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 $H$.

References in corpus (14)

• The ODE/IM Correspondence
• Time-dependent version of cryptohermitian quantum theory
• Time-Dependent Pseudo-Hermitian Hamiltonians Defining a Unitary Quantum System and Uniqueness of the Metric Operator
• PT-symmetric quantum mechanics
• MHD alpha^2-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator
• Scattering theory with localized non-Hermiticities
• Discrete PT-symmetric models of scattering
• Quantum Mechanics of Proca Fields
• A Positive-Definite Scalar Product for Free Proca Particle
• Propagation of Electromagnetic Waves in Linear Media and Pseudo-Hermiticity
• On the Role of the Normalization Factors ${\kappa }_n$ and of the Pseudo-Metric $\mathcal{P}\ne {\mathcal{P}}^{†}$ in Crypto-Hermitian Quantum Models
• Identification of observables in quantum toboggans
• Which operator generates time evolution in Quantum Mechanics?
• Relativistic vector bosons and PT-symmetry

Cited by in corpus (32)

• Gegenbauer-solvable quantum chain model
• Scattering theory using smeared non-Hermitian potentials
• Non-Hermitian Heisenberg representation
• Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics
• Three solvable matrix models of a quantum catastrophe
• Some results on the dynamics and transition probabilities for non self-adjoint hamiltonians
• Discrete quantum square well of the first kind
• Quantum star-graph analogues of PT-symmetric square wells
• Wheeler-DeWitt equation and the applicability of crypto-Hermitian interaction representation in quantum cosmology
• Hybrid form of quantum theory with non-Hermitian Hamiltonians
• Composite quantum Coriolis forces
• 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
• Non-stationary non-Hermitian "wrong-sign'' quantum oscillators and their meaningful physical interpretation
• Quantum Evolution of the Time-Dependent Non-Hermitian Hamiltonians: Real Phases
• Crypto-Hermitian Approach to the Klein-Gordon Equation
• Non-stationary quantum mechanics in hybrid non-Hermitian interaction representation
• Non-Hermitian-Hamiltonian-induced unitarity and optional physical inner products in Hilbert space
• A real expectation value of the time-dependent non-Hermitian Hamiltonians
• Quantum phase transitions mediated by clustered non-Hermitian degeneracies
• Intrinsic exceptional point -- a challenge in quantum theory
• Interference of non-Hermiticity with Hermiticity at exceptional points
• Construction of maximally non-Hermitian potentials under unbroken PT-symmetry constraint
• Three alternative model-building strategies using quasi-Hermitian time-dependent observables
• Comment on 'The operational foundations of PT-symmetric and quasi-Hermitian quantum theory'
• Time evolution of quantum systems with time-dependent non-Hermitian Hamiltonian and the pseudo Hermitian invariant operator
• Planarizable Supersymmetric Quantum Toboggans
• Emergence and localization of exceptional points in an exactly solvable toy model
• 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
• Non-Unitary evolution of quantum time-dependent non-Hermitian systems