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

TL;DR

This paper develops a formalism integrating de Broglie-Bohm trajectories with the Hilbert space framework, addressing measurement, spin, and relativity issues in quantum mechanics.

Contribution

It introduces a consistent formalism combining traditional quantum states with Bohmian trajectories, including discrete observables, and derives von Neumann's projection rule from this perspective.

Findings

Bohmian trajectories can be defined within the Hilbert space formalism.

The formalism extends to discrete observables like spin.

Derivation of von Neumann's projection rule from Bohmian evolution.

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 $\psi$. Given a Schrodinger wave function $\psi$, we use its value $\psi_0$ at some fixed time (say, $t = 0$) to define the probability measure $|\psi_0|^2 {\rm d}x$ on the system configuration space $M$ ($=\mathbb{R}^n$). On the resulting probability space $\mathcal{M}_0$, we introduce a stochastic process $\xi(t)$ corresponding to the Heisenberg position operator $X_H(t)$ such that, in the Heisenberg state $|\psi_h\rangle$ corresponding to $\psi_0$, the expectation value of $X_H(t)$ equals that of $\xi(t)$ in $\mathcal{M}_0$. This condition leads to the de Broglie-Bohm guidance equation for the sample paths of the process $\xi(t)$ which are, therefore, Bohmian trajectories supposedly representing time-evolutions of individual members of the $\psi_0$-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.