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Indivisible Effective Dynamics of Coarse-Grained Observers

Ryan Williams

Published May 28, 2026• 0 citations
Publisher

Abstract

Recent work has shown equivalence between the formal apparatus of quantum mechanics and a general class of non-Markovian systems called indivisible stochastic systems. This short paper demonstrates how these indivisible dynamical systems can be used to describe the effective dynamics of coarse-grained observers. The observation suggests that quantum mechanics may arise as the expected phenomenology of coarse-grained systems in a classical, deterministic substrate.

DOI
10.5281/zenodo.20428560

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Recent work has shown equivalence between the formal apparatus of quantum mechanics and a general class of non-Markovian systems called indivisible stochastic systems. This short paper demonstrates how these indivisible dynamical systems can be used to describe the effective dynamics of coarse-grained observers. The observation suggests that quantum mechanics may arise as the expected phenomenology of coarse-grained systems in a classical, deterministic substrate.
The origin of the phenomenological deterministic laws that approximately govern the quasiclassical domain of familiar experience is considered in the context of the quantum mechanics of closed systems such as the universe as a whole. A formulation of quantum mechanics is used that predicts probabilities for the individual members of a set of alternative coarse-grained histories that decohere, which means that there is negligible quantum interference between the individual histories in the set. We investigate the requirements for coarse grainings to yield decoherent sets of histories that are quasiclassical, i.e., such that the individual histories obey, with high probability, effective classical equations of motion interrupted continually by small fluctuations and occasionally by large ones. We discuss these requirements generally but study them specifically for coarse grainings of the type that follows a distinguished subset of a complete set of variables while ignoring the rest. More coarse graining is needed to achieve decoherence than would be suggested by naive arguments based on the uncertainty principle. Even coarser graining is required in the distinguished variables for them to have the necessary inertia to approach classical predictability in the presence of the noise consisting of the fluctuations that typical mechanisms of decoherence produce. We describe the derivation of phenomenological equations of motion explicitly for a particular class of models. Those models assume configuration space and a fundamental Lagrangian that is the difference between a kinetic energy quadratic in the velocities and a potential energy. The distinguished variables are taken to be a fixed subset of coordinates of configuration space. The initial density matrix of the closed system is assumed to factor into a product of a density matrix in the distinguished subset and another in the rest of the coordinates. With these restrictions, we improve the derivation from quantum mechanics of the phenomenological equations of motion governing a quasiclassical domain in the following respects: Probabilities of the correlations in time that define equations of motion are explicitly considered. Fully nonlinear cases are studied. Methods are exhibited for finding the form of the phenomenological equations of motion even when these are only distantly related to those of the fundamental action. The demonstration of the connection between quantum-mechanical causality and causality in classical phenomenological equations of motion is generalized. The connections among decoherence, noise, dissipation, and the amount of coarse graining necessary to achieve classical predictability are investigated quantitatively. Routes to removing the restrictions on the models in order to deal with more realistic coarse grainings are described.
Understanding how classical physics emerges from quantum mechanics remains a central problem in the foundations of physics. Here we derive a classical limit from finite-resolution measurements, modeled by continuous coarse-grained POVMs. When the resolved phase-space area is large compared with Planck's constant, the accessible statistics of any quantum state admit an effective classical description: coarse-grained observables become approximately jointly measurable and the induced probability density is positive. We derive the exact evolution equation for this density and show that, in the strong coarse-graining regime, its non-Liouville corrections are suppressed up to an Ehrenfest time, resulting in classical Hamiltonian flow generated by a Hamiltonian smoothed over the measurement cell. When the Hamiltonian varies negligibly across such a cell, the smoothed Hamiltonian reduces to the classical Hamiltonian whose quantization produced the quantum dynamics, thereby closing the quantization--classical-limit loop. Repeated finite-resolution measurements then generate stochastic records confined, with high probability, to tubes around classical trajectories. Our results provide a unified operational framework for the quantum-to-classical transition in microscopic and macroscopic systems, and establish the consistency of the quantization--classical-limit cycle.
Coarse-graining of some sort is a fundamental and unavoidable step in any attempt to derive the classical mechanical behavior from the quantum formalism. We utilize the two-mode Bose—Hubbard model to illustrate how different coarse-grained systems can be naturally associated with a fixed quantum system if it is compatible with different dynamical algebras. Alternative coarse-grained systems generate different evolutions of the same physical quantities, and the difference becomes negligible only in the appropriate macro-limit.
Constrained Hamiltonian dynamics is exploited to provide the mathematical framework of a coarse-grained description of the quantum system of two interacting qubits and of nonlinear interacting oscillators. The coarse-graining is treated as an equivalence relation on the set of quantum states resulting in the emergence of the classical phase-space. The equivalence relation imposes constraints on the Hamiltonian dynamics of the quantum system. It is seen that the evolution of the coarse-grained system preserves constant and minimal quantum fluctuations of the fundamental observables. This leads to the emergence of typical classical properties, like the relation between symmetry and integrability, and in the case of oscillators in the macro-limit implies the emergence of the classical system.
Starting from a coarse-grained map of a quantum many-body system, we construct the inverse map that assigns a microscopic state to a coarse-grained state based on the maximum entropy principle. Assuming unitary evolution in the microscopic system, we examine the resulting dynamics in the coarse-grained system using the assignment map. We investigate both a two-qubit system, with \swap\ and controlled-\textsc{not}\ gates, and nnn-qubit systems, configured either in an Ising spin chain or with all-to-all interactions. We demonstrate that these dynamics exhibit atypical quantum behavior, such as nonlinearity and non-Markovianity. Furthermore, we find that these dynamics depend on the initial coarse-grained state and establish conditions for general microscopic dynamics under which linearity is preserved. As the effective dynamics induced by our coarse-grained description of many-body quantum systems diverge from conventional quantum behavior, we anticipate that this approach could aid in describing the quantum-to-classical transition and provide deeper insights into the effects of coarse graining on quantum systems.