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

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 ψ₀ at some fixed time say, $t = 0$ to define the probability measure |ψ₀|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_hrangle corresponding to ψ₀, 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 ψ₀-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.