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A non-Hermitian operator may serve as the Hamiltonian for a unitary quantumsystem, if we can modify the Hilbert space of state vectors of the system sothat 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 ageneric conflict between the condition that the Hamiltonian is an observable ofthe system and that it generates a unitary time-evolution via the standardSchr\"odinger equation. We propose a geometric framework for addressing thisproblem. In particular we show that the Hamiltonian operator consists of ageometric part, which is determined by a metric-compatible connection on anunderlying Hermitian vector bundle, and a non-geometric part which we identifywith the energy observable. The same quantum system can be locally describedusing a time-dependent Hamiltonian that acts in a time-independent state spaceand is the sum of a geometric part and the energy operator. The full globaldescription of the system is achieved within the framework of a moderategeometric extension of quantum mechanics where the role of the Hilbert space ofstate vectors is played by a Hermitian vector bundle $\mathcal{E}$ endowed witha metric compatible connection, and observables are given by global sections ofa real vector bundle that is determined by $\mathcal{E}$. We examine theutility 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