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

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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 Schrodinger 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 E endowed with a metric compatible connection, and observables are given by global sections of a real vector bundle that is determined by E. We examine the utility of our proposal to describe a class of two-level systems where E is a Hermitian vector bundle over a two-dimensional sphere.

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American Physical Society (APS)

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Astronomy and astrophysics, Physics, particles and fields

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Physical Review D

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10.1103/PhysRevD.98.046022

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