Gene

Physics

Mathematical optimization

Hamiltonian (control theory)

Connection (principal bundle)

Quantum state

Repressor

Operator (biology)

Self-adjoint operator

POVM

Chemistry

SIC-POVM

Unitary operator

Open quantum system

Pure mathematics

Geometry

Quantum system

Observable

Mathematics

Quantum operation

Biochemistry

Quantum mechanics

Hilbert space

Quantum

Transcription factor

Hermitian matrix

Vector bundle

Physical review. D/Physical review. D.

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 Schr\"odinger 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 nongeometric 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.

Energy observable for a quantum system with a dynamical Hilbert space and a global geometric extension of quantum theory