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∣2dx on the system configuration space
M (
=Rn). On the resulting probability space
M0, we introduce a stochastic process
ξ(t) corresponding to the Heisenberg position operator
XH(t) such that, in the Heisenberg state
∣ψh⟩ corresponding to
ψ0, the expectation value of
XH(t) equals that of
ξ(t) in
M0. 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.
Quantum uncertainty emerged from two strikingly different routes to quantum mechanics. Heisenberg's matrix mechanics replaced the classical description of atomic orbits by relations among observable transition quantities and led, in 1927, to the uncertainty relation and its influential measurement-based interpretation. Schrödinger's wave mechanics grew instead from the Bose--Einstein and de Broglie developments and restored a continuous space-time description through matter waves. The two formalisms were soon shown to be closely related, but neither historical route by itself settles the ontological question of whether particles may possess underlying trajectories.
After reviewing these developments, including Einstein's criticism of a simple observability doctrine and Mott's wave-mechanical account of straight cloud-chamber tracks, we distinguish precise quantum uncertainty relations from the stronger claim that quantum mechanics excludes definite microscopic histories. We then compare Feynman paths, Bohmian trajectories, Nelson's stochastic mechanics, and finite-speed persistent Goldstein--Kac dynamics. The Kac process has continuous, piecewise differentiable finite-speed paths; its diffusion limit approaches Wiener kinematics, while analytic continuation of its two-sector dynamics is related to Dirac evolution. These examples show that quantum uncertainty relations constrain observable statistics without, by themselves, uniquely determining an underlying path geometry.
Pursuing the Hamiltonian formulation of the de Broglie-Bohm (deBB) theory presented in the preceding paper, the Hamilton-Jacobi (HJ) theory of the wave-particle system is developed. It is shown how to derive a HJ equationfor the particle, which enables trajectories to be computed algebraically using Jacobi's method. Using Liouville's equation in the HJ representation we find the restriction on the Jacobi solutions which implies the quantal distribution. This gives a first method for interpreting the deBB theory in HJ terms. A second method proceeds via an explicit solution of the field+particle HJ equation. Both methods imply that the quantum phase may be interpreted as an incomplete integral. Using these results and those of the first paper it is shown how Schrodinger's equation can be represented in Liouvillian terms, and vice versa. The general theory of canonical transformations that represent quantum unitary transformations is given, and it is shown in principle how the trajectory theory may be expressed in other quantum representations. Using the solution found for the total HJ equation, an explicit solution for the additional field containing a term representing the particle back-reaction is found. The conservation of energy and momentum in the model is established, and a weak form of the action-reaction principle is shown to hold. Alternative forms for the Hamiltonian are explored and it is shown that, within this theoretical context, the deBB theory is not unique. The theory potentially provides an alternative way of obtaining the classical limit.
We define the de Broglie-Bohm (dBB) weak interpretation as the dBB interpretation restricted to particles in unbound states whose wave function is defined in the three-dimensional physical space, and the dBB strong interpretation as the usual dBB interpretation applied to all wave functions, in particular to particles in bound states whose wave function is defined in a 3N-dimensional configuration space in which N is the number of particules. We show that the current criticisms of the dBB interpretation do not apply to this weak interpretation and that, furthermore, there are theoritical and experimental reasons to justify the weak dBB interpretation. Theoretically, the main reason concern the continuity existing for such particles between quantum mechanics and classical mechanics: we demonstrate in fact that the density and the phase of the wave function of a single-particle (or a set of identical particles without interaction), when the Planck constant tends to 0, converges to the density and the action of a set of unrecognizable prepared classical particles that satisfy the statistical Hamilton-Jacobi equations. As the Hamilton-Jacobi action pilots the particle in classical mechanics, this continuity naturally concurs with the weak dBB interpretation. Experimentally, we show that the measurement results of the main quantum experiments (Young's slits experiment, Stern and Gerlach, EPR-B) are compatible with the de Broglie-Bohm weak interpretation and everything takes place as if these unbounded particles had trajectories. In addition, we propose two potential solutions to complete the dBB weak interpretation.
Bohmian mechanics, developed by Bohm in 1952, provides an alternative interpretation to nonrelativistic quantum mechanics [1-3]. In the hydrodynamic formulation of quantum mechanics, the continuity equation and the quantum Hamilton-Jacobi equation (QHJE) are obtained by substituting the wave function expressed in terms of the real amplitude and the real action function into the time-dependent Schrödinger equation. Bohm’s analytical approach has been used to compute and interpret realvalued quantum trajectories from a precomputed wave function for a diverse range of physical processes [3-8]. In the synthetic approach, the quantum trajectory method has been developed as a computational tool to generate the wave function by evolving ensembles of real-valued quantum trajectories through the integration of the hydrodynamic equations on the fly [9]. Remarkable progress has been made in the use of T&F Cat # K11224, Chapter 18, Page 284, 15-9-2010 real-valued quantum trajectories for solving a wide range of quantum mechanical problems [10].