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Conference Publications

Stochastic deformation of classical mechanics

Jean-Claude Zambrini

Published Jan 1, 2013• 2 citations
Publisher

Abstract

We describe a method of stochastic deformation of classical mechanics preserving the time symmetry of this theory. It provides a new general strategy to deform stochastically Geometric Mechanics.
DOI
10.3934/proc.2013.2013.807

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A method is presented by which a geometrical interpretation can be given for boson models without leaving the domain of quantum mechanics. This approach turns out to be particularly useful in symmetry-preserving treatments of deformed boson systems. A U(3) boson model having an SO(2) symmetry, which is to be preserved throughout, is analyzed in detail. For a well-deformed system, rotational and vibrational modes are identified. A procedure is developed by which the parameters of the geometrical description for the deformed case can be calculated in a series expansion controlled by the inverse of the boson number. The method is compared with the corresponding random phase approximation approach.
This is an introduction to a stochastic version of E. Cartan's symplectic mechanics. A class of time-symmetric (Bern- stein) diusion processes is used to deform stochastically the ex- terior derivative of the Poincare-Cartan one-form on the extended phase space. The resulting symplectic two-form is shown to contain the (a.e.) dynamical laws of the diusions. This can be regarded as a geometrization of Feynman's path integral approach to quantum theory; when Planck's constant reduce to zero, we recover Cartan's mechanics. The underlying strategy is the one of Euclidean Quan- tum Mechanics.
This paper presents a complete and verifiable theoretical framework that unifies quantum mechanics and classical physics/general relativity under a single mathematical function: H(S,θ,K)=Fclassical(S)⋅Q(θ)+∇K⋅(ΠK[δS])H(S,θ,K)=Fclassical(S)⋅Q(θ)+∇K⋅(ΠK[δS]) We propose that quantum mechanics is not a fundamental separate theory, but rather a structural regulatory mechanism activated by specific boundary conditions within a fundamental continuous space-time. The theory introduces three core components: S: A unified continuous structural field representing both space-time and matter Q(θ): A context-dependent switching function that transitions between classical (Q=1) and quantum (Q=0) regimes K: An inherent geometric framework that governs all structural deformations The theory operates through three mathematically rigorous conditions: Pure Quantum Space (Q=0): Geometric deformations dominate via K Classical Physics + GR (Q=1): Reduces exactly to Einstein/Newton equations Local Quantum Effects (0<Q<1): Hybrid regime preserving fundamental constants Key innovations include: Resolution of the measurement problem through natural θ-transitions Preservation of all classical laws (c, G, ħ) as global constants Elimination of the quantum-gravity coupling problem Falsifiable predictions through approximate self-similarity analysis We propose three mathematical candidates for K (E₈ symmetry, Hodge-Laplacian, Spin Network) with specific experimental predictions in atomic spectra, optomechanics, and black hole entropy. The theory provides a concrete research program for 2026-2030, transitioning physics from fragmented descriptions to a unified dynamical system. This work represents not just an equation, but an ecosystem for unified physics—a testable speculative paradigm that enriches theoretical discourse regardless of its ultimate verification status.