Conditional expectations in Quantum Mechanics and causal interpretations: the Bohm momentum as a best predictor

TL;DR

This paper introduces a new perspective on quantum conditional expectations, especially the Bohm momentum, and develops a Bohm-type model for particles with spin, linking quantum and classical mechanics through conditional expectations.

Contribution

It defines quantum conditional expectations as best approximations and develops a Bohm-type model for spin particles using a classical-stochastic approach.

Findings

Bohm momentum as a conditional expectation of momentum given position.

Derived dynamics of momentum and spin conditional expectations.

Established a classical-stochastic particle dynamics compatible with quantum evolution.

Abstract

Given a normalized state-vector $\psi $, we define the conditional expectation $\mathbb{E }_{\psi } (A | B ) $ of a Hermitian operator $A $ with respect to a strongly commuting family of self-adjoint operators $B $ as the best approximation, in the operator mean square norm associated to $\psi $, of $A $ by a real-valued function of $B . $ A fundamental example is the conditional expectation of the momentum operator $P $ given the position operator $X $, which is found to be the Bohm momentum. After developing the Bohm theory from this point of view we treat conditional expectations with respect to general $B $, which we apply to non-relativistic spin 1/2-particles. We derive the dynamics of the conditional expectations of momentum and spin with respect to position and the third spin component. These dynamics can be interpreted in terms of classical continuum mechanics as a two-component fluid whose components carry intrinsic angular momentum. Interpreting the joint spectrum of the conditioning operators as a space of beables, we can introduce a classical-stochastic particle dynamics on this space which is compatible with the time-evolution of the Born probability, by combining the de Broglie-Bohm guidance condition with a Markov jump process, following an idea of J. Bell. This results in a new Bohm-type model for particles with spin. A basic problem is that such auxiliary particle dynamics are far from unique. We finally examine the relation of our conditional expectations with the conditional expectations of the theory of $C^* $-algebras and, as an application, derive a general evolution equation for conditional expectations for operators acting on finite dimensional Hilbert spaces. Two appendices re-interpret the classical Bohm model as an integrable constrained Hamiltonian system, and provide the details of the two-fluid interpretation.