Within the de Broglie–Bohm (dBB) theory, the measurement process and the determination of its outcome are usually discussed in terms of the effect of the Bohmian positions of the measured system S. This article shows that the Bohmian positions associated with the measurement apparatus M can actually play a crucial role in the determination of the result of measurement. Indeed, in many cases, the result is practically independent of the initial value of a Bohmian position associated with S, and determined only by those of M. The measurement then does not reveal the value of any pre-existing variable attached to S, but just the initial state of the measurement apparatus. Quantum contextuality then appears with particular clarity as a consequence of the dBB dynamics for entangled systems.
Abstract We will first define what is meant by “hidden variables”. Then, we will review theorems proving the impossibility of theories introducing such variables in a naive way. Next, we will outline the de Broglie-Bohm theory and show that it is not refuted by those theorems. To do that, we will explain how “measurements” work in that theory, using the examples of spin measurements and of momentum measurements. Those measurements do not reveal pre-existing properties of the quantum particle but are the result of interactions between that particle and the measuring apparatus. And this is why the de Broglie-Bohm theory is not refuted by the theorems on the impossibility of hidden variables. We will also explain the relationship between the no “hidden variables” theorems and nonlocality, with and without introducing Bell’s inequalities.
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.
There are three possible interpretations of the wave function in the de Broglie-Bohm theory: taking the wave function as corresponding to a physical entity or a property of the Bohmian particles or a law. In this paper, we argue that the first interpretation is favored by an analysis of protective measurements. The de Broglie-Bohm theory is an alternative to standard quantum mechan-ics initially proposed by de Broglie (1928) and later developed by Bohm (1952). According to the theory, a complete description of a quantum system is provided by the configuration defined by the positions of its particles together with its wave function. The wave function follows the linear Schrödinger equation and never collapses. The motion of the particles, which are usually called Bohmian particles, follows the so-called guiding equation. Although the de Broglie-Bohm theory is mathematically equivalent to quantum mechanics, there is no clear consensus with regard to its physical interpretation. In particular, the interpre-
Until now we have treated measurement apparatuses as devices taking quantum systems as their inputs and producing measurement outcomes as outputs. At this level of description, measurement apparatuses are fully described by observables, as discussed in Chapter 3. The measurement outcome distribution is determined by the particular input state and the observable describing the measurement device. In this chapter we explore a mathematical framework suitable for the description of measurement apparatuses for which, in addition to the measurement outcomes, the measured systems are available after the measurement is performed. In such cases we may try to obtain more information on the system by measuring some other observable, or, perhaps by filtering out some preferred output states we may use our measurement apparatus as a preparator. Clearly, in this kind of situation we need a description of measurement apparatuses that is more detailed than that of observables. Therefore, two more refined concepts, measurement models and instruments , will be investigated in this chapter. There is no customary name for a measurement model. As used here, it was first formalized in a general way by Ozawa [106], [107] who referred to the measurement process . A comprehensive general reference on quantum measurements is the monograph by Busch et al. [35], where measurement models are called premeasurements . Some other authors use the term indirect measurements . Three levels of description of measurements It is convenient to think of three levels of description when considering measurements.
In a recent paper, Brida et al (2002 J. Phys. B: At. Mol. Opt. Phys. 35 4751) reported the first experimental realization for testing of de Broglie–Bohm (dBB) theory against standard quantum mechanics (SQM), based on a suggested scheme presented in Golshani and Akhavan (2001 J. Phys. A: Math. Gen. 34 5259). However, some details of their experiment are not compatible with our suggestion. Therefore, their conclusion, in which the experimental data are in complete agreement with SQM but show some deviations from dBB predictions, cannot be considered as a final verdict.