Consistent treatment of quantum systems with a time-dependent Hilbert space

Abstract

We consider some basic problems associated with quantum mechanics of systems having a time-dependent Hilbert space. We provide a consistent treatment of these systems and address the possibility of describing them in terms of a time-independent Hilbert space. We show that in general the Hamiltonian operator does not represent an observable of the system even if it is a self-adjoint operator. This is related to a hidden geometric aspect of quantum mechanics arising from the presence of an operator-valued gauge potential. We also offer a careful treatment of quantum systems whose Hilbert space is obtained by endowing a time-independent vector space with a time-dependent inner product.

28 pages

References in corpus (37)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Complex Extension of Quantum Mechanics
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• PT-Symmetric Quantum Mechanics
• Pseudo-Hermitian Representation of Quantum Mechanics
• Pseudo-Hermiticity versus PT-Symmetry II: A complete characterizatio n of non-Hermitian Hamiltonians with a real spectrum
• Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries
• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• Exact PT-Symmetry Is Equivalent to Hermiticity
• Must a Hamiltonian be Hermitian?
• Physical Aspects of Pseudo-Hermitian and $PT$-Symmetric Quantum Mechanics
• PT-Symmetric Cubic Anharmonic Oscillator as a Physical Model
• 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
• Quantum Mechanics of Klein-Gordon-Type Fields and Quantum Cosmology
• Quantum Mechanics of Klein-Gordon Fields I: Hilbert Space, Localized States, and Chiral Symmetry
• Non-Hermitian interaction representation and its use in relativistic quantum mechanics
• Energy Observable for a Quantum System with a Dynamical Hilbert Space and a Global Geometric Extension of Quantum Theory
• Conceptual Aspects of PT-Symmetry and Pseudo-Hermiticity: A status report
• A Physical Realization of the Generalized PT-, C-, and CPT-Symmetries and the Position Operator for Klein-Gordon Fields
• Time-Dependent Hilbert Spaces, Geometric Phases, and General Covariance in Quantum Mechanics
• A quantum particle in a box with moving walls
• PT-Symmetric Quantum Mechanics: A Precise and Consistent Formulation
• Investigation of non-Hermitian Hamiltonians in the Heisenberg picture
• Time-Dependent Pseudo-Hermitian Hamiltonians and a Hidden Geometric Aspect of Quantum Mechanics
• Quantum Mechanics of Proca Fields
• Non-Hermitian Heisenberg representation
• Quantum Mechanics of a Photon
• Differential Realization of Pseudo-Hermiticity: A quantum mechanical analog of Einstein's field equation
• Crypto-unitary forms of quantum evolution operators
• Maxwell quantum mechanics
• Quantum Mechanics of Klein-Gordon Fields II: Relativistic Coherent States
• Quantum Canonical Transformations and Exact Solution of the Schreodinger Equation