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Physical Review A

Theory of fuzzy transitions between quantum and classical mechanics and proposed experimental tests

J. P. Hsu

Published Apr 1, 1991• 3 citations
Publisher

Abstract

In a generalized quantum-mechanical framework with fuzzy dynamical variables, there exists a small radical length R${10}^{\mathrm{\ensuremath{-}}17}$ cm, which is related to the physical impossibility of unlimited accuracy in measuring a particle's position. It sheds light on the problem of a detailed and gradual transition, as the mass increases, from the probabilistic behavior of microscopic objects (mcR\ensuremath{\ll}\ensuremath{\Elzxh}) to the deterministic behavior of macroscopic objects (mcR\ensuremath{\gg}\ensuremath{\Elzxh}), a problem unaddressable within the framework of ordinary quantum mechanics. A generalized equation of motion for a particle with an arbitrary mass m is postulated. It possesses a new local gauge symmetry and reduces to the Schr\"odinger and the Hamilton-Jacobi equations in the limits of small m and large m, respectively. For the double-slit experiment, the theory predicts that, as the mass increases with the momentum fixed, the interference pattern will have roughly half the usual intensity and two bright spots when mcR/\ensuremath{\Elzxh}\ensuremath{\sim}1.

DOI
10.1103/physreva.43.3227

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Physics Essays•Sep 1, 1996•8 citations
In a generalized quantum-mechanical framework with fuzzy dynamical variables, there exists a small radical length R10\ensuremath−17{10}^{\mathrm{\ensuremath{-}}17}10\ensuremath−17 cm, which is related to the physical impossibility of unlimited accuracy in measuring a particle's position. It sheds light on the problem of a detailed and gradual transition, as the mass increases, from the probabilistic behavior of microscopic objects (mcR\ensuremath{\ll}\ensuremath{\Elzxh}) to the deterministic behavior of macroscopic objects (mcR\ensuremath{\gg}\ensuremath{\Elzxh}), a problem unaddressable within the framework of ordinary quantum mechanics. A generalized equation of motion for a particle with an arbitrary mass m is postulated. It possesses a new local gauge symmetry and reduces to the Schr"odinger and the Hamilton-Jacobi equations in the limits of small m and large m, respectively. For the double-slit experiment, the theory predicts that, as the mass increases with the momentum fixed, the interference pattern will have roughly half the usual intensity and two bright spots when mcR/\ensuremath{\Elzxh}\ensuremath{\sim}1.
Based on Klauder's continuous representations in Hilbert space, we discuss a fundamental fuzziness of the coordinate and momentum variables for a quantum particle. These fuzzy dynamical variables are intimately related to a very small radical length R(\ensuremath{\lesssim}${10}^{\mathrm{\ensuremath{-}}18}$ cm) and a very large cosmological length S (\ensuremath{\gtrsim}1010{10}^{10}1010 light years). The uncertainty \ensuremath{\Delta}q for the measurement of a particle's position alone is assumed to be restricted by S\ensuremath{\gtrsim}\ensuremath{\Delta}q\ensuremath{\gtrsim}R. Physical properties of fuzzy base states (p\ensuremath{\Vert} and (q\ensuremath{\Vert} and their general cosmological and very-high-energy implications are discussed. In particular, a new density of states and its possible effects on decay rates and scattering cross sections at large momenta p\ensuremath{\gtrsim}\ensuremath{\Elzxh}/R\ensuremath{\sim}104{10}^{4}104 GeV/c (c is the velocity of light) are considered. We also discuss the superposition of waves in the presence of R and S.
Experiments violating Bell's inequality appear to indicate deterministic models do not correspond to a realistic theory of quantum mechanics. The theory of pilot waves seemingly overcomes this hurdle via nonlocality and statistical dependence, however it necessitates the existence of "ghost waves". This manuscript develops a deterministic dynamical system with local interactions. The aggregate behavior of the trajectories are reminiscent of a quantum particle evolving under the Schrödinger equation and reminiscent of Feynman's path integral interpretation in three canonical examples: motion in free space, double slit diffraction, and superluminal barrier traversal. Moreover, the system bifurcates into various dynamical regimes including a classical limit. These results illustrate a deterministic alternative to probabilistic interpretations and aims to shed light on the transition from quantum to classical mechanics.
It is shown that quantum mechanics can be regarded as what one might call a "fuzzy" mechanics whose underlying logic is the fuzzy one, in contradistinction to the classical "crisp" logic. Therefore classical mechanics can be viewed as a crisp limit of a "fuzzy" quantum mechanics. Based on these considerations it is possible to arrive at the Schroedinger equation directly from the Hamilton-Jacobi equation. The link between these equations is based on the fact that a unique ("crisp") trajectory of a classical particle emerges out of a continuum of possible paths collapsing to a single trajectory according to the principle of least action. This can be interpreted as a consequence of an assumption that a quantum "particle" "resides" in every path of the continuum of paths which collapse to a single(unique) trajectory of an observed classical motion. A wave function then is treated as a function describing a deterministic entity having a fuzzy character. As a consequence of such an interpretation, the complimentarity principle and wave-particle duality can be abandoned in favor of a fuzzy deterministic microoobject.
Geometric Derivation of Quantum Mechanics from a 5D Time-Delay Manifold This paper introduces the "Time-Delay Field" (TDF) theory, a novel geometric framework that derives the fundamental equations of quantum mechanics from a deterministic five-dimensional spacetime structure. By treating the fifth dimension as an intrinsic "delay" coordinate (τ\tauτ), we demonstrate that quantum behaviors—such as superposition and interference—are projections of continuous 5D null geodesics onto our 4D reality. Abstract:We present a geometric derivation of quantum mechanics based on the TDF theory. Extending the spacetime manifold to five dimensions (M4×S1M_4 \times S^1M4​×S1), where the fifth dimension τ\tauτ represents an intrinsic time delay, we postulate that all fundamental particles travel on null geodesics in the 5D bulk. We demonstrate that the rest mass of a particle arises from its momentum component in the delay dimension, leading to a helical trajectory. The quantum wave function ψ(x,t)\psi(x,t)ψ(x,t) is derived not as a probability amplitude, but as the Fourier projection of this 5D trajectory onto the 4D hypersurface. This framework rigorously recovers the Klein-Gordon and Schrödinger equations and provides a deterministic, geometric explanation for quantum interference and wave-particle duality. Furthermore, we address the measurement problem as a momentum transfer in the τ\tauτ-dimension and discuss consistency with recent experimental observations of symmetry-dependent quantum timing [Guo et al., 2026]. Key Contributions: Derivation of the Schrödinger Equation: Proves that the non-relativistic wave equation emerges naturally from the projection of 5D null paths. Resolution of Wave-Particle Duality: Explains wave-like interference as a topological phase accumulation (Moiré pattern) of helical trajectories. Objective Collapse Mechanism: Defines "measurement" as a momentum transfer in the delay dimension, eliminating the need for an observer. Mass-Dependent Decoherence: Introduces the "Quantumness Factor" (Q\mathcal{Q}Q), predicting an intrinsic transition from quantum to classical behavior for macroscopic masses. Experimental Alignment: Consistent with recent findings on geometry-dependent time delays in attosecond physics. Keywords: Quantum Foundations, Time-Delay Field, 5D Spacetime, Null Geodesics, Kaluza-Klein Theory, Measurement Problem, Emergent Quantum Mechanics, Geometric Phase.
It is shown that quantum mechanics can be regarded as what one might call a "fuzzy" mechanics whose underlying logic is the fuzzy one, in contradistinction to the classical "crisp" logic.Therefore classical mechanics can be viewed as a crisp limit of a "fuzzy" quantum mechanics.Based on these considerations it is possible to arrive at the Schroedinger equation directly from the Hamilton-Jacobi equation.The link between these equations is based on the fact that a unique ( "crisp") trajectory of a classical particle emerges out of a continuum of possible paths collapsing to a single trajectory according to the principle of least action.This can be interpreted as a consequence of an assumption that a quantum "particle" "resides" in every path of the continuum of paths which collapse to a single( unique) trajectory of an observed classical motion.A wave function then is treated as a function describing a deterministic entity having a fuzzy character.As a consequence of such an interpretation, the complimentarity principle and wave-particle duality can be abandoned in favor of a fuzzy deterministic microoobject.