Inertial Trajectories in de Broglie-Bohm Quantum Theory: An Unexpected Problem

ABSTRACT

A salient feature of de Broglie-Bohm (dBB) quantum theory is that particles have determinate positions at all times and in all physical contexts. Hence, the trajectory of a particle is a well-defined concept. One then may expect that the closely related notion of inertial trajectory is also unproblematically defined. I show that this expectation is not met. I provide a framework that deploys six different ways in which dBB theory can be interpreted, and I state that only in the canonical interpretation the concept of inertial trajectory is the customary one. However, in this interpretation the description of the dynamical interaction between the pilot-wave and the particles, which is crucial to distinguish inertial from non-inertial trajectories, is affected by serious difficulties, so other readings of the theory intend to avoid them. I show that in the alternative interpretations the concept at issue gets either drastically altered, or plainly undefined. I also spell out further conceptual difficulties that are associated to the redefinitions of inertial trajectories, or to the absence of the concept.

Acknowledgements

I thank two anonymous referees of this journal for their helpful criticisms on an earlier version of this work.

Notes

1 For simplicity, I gloss over the issue that position eigenstates are not normalizable, so that, strictly speaking, a quantum state is never an eigenstate of the position operator.

Modal interpretations do not accept the eigenstate eigenvalue link, so that a quantum system can possess a definite value for a certain observable even if its state is a superposition in the basis of the corresponding operator. However, in none of the different versions of the modal interpretation the value-state necessarily corresponds to a definite value for the position observable.

2 For a historical treatment of de Broglie's work, see Bacciagaluppi and Valentini (2009). De Broglie's original papers are collected in de Broglie and Brillouin (1928). For a conceptual analysis of Bohm's work and the historical and sociological conditions surrounding its reception, see Cushing (1994).

3 Skow (2010) identifies an interesting worry with respect to (P3). That the theory is not affected by the measurement problem strongly relies on the distribution postulate (P4), whose conservation over time is guaranteed by equation (3). Now, equation (1) is not the only equation of motion that complies with distribution conservation, any formula of the form , where is a solution of equation (1) and is divergence free, satisfies the conservation condition equation (3). Dürr, Goldstein, and Zanghì (1992, 1996) offer an argument to justify the choice of equation (1) in that, assuming certain symmetry conditions that implement a Galilean invariance constraint, equation (1) is the simplest law of motion that can be derived. Skow claims that the derivation is flawed because the symmetry assumptions it relies on cannot be justified. Thus, he claims, the symmetry argument cannot count as a reason to pick equation (1), among all possible choices, as the first-order equation of motion of the theory.

4 Equation (4) can be derived in the following way (Holland 1993, 74). We take equation (2) (for simplicity, in the single-particle system case), with . Rearranging terms and applying the gradient operator we get: . Now, since (equation (1)), and given the operator , we get .

5 I use ‘quasi-Newtonian’ instead of ‘Newtonian’ given the peculiar quantum features of dBB theory that are not present in classical physics (entanglement, non-locality, contextuality, etc.).

6 This tripartite classification of the interpretations of the wavefunction in dBB theory is advocated by Belot (2012). Esfeld et al. (2014) qualify the dispositionalist account of as a type of nomological interpretation, in which the nomological wavefunction is traced back to the expression of a dispositional property: I think that the divergence between Belot and Esfeld and co-authors is mainly a matter of terminology: Belot assumes a conception of scientific laws that does not make room for dispositionalism, whereas Esfeld and co-authors understand scientific laws in broader terms, with Humeanism and dispositionalism as the two main possibilities. For the purpose of this work, Belot's classification is more schematic, and it allows us to trace an important difference between the nomological view and the dispositionalist interpretation that is relevant for our subject.

7 One must be careful here, though. Due to entanglement and the associated non-local correlations, in dBB theory a particle is never strictly free. Anyhow, even if there were no free quantum particles in the universe, the notion of a particle following an inertial trajectory is conceptually well defined. Furthermore, there are (rather idealized) physical contexts in dBB in which particle motion is indeed inertial, see Allori et al. (2002). The situation is actually not substantially different than in classical mechanics. Even if there were no free Newtonian particles in the universe, the concept of a particle following an inertial trajectory is conceptually well defined, and it plays an important role in the foundations of the theory.

8 The problem may be alleviated by arguing that 3-space somehow supervenes on 3n-space, but Monton (2002) argues that this strategy does not work. The reason is that, in general, there are different ways in which the n objects in a system can evolve in 3-space that are compatible with a particular evolution of the corresponding objects in 3n-space: there is no way to specify which dimensions in configuration space correspond to which particle. In the case of the dBB theory, the evolution of the particles in 3-space is underdetermined by the evolution of the universal particle in 3n-space.

9 Norsen presents his toy theory with a first-order equation of motion, but he does not comment on the possibility of a second-order dynamics. We explore this possibility here.

10 To be fair, Belousek is aware of this problem, and because of it, he evaluates his interpretation as a provisional approach:

On this view quantum forces would not even have their origin in the quantum state itself, for it is just the interpretation of the quantum state as representing an entity subsisting in its own right that is being denied here. Instead, forces would simply exist on their own in addition to particles, and actual entities of both sorts would exist only in 3-dimensional space. One would have, then, a genuine dualistic ontology—equiprimordial particles and forces. Of course, one is left here without an account of the origin of such forces  …   So, because it is not completely satisfactorily ‘intuitive’, one might well regard the causal view proposed here as provisional  … , awaiting a better physical interpretation of the quantum potential. (Belousek 2003, 163)

11 I refer to ‘Valentini's approach’ to respect the authorship of the Aristotelian forces proposal. However, the following criticisms hold also for the first-order versions of Norsen's and Belousek's interpretations. That is, the problems I point out hold regardless of whether the Aristotelian forces are exerted by a quantum field in configuration space (Valentini), by a conditional field in 3-space (Norsen), or by nothing at all (Belousek).

12 Moreover, the difference between real and inertial forces is not a mere assumption or a consensual convention. The reason why fictitious forces are taken as unreal is dynamical: real forces are essentially connected to interactions resulting in motion, whereas in the case of fictitious forces there is no such interaction. The Machian strategy that Valentini mentions to conceive fictitious forces as real is not convincing. Just like in Mach's principle, we would demand for a complete description of how the fictitious forces are generated by acceleration with respect to distant matter.

13 To be fair, in the relativistic version of dBB theory this problem gets softened. In a Minkowskian setup, the non-locality of the theory introduces causal correlations between spacelike separated events in the case of entangled subsystems. Furthermore, the guidance equations of the field-version of the theory are not Lorentz invariant. Thus, it seems that a privileged hyperplane of simultaneity is dynamically suggested, a hyperplane that in turn picks a privileged frame. Anyhow, the dynamic grounds to introduce a preferred frame in dBB theory are connected to non-locality and to the failure of Lorentz invariance, not to Aristotelian forces and a state of absolute rest.

14 Dürr, Goldstein, and Zanghì (1992) explain the meaning of in the following way. In standard quantum mechanics, given a system that has been measured by an apparatus, the composite wavefunction is of the form , where the different are the possible experiment outcomes, as given by apparatus pointer positions, for example. In dBB theory, only one of those , say, , is selected—depending deterministically on the initial configuration of the particle(s). To emphasize this feature of the theory, we can write the post-measurement composite state as , where , , and . In simple words, represents the ‘empty’ zones of the wavefunction of the composite system.

The effective wavefunction is equivalent to Holland's notion of effective factorization (Holland 1993, 287–289). Let us say that a wavefunction is strictly factorizable if and only if —so that the subsystem strictly factorizes (it gets dynamically isolated) from the rest of the universe. Strict factorizability is a highly unrealistic assumption, for interaction between particles in the subsystems typically result in entanglement correlations. However, if observes the condition expressed in equation (21) and if , the -subsystem effectively factorizes from the rest of the universe. Now, decoherence processes determining the interactions between a subsystem and its environment typically result in effective factorization. That is, for all practical purposes at least, the entanglement correlations between the -subsystem and its environment are dynamically idle—and thus, whenever it exists, the effective wavefunction of a subsystem evolves according to the Schrödinger equation.

15 The difficulties with a fully nomological reading that Goldstein and Zanghì refer to are that evolves over time, and that it is experimentally controllable. The first issue is troubling because laws are not supposed to change over time according to another dynamical law—in this case, according to the Schrödinger equation. This problem affects the universal wavefunction as well, and Goldstein and Zanghì (2013, 268–270) propose a tentative argument to face it—see also Solé (2013, 372). That is experimentally controllable is worrisome because the form of a physical law is not supposed to be under our control, but we can prepare a quantum system with a specific . For the sake of the argument, we can dodge these problems and accept the view that the effective wavefunction is a quasi-nomological term.

The nomological interpretation of the wavefunction in dBB theory can adopt different forms depending on the particular conception of the laws of nature that is assumed (Humeanism, universalism, primitivism, etc.). However, those differences are not relevant for our subject. The important point here is that none of the different accounts of laws of nature states that a law is an entity in the physical world that dynamically interacts with other physical systems. This view that laws themselves are not elements of concrete physical reality holds also for the effective wavefunction, regardless of the metaphysics and epistemology of laws of nature one may assume.

16 I thank Michael Esfeld (private communication) for this remark.

17 Notice that this view does not consider the force itself as a beable, the beables are the object that exerts the force and the body on which it is exerted. A force is the measure of the interaction between these beables resulting in a change of state of motion.

18 Pitowski did not present his proposal as a reformulation of the theory, but as a new generally covariant theory written in the spirit of Bohm's approach:

The idea is as following: for each quantum state , we absorb the effects of the ‘quantum potential’ associated with into the metric , while, at the same time, we demand that satisfies a covariant equation with respect to that same metric. In that way and are coupled in (essentially) 11 partial differential equations in 11 unknowns. (Pitowski 1991, 343–344)

Pitowski does not address the question of whether the proposed theory is predictively equivalent or not to standard quantum mechanics and to dBB theory, neither its formal and conceptual connection with general relativity.

19 Notice that in the nomological first-order interpretation of dBB theory the concept of energy plays no role either. But then one wonders about the meaning of the term in the Schrödinger equation—it can hardly represent a potential in this interpretation. Perhaps it has to be understood in quasi-nomological terms as well, but Dürr, Goldstein, and Zanghì do not address this worry.

20 A worry concerning Suárez's manoeuvre is that we could define a third-order dispositional property by differentiating equation (4) with respect to time, a property that would come to dynamically explain and its evolution—and we could then define a fourth-order property, and so on (perhaps until the derivative is 0). True that such a manoeuvre would be against formal economy, but so is to include equation (4) as a part of the formalism. Besides, Suárez states that the justification of including a second-order dispositional property is explanatory, for the property expressed by in equation (4) explains , but the putative higher-order properties would in turn explain the lower-order ones.

21 And just as in the case of the second-order nomological interpretation, the manoeuvre of stipulating that both terms must vanish would not do the trick. Even in a dispositional reading of the wavefunction and of the classical potential, that and are epistemically on a par (dispositional terms) does not mean that they have the same dynamical significance ( involves an interaction, does not). Furthermore, in the nomological approach we saw that it was possible to directly reify , interpreting it as the expression of a primitive quantum force à la Belousek (on the pain of sacrificing the basic motivation of the approach); but such a manoeuvre does not make sense in the dispositional framework. If the dispositional property already determines the motion (no interaction present), why would we add a primitive quantum force?