We explore Poincar´e recurrence as a mechanism for generating quantum-likestatistics from recurrent classical dynamics. Modeling the universe as a singlerecurrent, ergodic trajectory on a compact phase space, we show that an embedded observer’s ignorance of their location along the recurrent trajectory definesa natural ensemble. Using the Kolmogorov Extension Theorem, we constructa Kolmogorov-consistent, normalizable measure on the induced path space. Wefurther show that recurrent loop topology fixes the complex phase weighting ofhistories, reconstructing the Feynman path integral. The Born rule emerges fromthe same consistency requirements.
We explore Poincar´e recurrence as a mechanism for generating quantum-likestatistics from recurrent classical dynamics. Modeling the universe as a singlerecurrent, ergodic trajectory on a compact phase space, we show that an embedded observer’s ignorance of their location along the recurrent trajectory definesa natural ensemble. Using the Kolmogorov Extension Theorem, we constructa Kolmogorov-consistent, normalizable measure on the induced path space. Wefurther show that recurrent loop topology fixes the complex phase weighting ofhistories, reconstructing the Feynman path integral. The Born rule emerges fromthe same consistency requirements.
Poincaré's recurrence theorem, which states that every Hamiltonian dynamics enclosed in a finite volume returns to its initial position as close as one wishes, is a mathematical basis of statistical mechanics. It is Liouville's theorem that guarantees that the dynamics preserves the volume on the state space. A quantum version of Poincaré's theorem was obtained in the middle of the 20th century without any volume structures of the state space (Hilbert space). One of our aims in this paper is to establish such properties of quantum dynamics from an analog of Liouville's theorem, namely, we will construct a natural probability measure on the Hilbert space from a Hamiltonian defined on the space. Then we will show that the measure is invariant under the corresponding Schrödinger flow. Moreover, we show that the dynamics naturally causes an infinite-dimensional Weyl transformation. It also enables us to discuss the ergodic properties of such dynamics.
An analytical consideration of quantum walks in the Hilbert space is suggested for dynamical systems. It is shown that in the semiclassical limit, statistics of the quantum recurrences relates to statistics of the Poincare recurrences of the classical counterpart. It is shown that the statistics of the quantum recurrences is sensitive to the statistics of the corresponding quantum spectrum. The difference in the statistics of quantum recurrences in the Hilbert space for the chaotic and integrable systems follows from the essential difference between the level statistics of integrable and chaotic systems. In particular, when the integrable part of the phase space emerges due to bifurcation, and the exponential distribution of the Poincar´e recurrences of chaotic trajectories is changed into the power law, the statistics of the quantum walks in the Hilbert space follows exactly its classical counterpart.
This paper develops a neural architecture capable of reproducing several structural aspects of quantum mechanics. The proposed model, termed the Born Perceptron, extends the classical perceptron by structuring outputs as value–index pairs and interpreting normalized activations as probability distributions over indexed outcomes. By allowing indices to inhabit geometric manifolds rather than discrete labels, the perceptron naturally represents probabilistic sampling over state spaces. When stacked sequentially, such perceptrons generate stochastic trajectories reminiscent of quantum state evolution under repeated measurement. Introducing complex-valued activations yields probabilities determined by squared magnitudes, reproducing the Born rule. The framework is further extended to treat measurements distributed in time, particularly position samplings. In this regime, the neural system produces sequences of sampled positions whose joint probabilities resemble those derived in path-integral formulations of quantum mechanics. The resulting architecture offers a classical computational model that captures several conceptual features of quantum theory including measurement, collapse, trajectory evolution, and amplitude-based probability rules.
I prove a local uniqueness theorem for the Born rule in the setting of quiver-generated categories equipped with complex morphism weights and path-amplitude probability functionals. Given (i) non-negativity, (ii) invariance of bounded total degree, (iii) global U(1) invariance, (iv) classical-limit additivity over mutually exclusive paths, and (v) normalization, I show that the probability assignment P: C -> R>=0 is uniquely determined to be P(z) = |z|^2. The notion of mutually exclusive paths is given a precise categorical formulation as the absence of a shared factorization through any common morphism. I relate the result to reconstructions of quantum probability due to Gleason, Hardy, Chiribella-D'Ariano-Perinotti, and several further, more recent reconstructions, and identify the extension to global coherence under morphism composition as an open problem connected to synthetic probability theory in Markov categories. The Born rule emerges as the unique locally consistent probability law on complex amplitudes, fixed by phase invariance and classical-limit behavior together with polynomiality and normalization, independent of any Hilbert-space framework.
We present a formulation for the emergence of classical dynamics in a quantum world using a path integral approach that incorporates continuous measurements. Our approach complements decoherence and coarse-grained quantum-to-classical transition frameworks. The path-integral formulation provides the joint statistics of a sequence of measurements, with each Feynman path picking up an additional random phase. Its magnitude is proportional to the measurement strength, and we give conditions under which the dominant contribution to the probability amplitude comes from trajectories near classical paths. Information proliferates across the environment, a key feature of quantum Darwinism, via plane-wave probe scattering. Extending to repeated measurements, we show that in the continuous limit each system trajectory picks up an additional phase due to momentum kicks from the probes--the origin of the back-action force. We provide conditions under which measurements yield enough ``which-path'' information while keeping the wave packet localized. This allows the quantum-to-classical transition to be described from individual measurement records, complementing the ensemble description from density matrices. We further show that the same scattering that decoheres a trajectory heats it, tying decoherence and measurement back-action together. This bounds how redundantly an individual classical trajectory can be recorded before back-action randomises it into Brownian motion. For a trapped particle, the ceiling is fixed by the resolution measured in units of the zero-point motion. It is not restrictive for macroscopic systems; it collapses to a single record where the semiclassical description of a trajectory fails, delimiting the regime in which objective classical trajectories exist. The deterministic-to-Brownian crossover is accessible in levitated optomechanics.
Abstract. I prove a local uniqueness theorem for the Born rule in the setting of small categories equipped with complex morphism weights and path-amplitude probability functionals. Given (i) non-negativity, (ii) polynomiality of bounded total degree, (iii) global U(1) invariance, (iv) classical-limit additivity over mutually exclusive paths, and (v) normalization, I show that the probability assignment P : ℂ → ℝ≥₀ is uniquely determined to be P(z) = |z|^2. The notion of mutually exclusive paths is given a precise categorical formulation as the absence of a shared factorization through any common morphism. I relate the result to reconstructions of quantum probability due to Gleason, Hardy, and Chiribella–D’Ariano–Perinotti, and identify the extension to global coherence under morphism composition as an open problem connected to synthetic probability theory in Markov categories. The Born rule emerges as the unique locally consistent probability law on complex amplitudes, fixed solely by phase invariance and classical-limit behavior, independent of any Hilbert-space framework.