On convergence of observables in quantum mechanics

Frank J. Tipler

The Born Interpretation of the wave function gives only the relative frequencies as the number of observations approaches infinity. Using the Many-Worlds Interpretation of quantum mechanics, specifically the fact that there must exist other versions of ourselves in the multiverse, I show that the observed frequencies should approach the Born frequencies as 1/N, where N is the number of observations. In the body of the paper I state this convergence rate precisely as AN EASILY TESTABLE FORMULA. We can therefore test the central claim of the MWI by measuring the convergence rate to the final Born frequency. Conversely, the MWI allows us to calculate this convergence rate.

K.Kong Wang, Tony Jackson

Jincheng Zhang

Quantum-Classical convergence describes the inevitable merging of quantum and classical systems, often observed in complex systems exhibiting chaotic behavior. This paper explores a novel mathematical framework for understanding this convergence, proposing a hybrid approach that integrates concepts from quantum mechanics (superposition, entanglement) and classical chaos theory. We introduce a model where classical elements are intrinsically influenced by quantum fluctuations, aiming to illuminate the dynamic interplay between these two fundamental scales. The paper details the core mechanism, outlining the integration of quantum and chaotic elements into a unified system. The resulting model offers a new perspective on the emergence of patterns and stability within complex dynamical systems, particularly in the context of chaotic systems. This work provides a foundation for advanced theoretical investigations into the fundamental principles underlying the convergence of these seemingly disparate realms.

Jorge Bernal, A. Martín-Ruiz, J. C. García-Melgarejo

C. D. Cushen, R. L. Hudson

The concepts of distribution operator, stochastic independence, convergence in distribution and normal distribution are formulated for pairs of canonically conjugate quantum-mechanical momentum and position operators. It is shown that if the sequence (pn, qn), n = 1, 2, ··· is stochastically independent and identically distributed with finite covariance and zero mean then the sequence of pairs of canonical observables converges in distribution to a normal limit distribution.

Holger Neumann