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

Abstract: de Broglie-Bohm theory (dBBT), treating quantum particles as point objectsmoving along well defined (Bohmian) trajectories, offers an appealingsolution 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. Given a Schrodinger wave function, we use its value u at some fixed time (say, t = 0) to define the probability measure |u|sqdx on the system configuration space. On the resulting probability space M0, we introduce a stochastic process xi(t) corresponding to the Heisenberg position operator XH(t) such that, in the Heisenberg state phi corresponding to u, the expectation value of XH(t) equals that of xi(t) in M0. This condition leads to the de Broglie-Bohmguidance equation for the sample paths of the process xi(t) which are, therefore, Bohmian trajectories supposedly representing time-evolutions of individual members of the u-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.