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arXiv:2611.02138

Classical and Quantum Dynamics in State Space: Projective Hilbert Space, Measurement, and Classical Emergence

Alexey A. Kryukov

quant-phmath-phSubmitted Nov 1, 2026• Announced Nov 3, 2026
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Abstract

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.

Comments
240 pages. Forthcoming monograph

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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 develop a state-space geometric framework for measurement, classicality, and quantum paradoxes, based on one dynamical conjecture. Classical configuration space and classical phase space for a mechanical system arise as distinguished submanifolds of projective quantum state space. On these submanifolds, the Fubini--Study geometry induces Euclidean classical geometry, and the tangent component of Schrödinger evolution reproduces Newtonian dynamics. Within this framework, interactions with measuring devices and environments are described by random-matrix dynamics on projective state space, generated by matrices drawn from the Gaussian Unitary Ensemble. We show that this random-matrix dynamics yields isotropic diffusion, giving Born-rule transition probabilities in microscopic measurements and stabilizing classical behavior in macroscopic systems. We further argue that the random-matrix conjecture is not an independent ad hoc assumption: under natural translation-invariance assumptions on the distribution of state-space steps originating on the classical submanifold, the unitary lift of homogeneous and isotropic Brownian motion on that submanifold is uniquely given by the Gaussian Unitary Ensemble, up to scale and an irrelevant scalar part. The resulting framework provides a unitary account of measurement and the quantum-to-classical transition and, if accepted, offers a dynamical resolution of standard quantum paradoxes.
We develop a dynamical framework for quantum measurement based on stochastic but unitary evolution in projective state space. Random Hamiltonians drawn from the Gaussian Unitary Ensemble generate stochastic unitary dynamics of the quantum state, while equivalence classes reflecting finite detector resolution define classical observables as well as classical configuration-space and phase-space submanifolds. When the evolution is constrained to the phase-space submanifold, free Schrödinger dynamics reduces to Newtonian motion, while stochastic motion constrained to the classical configuration-space submanifold yields ordinary Brownian motion in classical space. Transition probabilities under the stochastic dynamics satisfy the Born rule, whereas the constrained classical evolution produces the normal probability distributions characteristic of classical measurements. We show that, in this setting, measurement, state reduction, and the quantum-classical transition emerge from unitary dynamics alone, without invoking nonunitary collapse or additional postulates. Entanglement and EPR correlations arise geometrically from the evolution of joint states in composite state space, preserving locality in spacetime. The framework provides a unified dynamical account of measurement and classicality compatible with the structure of quantum mechanics.
It has been recently shown that Newtonian dynamics is the Schrodinger dynamics of the system whose state is constrained to a submanifold in the space of states of the system. Thus defined, the submanifold can be identified with the classical phase space of the system. The classical space is then also embedded into the space of states in a physically meaningful way. The resulting unified geometric framework establishes a new connection between classical and quantum physics. The framework is rigid in the sense that the Schrodinger dynamics is a unique extension of the Newtonian one from the classical phase space submanifold to the space of states. Quantum observables in the framework are identified with vector fields on the space of states. The commutators of canonical conjugate observables are expressed through the curvature of the sphere of normalized states. The velocity and acceleration of a particle in Newtonian dynamics are components of the velocity of state under the corresponding Schrodinger evolution. The metric properties of the embedding of the classical space into the space of states result in a relationship between the normal distribution of the position of a particle and the Born rule for the probability of transition of quantum states. In this paper the implications of the obtained mathematical results to the process of measurement in quantum physics are investigated. It is argued that interaction with the environment constrains the state of a macroscopic body to the classical space. The notion of collapse of a quantum state is analyzed. The double-slit, EPR and Schrodinger cat type experiments are reviewed anew. It is shown that, despite reproducing the usual results of quantum theory, the framework is not simply a reformulation of the theory. New experiments to discover the predicted effects are proposed.
It was recently shown that the Born rule for transition probabilities of a microscopic particle can be derived from the linear Schr{\"o}dinger equation with a Hamiltonian represented by random matrices from the Gaussian Unitary Ensemble, taken to be independent at different time points. Under such an evolution, the particle's state undergoes an isotropic random walk on the projective state space, and the relative frequencies of reaching different regions obey the Born rule. In this work, we extend these results to demonstrate how the same framework accounts for the emergence of Newtonian dynamics for macroscopic bodies on a submanifold of state space, thereby providing a unified geometric account of the quantum-classical transition.
It was recently shown that Newtonian dynamics of macroscopic particles can be derived from unitary Schrödinger evolution under a random-matrix assumption on the system-environment interaction. In that framework, classical phase space is realized geometrically as a manifold of localized equivalence classes in quantum state space, the tangent component of Schrödinger evolution reproduces Newtonian motion, and environmental interactions stabilize the state near this manifold. We extend this framework to quantum fields. The field itself is not assumed to become classical. Instead, macroscopic particles stabilized near the classical particle manifold interact with the field through the sector of field state space accessible to localized particle dynamics. The classical field is represented by the corresponding localized sector, and finite probe resolution leads to a quotient description in terms of localized equivalence classes of field states. The tangent component of the quantum-field Schrödinger dynamics on this localized quotient sector yields the corresponding classical field equations. Finite-dimensional simulations illustrate the mechanism for scalar and electromagnetic fields. The accessible field coordinates satisfy the sourced Klein--Gordon and Maxwell equations, and a localized test charge responds to the electromagnetic field through the Lorentz force. Thus classical field behavior emerges within unitary Schrödinger dynamics, without identifying the classical field with an expectation value, without relying on coherent states as special physical states, and without introducing a nonunitary collapse postulate.
Newtonian and Schrodinger dynamics can be formulated in a physically meaningful way within the same Hilbert space framework. This fact was recently used to discover an unexpected relation between classical and quantum motions that goes beyond the results provided by the Ehrenfest theorem. The Newtonian dynamics was shown to be the Schrodinger dynamics of states constrained to a submanifold of the space of states, identified with the classical phase space of the system. Quantum observables are identified with vector fields on the space of states. The commutators of observables are expressed through the curvature of the space. The resulting embedding of the Newtonian and Schrodinger dynamics into a unified geometric framework is rigid in the sense that the Schrodinger dynamics is a unique extension of the Newtonian one. Furthermore, under the embedding, the normal distribution of measurement results associated with a classical measurement implies the Born rule for the probability of transition of quantum states. In this paper, the implications of the obtained theory to the process of measurement in quantum theory are analyzed. The double-slit, EPR and Schrodinger cat type experiments are reviewed anew. It is shown that, despite reproducing the usual results of quantum theory, the framework is not simply a reformulation of the theory. New experiments to discover the predicted effects are proposed.