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Emergence of Classical Dynamics from a Random Matrix Schrödinger Model

Alexey A. Kryukov

Abstract

The Newtonian motion of a macroscopic particle is derived from the linear Schrödinger equation with a Hamiltonian consisting of the free-particle term and a random Hamiltonian drawn from the Gaussian Unitary Ensemble. The random term models interaction with the environment. We show that the parameters governing the resulting state-space random walk, together with the treatment of experimentally indistinguishable states as equivalence classes, explain the contrasting behavior of microscopic and macroscopic systems. The analysis extends previous work deriving the Born rule for microscopic particles when the free-particle term is negligible.

Emergence of Classical Dynamics from a Random Matrix Schrödinger Model

Abstract

The Newtonian motion of a macroscopic particle is derived from the linear Schrödinger equation with a Hamiltonian consisting of the free-particle term and a random Hamiltonian drawn from the Gaussian Unitary Ensemble. The random term models interaction with the environment. We show that the parameters governing the resulting state-space random walk, together with the treatment of experimentally indistinguishable states as equivalence classes, explain the contrasting behavior of microscopic and macroscopic systems. The analysis extends previous work deriving the Born rule for microscopic particles when the free-particle term is negligible.
Paper Structure (19 sections, 92 equations)

This paper contains 19 sections, 92 equations.

Table of Contents

  1. Motivation and Overview
  2. Dynamics on state space
  3. Main Result: Emergence of Newtonian Dynamics for Macroscopic Bodies
  4. Tangent displacement during one collision.
  5. Orthogonal spreading during one collision.
  6. Effective time step.
  7. Step variance and diffusion coefficient.
  8. Magnitude of stochastic corrections.
  9. Stroboscopic Newtonian motion.
  10. Comparison with existing approaches
  11. Relation to decoherence theory.
  12. Relation to continuous measurement and unravelings.
  13. Difference from the Ehrenfest limit.
  14. Relation to collapse models
  15. Supplementary material
  16. ...and 4 more sections