Time-Dependent Pseudo-Hermitian Hamiltonians and a Hidden Geometric
Aspect of Quantum Mechanics
- URL: http://arxiv.org/abs/2004.05254v1
- Date: Fri, 10 Apr 2020 23:00:43 GMT
- Title: Time-Dependent Pseudo-Hermitian Hamiltonians and a Hidden Geometric
Aspect of Quantum Mechanics
- Authors: Ali Mostafazadeh
- Abstract summary: A non-Hermitian operator $H$ defined in a Hilbert space with inner product $langlecdot|cdotrangle$ may serve as the Hamiltonian for a unitary quantum system.
Such quantum systems, which are also encountered in the study of quantum mechanics in cosmological backgrounds, suffer from a conflict between the unitarity of time evolution and the unobservability of the Hamiltonian.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: A non-Hermitian operator $H$ defined in a Hilbert space with inner product
$\langle\cdot|\cdot\rangle$ may serve as the Hamiltonian for a unitary quantum
system, if it is $\eta$-pseudo-Hermitian for a metric operator
(positive-definite automorphism) $\eta$. The latter defines the inner product
$\langle \cdot|\eta\cdot\rangle $ of the physical Hilbert space $\mathscr{H}
_{\eta}$ of the system. For situations where some of the eigenstates of $H$
depend on time, $\eta$ becomes time-dependent. Therefore the system has a
non-stationary Hilbert space. Such quantum systems, which are also encountered
in the study of quantum mechanics in cosmological backgrounds, suffer from a
conflict between the unitarity of time evolution and the unobservability of the
Hamiltonian. Their proper treatment requires a geometric framework which
clarifies the notion of the energy observable and leads to a geometric
extension of quantum mechanics (GEQM). We provide a general introduction to the
subject, review some of the recent developments, offer a straightforward
description of the Heisenberg-picture formulation of the dynamics for quantum
systems having a time-dependent Hilbert space, and outline the
Heisenberg-picture formulation of dynamics in GEQM.
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