Bohmian Trajectories Within Hilbert Space Based Quantum Mechanics. Solution of the Measurement Problem

TL;DR

This paper tackles the measurement problem by embedding Bohmian trajectories inside a Hilbert space–based quantum framework using the ensemble interpretation of the wave function. It constructs a stochastic process $\xi(t)$ tied to the Heisenberg position operator $X_H(t)$ whose sample paths are Bohmian trajectories, and extends the configuration space to include spin observables via a spectral space, enabling Pauli–equation analyses. A Schrödinger–Bohm evolution framework is developed to derive von Neumann’s projection rule directly for the system–apparatus density operator, with explicit consideration of macroscopic pointer distinguishability and decoherence within this be-able–observables setting. The approach preserves the predictive power of standard QM while restoring a clear ontological status for beables, and it opens the door to quantum cosmology applications by providing objective trajectories without invoking many-worlds. Overall, the work provides a consistent, probabilistic, and ontologically transparent route to unify pilot-wave ideas with Hilbert-space quantum mechanics and to address foundational questions about measurement and symmetry.

Abstract

de Broglie-Bohm theory (dBBT), treating quantum particles as point objects moving along well defined (Bohmian) trajectories, offers an appealing solution of the measurement problem in quantum mechanics; it has, however, problems relating to spin, relativity and lack of proper integration with the Hilbert space based framework. In this work, we present a consistent formalism which has the traditional state-observable framework integrated with the desirable features of dBBT. We adopt ensemble interpretation for the Schrodinger wave function $ψ$. Given a Schrodinger wave function $ψ$, we use its value $ψ_0$ at some fixed time (say, $t = 0$) to define the probability measure $|ψ_0|^2 {\rm d}x$ on the system configuration space $M$ ($=\mathbb{R}^n$). On the resulting probability space $\mathcal{M}_0$, we introduce a stochastic process $ξ(t)$ corresponding to the Heisenberg position operator $X_H(t)$ such that, in the Heisenberg state $|ψ_h\rangle$ corresponding to $ψ_0$, the expectation value of $X_H(t)$ equals that of $ξ(t)$ in $\mathcal{M}_0$. This condition leads to the de Broglie-Bohm guidance equation for the sample paths of the process $ξ(t)$ which are, therefore, Bohmian trajectories supposedly representing time-evolutions of individual members of the $ψ_0$-ensemble. Stochastic processes and Bohmian trajectories corresponding to observables with discrete eigenvalues (in particular spin) are treated by extending the configuration space to the spectral space of the commutative algebra obtained by adding appropriate discrete observables to the position observables. Pauli's equation is treated as an example. A straightforward derivation of von Neumann's projection rule employing the Schrodinger-Bohm evolution of individual systems along their Bohmian trajectories is given. Some comments on the potential application of the formalism developed here to quantum mechanics of the universe are included.