The Attractor and the Quantum States

TL;DR

This paper constructs a dissipative dynamics model that reproduces quantum behavior from a deterministic classical framework, demonstrating how quantum states emerge as attractors in an extended phase space.

Contribution

It explicitly develops a dissipative extension of Liouville's equation that models the emergence of quantum states and dynamics, generalizing classical phase space concepts.

Findings

Quantum states emerge as stable attractors in the extended phase space.

Expectations of observables align with the Born rule without being imposed.

The model applies to interacting systems, including field theories.

Abstract

The dissipative dynamics anticipated in the proof of 't Hooft's existence theorem -- "For any quantum system there exists at least one deterministic model that reproduces all its dynamics after prequantization" -- is constructed here explicitly. We propose a generalization of Liouville's classical phase space equation, incorporating dissipation and diffusion, and demonstrate that it describes the emergence of quantum states and their dynamics in the Schroedinger picture. Asymptotically, there is a stable ground state and two decoupled sets of degrees of freedom, which transform into each other under the energy-parity symmetry of Kaplan and Sundrum. They recover the familiar Hilbert space and its dual. Expectations of observables are shown to agree with the Born rule, which is not imposed a priori. This attractor mechanism is applicable in the presence of interactions, to few-body or field theories in particular.