Energy Observable for a Quantum System with a Dynamical Hilbert Space and a Global Geometric Extension of Quantum Theory

Abstract

A non-Hermitian operator may serve as the Hamiltonian for a unitary quantum system, if we can modify the Hilbert space of state vectors of the system so that it turns into a Hermitian operator. If this operator is time-dependent, the modified Hilbert space is generally time-dependent. This in turn leads to a generic conflict between the condition that the Hamiltonian is an observable of the system and that it generates a unitary time-evolution via the standard Schrödinger equation. We propose a geometric framework for addressing this problem. In particular we show that the Hamiltonian operator consists of a geometric part, which is determined by a metric-compatible connection on an underlying Hermitian vector bundle, and a non-geometric part which we identify with the energy observable. The same quantum system can be locally described using a time-dependent Hamiltonian that acts in a time-independent state space and is the sum of a geometric part and the energy operator. The full global description of the system is achieved within the framework of a moderate geometric extension of quantum mechanics where the role of the Hilbert space of state vectors is played by a Hermitian vector bundle $\mathcal{E}$ endowed with a metric compatible connection, and observables are given by global sections of a real vector bundle that is determined by $\mathcal{E}$. We examine the utility of our proposal to describe a class of two-level systems where $\mathcal{E}$ is a Hermitian vector bundle over a two-dimensional sphere.

31 pages, 1 figure, slightly expanded version, to appear in Phys. Rev. D

References in corpus (9)

• Physical realization of $\mathcal{P}\mathcal{T}$-symmetric potential scattering in a planar slab waveguide
• Time-dependent version of cryptohermitian quantum theory
• Time-Dependent Pseudo-Hermitian Hamiltonians Defining a Unitary Quantum System and Uniqueness of the Metric Operator
• Time evolution of non-Hermitian Hamiltonian systems
• Exact analytical solutions for time-dependent Hermitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians
• Solvable two dimensional time-dependent non-Hermitian quantum systems with infinite dimensional Hilbert space in the broken PT-regime
• Non-Hermitian Heisenberg representation
• Hermitian--to--quasi-Hermitian quantum phase transitions
• Quantum Mechanics With Noncommutative Amplitudes

Cited by in corpus (31)

• Quantum Geometric Tensor in $\mathcal{P}\mathcal{T}$-Symmetric Quantum Mechanics
• Time-dependent $\mathcal{P}\mathcal{T}$-symmetric quantum mechanics in generic non-Hermitian systems
• Eternal life of entropy in non-Hermitian quantum systems
• $\mathcal{P}\mathcal{T}$-symmetric, non-Hermitian quantum many-body physics -- a methodological perspective
• Time-Dependent Pseudo-Hermitian Hamiltonians and a Hidden Geometric Aspect of Quantum Mechanics
• Exactly solvable time-dependent non-Hermitian quantum systems from point transformations
• Time-dependent Darboux (supersymmetric) transformations for non-Hermitian quantum systems
• Non-Hermitian coupled cluster method for non-stationary systems and its interaction-picture reinterpretation
• Quantum nonstationary oscillators: Invariants, dynamical algebras and coherent states via point transformations
• Quantum Metric Unveils Defect Freezing in Non-Hermitian Systems
• Quasi-exactly solvable quantum systems with explicitly time-dependent Hamiltonians
• Spectrally equivalent time-dependent double wells and unstable anharmonic oscillators
• Anomalous dynamical response of non-Hermitian topological phases
• Observing a Changing Hilbert-Space Inner Product
• Observables in non-Hermitian systems: A methodological comparison
• Holonomic Gates in Pseudo-Hermitian Quantum Systems
• Time-dependent metric for the two dimensional, non-Hermitian coupled oscillator
• The effect of thermal photons on exceptional points in coupled resonators
• Hybrid Geometrodynamics: A Hamiltonian description of classical gravity coupled to quantum matter
• Real energies and Berry phases in all PT-regimes in time-dependent non-Hermitian theories
• Topological invariants of complex energy plane in non-Hermitian systems
• Non-Hermitian quantum walks and non-Markovianity: the coin-position interaction
• Entanglement between uncoupled modes with time-dependent complex frequencies
• Towards quantum Mechanics on the curved cotangent bundle
• Consistent treatment of quantum systems with a time-dependent Hilbert space
• Anti-$\mathcal{P}\mathcal{T}$-symmetric Qubit: Decoherence and Entanglement Entropy
• Decomposition of a system in pseudo-Hermitian quantum mechanics
• Topological invariant of velocity field in quantum systems
• Covariant formulation of the Berry connection in non-Hermitian systems
• The operational foundations of PT-symmetric and quasi-Hermitian quantum theory
• Geometric criterion of topological phase transition for non-Hermitian systems