Quantum, Stochastic, and Classical Dynamics Within A Single Geometric Framework

• URL: http://arxiv.org/abs/2510.27170v1
• Date: Fri, 31 Oct 2025 04:46:39 GMT
• Title: Quantum, Stochastic, and Classical Dynamics Within A Single Geometric Framework
• Authors: Partha Ghose,
• Abstract summary: We show that the Koopman--von Neumann (KvN) phase of classical mechanics emerges naturally as the $lambda to 1$ limit of this $sigma$--$lambda$ hierarchy.<n>This unified picture links quantum, and classical dynamics within a single continuous framework.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Nelson's stochastic mechanics links quantum mechanics to an underlying Brownian motion with the identification $\hbar = m\sigma$. Ghose's interpolating equation introduces a continuous parameter $\lambda$ that suppresses the quantum potential $Q[\psi]$ and yields a smooth transition between quantum ($\lambda=0$) and classical ($\lambda=1$) regimes. In this short note, we show that the Koopman--von Neumann (KvN) Hilbert-space formulation of classical mechanics emerges naturally as the $\lambda \to 1$ limit of this stochastic $\sigma$--$\lambda$ hierarchy. The KvN phase-space amplitude provides an operator representation of the classical Liouville equation, while the $\lambda$ parameter acts as a projection flow from the complex projective Hilbert manifold $\mathbb{C}P^n$ to its classical quotient $\mathbb{C}P^*/U(1)$, implementing phase superselection. This unified picture links quantum, stochastic, and classical dynamics within a single continuous framework.

Related papers

• Transmutation based Quantum Simulation for Non-unitary Dynamics [35.35971148847751]
  We present a quantum algorithm for simulating dissipative diffusion dynamics generated by positive semidefinite operators of the form $A=Ldagger L$.<n>Our main tool is the Kannai transform, which represents the diffusion semigroup $e-TA$ as a Gaussian-weighted superposition of unitary wave propagators.
  arXiv  Detail & Related papers  (2026-01-07T05:47:22Z)
• Discrete symmetries in classical and quantum oscillators [51.56484100374058]
  We show the eigenfunctions $_n=zn$ of the quantum Hamiltonian in the complex Bargmann-Fock-Segal representation.<n>The superposition $=sum_n c_n_n$ arises only with incomplete knowledge of the initial data for solving the Schrdinger equation.
  arXiv  Detail & Related papers  (2026-01-05T10:04:39Z)
• The threshold for quantum-classical correspondence is $D \sim \hbar^{\rac43}$ [0.3623365995586146]
  In chaotic quantum systems, an initially localized quantum state can deviate strongly from the corresponding classical phase-space distribution.<n>We show that the scaling $D sim hbarfrac43$ is indeed the threshold for quantum-classical correspondence beyond the Ehrenfest time.
  arXiv  Detail & Related papers  (2025-12-19T14:28:41Z)
• Quantum quenches across continuous and first-order quantum transitions in one-dimensional quantum Ising models [0.0]
  We investigate the quantum dynamics generated by quantum quenches (QQs) of the Hamiltonian parameters in many-body systems.<n>We consider the quantum Ising chain in the presence of homogeneous transverse ($g$) and longitudinal ($h$) magnetic fields.
  arXiv  Detail & Related papers  (2025-12-19T08:24:50Z)
• Slow Mixing of Quantum Gibbs Samplers [47.373245682678515]
  We present a quantum generalization of these tools through a generic bottleneck lemma.<n>This lemma focuses on quantum measures of distance, analogous to the classical Hamming distance but rooted in uniquely quantum principles.<n>We show how to lift classical slow mixing results in the presence of a transverse field using Poisson Feynman-Kac techniques.
  arXiv  Detail & Related papers  (2024-11-06T22:51:27Z)
• Scattering Neutrinos, Spin Models, and Permutations [42.642008092347986]
  We consider a class of Heisenberg all-to-all coupled spin models inspired by neutrino interactions in a supernova with $N$ degrees of freedom. These models are characterized by a coupling matrix that is relatively simple in the sense that there are only a few, relative to $N$, non-trivial eigenvalues.
  arXiv  Detail & Related papers  (2024-06-26T18:27:15Z)
• Klein-Gordon oscillators and Bergman spaces [55.2480439325792]
  We consider classical and quantum dynamics of relativistic oscillator in Minkowski space $mathbbR3,1$. The general solution of this model is given by functions from the weighted Bergman space of square-integrable holomorphic (for particles) and antiholomorphic functions on the K"ahler-Einstein manifold $Z_6$.
  arXiv  Detail & Related papers  (2024-05-23T09:20:56Z)
• Classical correspondence beyond the Ehrenfest time for open quantum systems with general Lindbladians [1.497411457359581]
  Quantum and classical systems evolving under the same formal Hamiltonian $H$ may dramatically differ after the Ehrenfest timescale.<n>We prove a version of this correspondence, bounding the error between the quantum and classical evolutions for any sufficiently regular Hamiltonian $H(x,p)$ and Lindblad functions $L_k(x,p)$.<n>In a shorter companion paper, we treat the special case of Hamiltonians that decompose into kinetic and potential energy with linear Lindblad operators.
  arXiv  Detail & Related papers  (2023-07-07T17:01:23Z)
• Composite quantum Coriolis forces [0.0]
  Coriolis force finds its quantum analogue in the difference $Sigma(t)=H(t)-G(t)$ where the true'', observable Hamiltonian $H(t)$ represents the instantaneous energy. The other, false'' Hamiltonian $G(t)$ generates the time-evolution of wave functions.
  arXiv  Detail & Related papers  (2023-03-07T22:18:20Z)
• On the Operator Origins of Classical and Quantum Wave Functions [0.0]
  We introduce a formalism of Operator Mechanics based on a noncommutative Poisson, symplectic and noncommutative differential structures. We show that the Schr"odinger equation is obtained from the Koopman-von Neumann equation. What this suggests is that neither the Schr"odinger equation nor the quantum wave function are fundamental structures.
  arXiv  Detail & Related papers  (2022-11-03T14:19:37Z)
• On quantum algorithms for the Schr\"odinger equation in the semi-classical regime [27.175719898694073]
  We consider Schr"odinger's equation in the semi-classical regime. Such a Schr"odinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics.
  arXiv  Detail & Related papers  (2021-12-25T20:01:54Z)
• Speeding up Learning Quantum States through Group Equivariant Convolutional Quantum Ans\"atze [13.651587339535961]
  We develop a framework for convolutional quantum circuits with SU$(d)$symmetry. We prove Harrow's statement on equivalence between $nameSU(d)$ and $S_n$ irrep bases.
  arXiv  Detail & Related papers  (2021-12-14T18:03:43Z)
• Quantum dynamics and relaxation in comb turbulent diffusion [91.3755431537592]
  Continuous time quantum walks in the form of quantum counterparts of turbulent diffusion in comb geometry are considered. Operators of the form $hatcal H=hatA+ihatB$ are described. Rigorous analytical analysis is performed for both wave and Green's functions.
  arXiv  Detail & Related papers  (2020-10-13T15:50:49Z)
• Classical Dynamics from Self-Consistency Equations in Quantum Mechanics -- Extended Version [0.0]
  We propose a new mathematical approach to Bona's non-linear generalization of quantum mechanics. It highlights the central role of self-consistency. Some new mathematical concepts are introduced, which are possibly interesting by themselves.
  arXiv  Detail & Related papers  (2020-09-10T16:20:25Z)
• Quantum dynamics of the classical harmonic oscillator [0.0]
  A correspondence is established between measure-preserving, ergodic dynamics of a classical harmonic oscillator and a quantum mechanical gauge theory on two-dimensional Minkowski space.
  arXiv  Detail & Related papers  (2019-12-27T21:00:10Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.