quantum systems
• Relativistic supersymmetric quantum mechanics based on Klein-Gordon equation
• Quantum Mechanics of Proca Fields
• A Positive-Definite Scalar Product for Free Proca Particle
• The problem of coexistence of several non-Hermitian observables in PT-symmetric quantum mechanics
• A non self-adjoint model on a two dimensional noncommutative space with unbound metric
• Turning big bang into big bounce: II. Quantum dynamics
• Non-Hermitian coupled cluster method for non-stationary systems and its interaction-picture reinterpretation
• On the Role of the Normalization Factors ${\kappa }_n$ and of the Pseudo-Metric $\mathcal{P}\ne {\mathcal{P}}^{†}$ in Crypto-Hermitian Quantum Models
• Spectrally equivalent time-dependent double wells and unstable anharmonic oscillators
• Hybrid form of quantum theory with non-Hermitian Hamiltonians
• Composite quantum Coriolis forces
• Quantization of time and the big bang via scale-invariant loop gravity
• Real energies and Berry phases in all PT-regimes in time-dependent non-Hermitian theories
• Systematics of quasi-Hermitian representations of non-Hermitian quantum models
• Non-stationary non-Hermitian "wrong-sign'' quantum oscillators and their meaningful physical interpretation

Cited by in corpus (7)

• Non-Hermitian Generalization of Rayleigh-Schrödinger Perturbation Theory
• Quantum singularities in a solvable toy model
• Few-grid-point simulations of Big Bang singularity in quantum cosmology
• Features, paradoxes and amendments of perturbative non-Hermitian quantum mechanics
• Resonances and continued-fraction Green's functions in non-Hermitian Bose-Hubbard-like quantum models
• Three alternative model-building strategies using quasi-Hermitian time-dependent observables
• Anisotropy and asymptotic degeneracy of the physical-Hilbert-space inner-product metrics in an exactly solvable crypto-unitary quantum model