Justifying Born’s Rule Pα = |Ψα|2 Using Deterministic Chaos, Decoherence, and the de Broglie–Bohm Quantum Theory

Abstract

In this work, we derive Born’s rule from the pilot-wave theory of de Broglie and Bohm. Based on a toy model involving a particle coupled to an environment made of “qubits” (i.e., Bohmian pointers), we show that entanglement together with deterministic chaos leads to a fast relaxation from any statistical distribution ρ(x) of finding a particle at point x to the Born probability law |Ψ(x)|2. Our model is discussed in the context of Boltzmann’s kinetic theory, and we demonstrate a kind of H theorem for the relaxation to the quantum equilibrium regime.

A Decoherence-Based Approach to the Classical Limit in Bohm’s Theory

“…The effective wave functions produced by the interaction with the environment, which we may call environmentally-selected effective wave functions (ES-EWFs) are well-localized states. This follows from the diagonalization of the reduced density matrix described by the subsystem master equations 7 , which trans-forms an initial pure state into an (improper) mixture of well-localized states. 8 Such states are generally called pointer states: these are the states that survive the decoherence process and that remain stable during the interaction with the environment.…”

“…A comparative review of the two approaches has been made by Norsen (2018). See also Drezet (2021), for a recent proposal to justify the Born's rule using a decoherence framework. 14 For example: the quantum force is what makes the particles' trajectories deviate from straight lines in the two-slit experiment, even if there is no classical force acting on the Bohmian particles between the slits and the final screen.…”

Whence Nonlocality? Removing Spooky Action-at-a-Distance from the de Broglie Bohm Pilot-Wave Theory Using a Time-Symmetric Version of the de Broglie Double Solution

“…As an illustration, consider the case of a single dBB particle interacting with a 50/50 beam splitter as studied for example in [65] (see Figure 2a).…”

Chaos and ergodicity in a partially integrable 3d Bohmian system: a comparison with 2d systems

“…The existence of ergodicity led us to interesting results [15] on the establishment of Born's Rule (BR), which states that the probability density of finding a particle in a certain region of space is given by P(x, t) = |Ψ(x, t)| 2 (for the origin of BR in BQM see [18][19][20][21][22][23][24][25][26][27]). If BR is initially satisfied (P 0 = |Ψ 0 | 2 ) then it remains valid at all times.…”