Three-Hilbert-Space Formulation of Quantum Mechanics

Abstract

In paper [Znojil M., Phys. Rev. D 78 (2008), 085003, 5 pages, arXiv:0809.2874] the two-Hilbert-space (2HS, a.k.a. cryptohermitian) formulation of Quantum Mechanics has been revisited. In the present continuation of this study (with the spaces in question denoted as ${\mathcal{H}}^{(\mathrm{auxiliary})}$ and ${\mathcal{H}}^{(\mathrm{standard})}$) we spot a weak point of the 2HS formalism which lies in the double role played by ${\mathcal{H}}^{(\mathrm{auxiliary})}$. As long as this confluence of roles may (and did!) lead to confusion in the literature, we propose an amended, three-Hilbert-space (3HS) reformulation of the same theory. As a byproduct of our analysis of the formalism we offer an amendment of the Dirac's bra-ket notation and we also show how its use clarifies the concept of covariance in time-dependent cases. Via an elementary example we finally explain why in certain quantum systems the generator $H_{(\mathrm{gen})}$ of the time-evolution of the wave functions may differ from their Hamiltonian $H$.

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• Scattering theory using smeared non-Hermitian potentials
• From self-adjoint to non self-adjoint harmonic oscillators: physical consequences and mathematical pitfalls
• Unitarity corridors to exceptional points
• The problem of coexistence of several non-Hermitian observables in PT-symmetric quantum mechanics
• Non-Hermitian Heisenberg representation
• Crypto-unitary forms of quantum evolution operators
• Admissible perturbations and false instabilities in PT-symmetric quantum systems
• Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics
• Non-Hermitian coupled cluster method for non-stationary systems and its interaction-picture reinterpretation
• Some results on the dynamics and transition probabilities for non self-adjoint hamiltonians
• Three solvable matrix models of a quantum catastrophe
• Discrete quantum square well of the first kind
• Quantum phase transitions in nonhermitian harmonic oscillator
• Quantizations from reproducing kernel spaces
• Complex symmetric Hamiltonians and exceptional points of order four and five
• Quantum star-graph analogues of PT-symmetric square wells
• Quantum inner-product metrics via recurrent solution of Dieudonne equation
• Wheeler-DeWitt equation and the applicability of crypto-Hermitian interaction representation in quantum cosmology
• Quantum Big Bang without fine-tuning in a toy-model
• Phase space representation of non-Hermitian system with $\mathcal{P}\mathcal{T}$-symmetry
• Composite quantum Coriolis forces
• Pseudo-invariant approach for a particle in a complex time-dependent linear potential
• Hybrid form of quantum theory with non-Hermitian Hamiltonians
• Hilbert space inner products for PT-symmetric Su-Schrieffer-Heeger models
• Damping and Pseudo-fermions
• Cryptohermitian Picture of Scattering Using Quasilocal Metric Operators
• Discrete-coordinate crypto-Hermitian quantum system controlled by time-dependent Robin boundary conditions
• Quasi-Hermitian formulation of quantum mechanics using two conjugate Schrödinger equations
• The coupled-cluster approach to quantum many-body problem in a three-Hilbert-space reinterpretation
• Broken Hermiticity phase transition in Bose-Hubbard model
• Hermitian--to--quasi-Hermitian quantum phase transitions
• Non-stationary non-Hermitian "wrong-sign'' quantum oscillators and their meaningful physical interpretation
• Hermitian - non-Hermitian interfaces in quantum theory
• Non-Hermitian-Hamiltonian-induced unitarity and optional physical inner products in Hilbert space
• Crypto-Hermitian Approach to the Klein-Gordon Equation
• Non-stationary quantum mechanics in hybrid non-Hermitian interaction representation
• Quantum Evolution of the Time-Dependent Non-Hermitian Hamiltonians: Real Phases
• A real expectation value of the time-dependent non-Hermitian Hamiltonians
• Unitary unfoldings of Bose-Hubbard exceptional point with and without particle number conservation
• Non-isospectral Hamiltonians, intertwining operators and hidden hermiticity
• Interference of non-Hermiticity with Hermiticity at exceptional points
• Intrinsic exceptional point -- a challenge in quantum theory
• Quantum phase transitions mediated by clustered non-Hermitian degeneracies
• Comment on 'The operational foundations of PT-symmetric and quasi-Hermitian quantum theory'
• Three alternative model-building strategies using quasi-Hermitian time-dependent observables
• New Concept of Solvability in Quantum Mechanics
• Time evolution of quantum systems with time-dependent non-Hermitian Hamiltonian and the pseudo Hermitian invariant operator
• Construction of maximally non-Hermitian potentials under unbroken PT-symmetry constraint
• Planarizable Supersymmetric Quantum Toboggans
• PT-symmetric quantum models living in an auxiliary Pontryagin space
• The dynamical problem for a non self-adjoint Hamiltonian
• Pseudo-bosons and bi-coherent states out of $\Lc^2(\mathbb{R})$
• Identification of the metric for diagonalizable (anti-)pseudo-Hermitian Hamilton operators represented by two-dimensional matrices
• Anisotropy and asymptotic degeneracy of the physical-Hilbert-space inner-product metrics in an exactly solvable crypto-unitary quantum model
• Coulomb potential and the paradoxes of PT-symmetrization
• Non-Unitary evolution of quantum time-dependent non-Hermitian systems
• Emergence and localization of exceptional points in an exactly solvable toy model