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arXiv:2609.06800

Hilbert space geometry change by a nonadiabatic electromagnetic field

Ivan Georgiev Koprinkov

quant-phSubmitted Sep 6, 2026• Announced Sep 8, 2026• 0 citations
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Abstract

A change in the geometry of the Hilbert space of a quantum system interacting with a nonadiabatic electromagnetic field is found. To identify its physical origin, the evolution of orthogonality to non-orthogonality of the states of the quantum system has been investigated based on the matrix elements of the quantum states in three main physical cases: a quantum system in zero external field (closed quantum system), a quantum system in slowly changing (adiabatic) field, and a quantum system in rapidly changing (nonadiabatic) field. The quantum states are orthogonal in the first two cases and change to non-orthogonal in a nonadiabatic field, following continuously the parameters of the field. A time-dependent Fubini-Study metric of the Hilbert space can be determined based on the matrix elements of the quantum states in the nonadiabatic case. Such a dynamical metric of the Hilbert space in quantum mechanics resembles the dynamical metric of spacetime in general relativity.

Comments
5 pages, 1 figure, Submitted to the AMEE'26 Conference Proceedings

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A change in the geometry of the Hilbert space of a quantum system interacting with a nonadiabatic electromagnetic field is found. To identify its physical origin, the evolution of orthogonality to non-orthogonality of the states of the quantum system has been investigated based on the matrix elements of the quantum states in three main physical cases: a quantum system in zero external field (closed quantum system), a quantum system in slowly changing (adiabatic) field, and a quantum system in rapidly changing (nonadiabatic) field. The quantum states are orthogonal in the first two cases and change to non-orthogonal in a nonadiabatic field, following continuously the parameters of the field. A time-dependent Fubini-Study metric of the Hilbert space can be determined based on the matrix elements of the quantum states in the nonadiabatic case. Such a dynamical metric of the Hilbert space in quantum mechanics resembles the dynamical metric of spacetime in general relativity.
We show that the quadratic short time behavior of transition probability is a natural consequence of the inner product of the Hilbert space of the quantum system. We provide a relation between the survival probability and the underlying geometric structure such as the Fubini-Study metric defined on the projective Hilbert space of the quantum system. This predicts the quantum Zeno effect even for systems described by nonlinear and nonunitary evolution equations, within the collapse mechanism of the wave function during measurement process.
In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning. Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.Concretely, we consider a system of N relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the N particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.
We develop a geometric formulation of classical and quantum dynamics in which projective Hilbert space is taken as the primary state space and classical configuration and phase spaces arise as submanifolds of localized quantum states. The Fubini-Study metric induces the Euclidean metric on these submanifolds, while the tangential component of Schrödinger evolution reproduces Hamiltonian and Newtonian dynamics. We then extend the construction to stochastic motion. Brownian dynamics on the localized classical manifold is lifted to a common stochastic unitary evolution in projective Hilbert space; under homogeneity, isotropy, and independence assumptions, the corresponding Hermitian increments have Gaussian Unitary Ensemble statistics. This yields a random-matrix Schrödinger dynamics whose transition probabilities depend only on Fubini-Study distance. Through the embedding of localized states, the classical Gaussian transition law becomes the Born rule in projective state space. Measurement outcomes are identified with first arrival at detector-defined finite-resolution equivalence classes, while each realized state-space evolution remains linear and unitary. The framework is applied to classical emergence, macroscopic stabilization, double-slit experiments, composite systems, and entangled states. A spatially local Caldeira-Leggett-type system-environment model is also shown, under explicit thermal, symmetry, Ohmic, and coarse-graining assumptions, to generate the Brownian family required by the stochastic lift. The remaining open problem is to establish the range and universality of this mechanism for realistic many-body measuring environments.
We provide explicit geometric description of state manifolds obtained from evolution governed by a four-parameter family of time-independent Hamiltonians. We cover most cases related to the real interacting two-qubit systems and discuss possible types of evolutions in terms of the defining parameters. The relevant description of the pure state spaces and their Riemannian geometry with the Fubini–Study metric is given. In particular, we analyze the modification of known geometry of quantum state manifold by the linear noncommuting perturbation of the Hamiltonian. Finally, we investigate the behavior of the entanglement for obtained families of states resulting from the unitary evolution.
We study metric representations in finite-dimensional non-Hermitian quantum mechanics. The main purpose of this work is to emphasize that the physical description of a non-Hermitian system may be formulated in different, but isomorphic, Hilbert spaces. In particular, within the Krein-space formalism, we show that the vector space endowed with an indefinite Krein metric can be explicitly related to the standard Hilbert space through a suitable isomorphism. This observation is essential for a consistent description of non-Hermitian Hamiltonians. Physical states, metrics, and operators must be transported through the corresponding Hilbert-space isomorphism. In this way, equivalent representations of the same system can be used without changing the physical content of the theory. We illustrate these theoretical aspects by studying a two-level non-Hermitian spin model. We use the Robertson uncertainty relation as a consistency test. Apparent violations can arise when operators and states are kept fixed while the metric is changed, and therefore reflect a mismatch of representations rather than a failure of the uncertainty principle.