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On the Conceptual Issues Surrounding the Notion of Relational Bohmian Dynamics

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Abstract

The paper presents a program to construct a non-relativistic relational Bohmian theory, that is, a theory of N moving point-like particles that dispenses with space and time as fundamental background structures. The relational program proposed is based on the best-matching framework originally developed by Julian Barbour. In particular, the paper focuses on the conceptual problems that arise when trying to implement such a program. It is argued that pursuing a relational strategy in the Bohmian context leads to a more parsimonious ontology than that of standard Bohmian mechanics without betraying the original motivations for adopting a primitive ontology approach to quantum physics. It is also shown how a relational Bohmian approach might clarify the issue of the timelessness of the dynamics resulting from the quantization of a classical relational system of particles.

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Notes

  1. We assume for simplicity’s sake that \(\hbar =1\).

  2. It is important to highlight that equations (1) can be generalized to whatever configuration space endowed with a non-trivial Riemannian structure \(g_{ij}\ne \delta _{ij}\). See [27, Sect. 2] for the technical details.

  3. However, the theory can be easily generalized in order to account for phenomena involving spin, as shown, for example, in [53].

  4. See [28] for a rigorous justification of these claims.

  5. Roughly, the analogy resides in the fact that both the wave function in BM and the Hamiltonian in the phase space formulation of mechanics “generate” through the equations of motion a vector field in the appropriate space whose integral curves are in fact the dynamical trajectories of the physical system under scrutiny.

  6. It is worth noting that many authors are not sympathetic to BM exactly because it brings - paraphrasing [22]—classical terms into the equations. For these authors such a move amounts to forcing a “folk” metaphysical reading on quantum phenomena, which instead should be understood as a radical departure from our everyday picture of reality. However, this is just a declaration of metaphysical tastes since it does not entail in any way that the Bohmian approach to quantum physics, because of its intuitive character, should be empirically inadequate.

  7. In particular, Edward Anderson has later refined and expanded Barbour’s relational framework while developing a new quantum gravity program. Cf. [8] for his monumental review, which presents the state-of-the-art in relational mechanics and describes its developments since the inception of the first models.

  8. That is, a system subjected to a scalar potential that can be a function of coordinates, velocities, and time.

  9. Usually, the Lagrangian is taken to be the difference \(T-V\) between the kinetic and the potential energies of the system. However, as we will see later, other choices are possible.

  10. It is worth noting that standard quantum mechanics is already formulated over the configuration space but only in BM do we see clearly the significance of this fact. This is because in BM, quantum dynamics is presented in such a way that the parallels between quantum and classical dynamics can be seen clearly. Of course, this is due to the fact that BM can be interpreted using beables only, which makes it possible to establish a fundamental continuity between quantum and classical dynamics on the meaning of dynamics—i.e. both are dynamics of point-like particles. The difference between them thus lies only in the form of the law of dynamics.

  11. The language of “perfect” dynamics is due to the authors and not to the originators of the theory. In our view, this language is helpful to bring out the fundamental physical motivation underlying Barbour’s program, namely, that it is looking for some criteria to evaluate what constitutes a “perfect” dynamics.

  12. We observe ratios of relative distances, not relative distances themselves since in measuring an object with a ruler, we are really comparing it to the ruler.

  13. See [18, 51], for a technical introduction to Barbour’s program, [14] for an extensive survey of the philosophical motivations behind the program, and [10] for a conceptual and technical expansion of the program.

  14. This requirement is what Barbour calls Poincaré’s principle. For a formulation of this principle, see [19, p. 302].

  15. The shrewd reader might already object that no “perfect” universal classical theory of moving point-like particles can recover the full empirical predictions of Newtonian mechanics, for that would mean specifying among the initial data the rate of change (in absolute time) of the orientation of the overall configuration of particles with respect to absolute space. That would obviously violate the minimalist requirement. We will discuss this point later.

  16. It is not obvious why this is the case. For instance, the wave function is not straightforwardly physically observable but according to the most common understanding of standard quantum mechanics, it is an essential theoretical structure for a successful formulation of the theory.

  17. See [56] for a preliminary discussion of the compatibility between Bohmian dynamics and the requirement of background independence.

  18. This point is nicely illustrated in [1].

  19. See [17, especially Sect. 1] for this line of interpretation and the historical details.

  20. To be precise, uniform scalings are particular cases of homothety transformations. However, this level of precision is not essential for our purposes.

  21. See [11, especially Sect. 2 and Appendix B], for a self-contained technical discussion of the topic, including field theories. That article makes also clear that the stratified structure of \(\mathcal {Q}_0\) carries physical import, so it should be accepted as an unavoidable element of (scale-free) relational theories.

  22. This remark makes manifest the fact that the pursuit of a “perfect” dynamics amounts to an implementation of spatial and temporal relationalism.

  23. See [47, pp. 132–140] for a short introduction.

  24. In the remainder of this section, we will refer to a “timeless” dynamics in the weak sense of “without absolute time”. We will postpone to Sect. 4 (in particular 4.2) the discussion of whether this kind of dynamics should be interpreted as giving up time entirely or still retaining some (weak) temporal structure.

  25. In other words, \((E-V)\) plays the role of a conformal factor. This of course means that, in order for the formalism to make sense, such a factor should be well-behaved enough (e.g. no zeros, infinities, or non-smothnesses).

  26. See, for example, [47, Chap. I-Sect. 5, Chap. V-Sect. 7, and Chap. VIII-Sect. 9].

  27. See [8, Sect. 9.6] for a critique of the “marching in step” criterion.

  28. One could attempt to formulate such a dynamics via relative coordinates such as inter-particle distances \(r_{ij} = |\mathbf {q}_i - \mathbf {q}_j|\). However, such theories notoriously suffer from predicting anisotropic masses that disagree with empirical observations. See [55, especially Sect. 6] for a discussion and references.

  29. See [8, Sects. 1.5–1.8, 2.A] for a technical overview on the evolution of the formalism (from the original one in [19], which spoiled the reparametrization invariance of the action, to the one presented here, which fixed the issue), and [6] for a detailed technical discussion.

  30. Besides the already mentioned [3, 5], see also [8, Sects. 2.2.1, 2.3.1, 2.3.2] for a discussion of this kind of “differential” and its relation to Lie derivatives.

  31. More pictorially, carrying out the procedure for which \({\varvec{\updelta }} \mathcal {D}(\delta \mathbf {q}_i) = 0\) amounts to “juxtapose” \(q_0^2\) with \(q_0^1\) such that they fit in the best way possible. Clearly, in the limit case where \(q_0^1\) and \(q_0^2\) represent the same shape, this “juxtaposition” will make them perfectly overlap: their “distance” is then zero.

  32. \(T_{JBB}\) is homogeneous of degree 2, while E-V must be homogeneous of degree \(-2\) (see (14)). As Anderson notes [8, Sect. 2.3.2], since we are dealing with a scale-invariant theory, it would be far more geometrically natural to render both terms homogeneous of degree 0 by, respectively, dividing and multiplying by the total moment of inertia I. However, as Anderson himself acknowledges, the form (10) is the mechanically-natural one, since \(T_{JBB}, E\), and V bear the usual physical units (I acting in this context just as a constant “conversion factor” between the two formulations). We then prefer to stick to this latter representation, which will make clearer the extension of the framework to BM discussed in Sect. 3.

  33. See [4, 9] for two fully worked-out models involving this formalism. Earlier work on “Barbour–Bertotti” models include, notably, [32, 33].

  34. All the three relations (12, 13, 14), hold in the center-of-mass frame.

  35. The concrete calculations are carried out in [17, Sect. 2].

  36. We note that nothing speaks against the possibility of constructing a classical theory that accounts for global \(J\ne 0\) effects in terms of change in the spatial relations of particles only. Of course, such a theory would exhibit a mathematical structure far more complicated than the present one.

  37. See [15], [45, Sects. 3.1 and 3.4]. However, as we will see in Sect. 4, not everybody agrees on this.

  38. Unless they are solved prior to quantization, which would lead to the same result considered above.

  39. For a detailed technical discussion of the quantization of theories that dispense with space and time as fundamental notions, see [26].

  40. A presentation and defense of this view can be found in [36].

  41. [35] provide a clear review of the conceptual pros of going Bohmian in the context of a canonically quantized theory, especially in quantum canonical general relativity.

  42. More precisely, the velocity field should be chosen such that the probability distribution \(|\Psi |^{2}\) is equivariant with respect to it.

  43. Here we gloss over the boundary and continuity conditions that must be placed to ensure that \(\Psi \) - and hence also S and R—is physically meaningful.

  44. It is possible to arrive at the very same expression by differentiating (1b) with respect to time, which stresses the fact that the two formalisms are equivalent.

  45. Otherwise, one of our key motivations for pursuing this program, i.e. finding a Bohmian theory of N-particles with a more parsimonious and coherent ontology than the standard one, would be betrayed.

  46. See [40, Sect. 3.11], for a discussion of the invariance properties of the theory (17).

  47. Nonetheless, R does have a general distinctive feature that might be interesting in this context, that is, the fact that it influences the form of the quantum potential (18) modulo a multiplicative constant (i.e. \(\mathcal {V}\) does not change under transformations \(R\rightarrow kR, k\in \mathbb {R}\)). This means that the physical information encoded in R which determines (18) is insensitive to scaling transformations.

  48. Actually, there is no consensus over whether the spatial and temporal structures entering the dynamics of BM should be best understood as standard absolute space and time or a neo-Newtonian 4-dimensional structure. For simplicity’s sake, we gloss over this further issue.

  49. Of course, such a function is not unique nor objective, since it depends on an arbitrary fixing of the spatial scale.

  50. This terminology is of course borrowed from [50].

  51. Formally speaking, this amounts to recovering (19) from the RBM’s version of (5).

  52. Which by no means exhaust the list of strategies for accounting for time in a quantum relational setting. [8, Sects. 20–26] gives a detailed overview of the state-of-the-art in this field: interestingly enough, primitive ontology approaches seem not to be particularly considered in current research, which further motivates the present article.

  53. It is worth noting that Barbour’s proposal was made at a stage in which there was not enough knowledge of shape spaces’ geometry (i.e. before work like in [11] was carried out). For this reason, we intend Barbour’s proposal as a heuristically presented possibility with no detailed mechanism offered (see [7] for a discussion on the general approach to records theory in physics). However, what interest us here are the metaphysical implications of Barbour’s proposal, independently on its actual (or possible) technical implementation.

  54. Or degrees of belief, in the Bayesian case.

  55. See also [54] for a more general discussion of the problem.

  56. But see [38] for a new implementation of this framework in terms of a generalized Hamilton–Jacobi formalism.

  57. This alternative taxonomy is presented and explained in a philosophical fashion in [39, Sect. 2].

  58. These authors consider the issue in the context of quantum theories of gravity that do not posit space and time as fundamental entities; however, their reasoning easily applies to the program considered here.

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Acknowledgments

We are very grateful to an anonymous referee, Sean Gryb, and Roderich Tumulka for detailed comments on an earlier draft of this paper. Antonio Vassallo acknowledges support from the Swiss National Science Foundation, Grant No. \(105212\_149650\).

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  1. Department of Philosophy, University of Lausanne, 1015, Lausanne, Switzerland

    Antonio Vassallo

  2. Magdalene College, Cambridge, CB3 0AG, UK

    Pui Him Ip

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Vassallo, A., Ip, P.H. On the Conceptual Issues Surrounding the Notion of Relational Bohmian Dynamics. Found Phys 46, 943–972 (2016). https://doi.org/10.1007/s10701-016-9992-z

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