Abstract
The feasibility of establishing a proper notion of a distinguishable object in the context of the de Broglie–Bohm approach to quantum mechanics seems, at first sight, uncontroversial by virtue of the fact that this theory can supposedly be interpreted in terms of a system of objective particles distinguished by individuating properties. However, after conducting a critical revision and evaluation of this trivial interpretation, and having assessed different alternatives that have been proposed in recent literature, I argue that within this theory an appropriate notion of a distinguishable object can only be articulated by means of the following theoretical and metaphysical strategies: firstly, by appealing to a pre-existing, symmetrized Bohmian framework that is empirically indistinguishable but physically distinguishable from the standard Bohmian formulation; and secondly, by suggesting a different interpretation of this symmetrized formulation based upon a relational notion of the distinguishable object that can only be appropriately conceived from a structuralist point of view.
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Notes
The standard view of QM will be confined to von Neumann’s formulation, as usually reviewed in physics textbooks.
Definitions: I shall define objects and properties according to the Aristotelian primary-secondary substance distinction, and in modern terminology (following Gracia 1988), as (non-instantiable) particulars and (instantiable) universals, respectively. Furthermore, I shall define intrinsic properties (contextualized to QM) as those that are constitutive of the object in question independently of its relationship to other objects. In particular, any property is intrinsic to a particle at a given time if it is instantiated in that particle at that time as its unique bearer. Formally speaking, intrinsic property means a 1-place predicate F, which is instantiated by object p, say F(p). However, we shall see that an extrinsic property F is interpreted as a relational property, in the sense that any property of one particle x is a property of any other particle y, say F(x,y) (Muller and Saunders 2008, p. 527). For the sake of analysis, the deep debate about intrinsicality in the metaphysical literature shall not be considered. Additionally, I shall follow French and Krause (2006) and define individuals as objects that are not non-individuals (i.e., objects within a metamathematical formal framework, such as quasi-set theory, in which self-identity is not always well-defined). Finally, as we shall see in Sect. 2, individuality shall be defined in terms of a particular interpretation of (ontological) distinguishability based on two principles of individuation: the bundle theory of intrinsic properties and the transcendental theory of individuation (to be defined in Sect. 3). A more general notion of a distinguishable object (as opposed to an individual object), whether defined in terms of intrinsic properties, extrinsic properties or transcendental notions shall be considered from the start.
According to a naturalistic approach to metaphysics, the world is epistemically accessible (presented) to us by metaphysical inferences obtained from QM and not necessarily from sense data. For the sake of precision, the possibility of acquiring knowledge of the world from physics will be called ‘physical admissibility’.
I shall define (non)-identical particles as particles that (do not) possess all their attributed parameters in common, e.g., mass, charge, etc. The state associated with identical particles, however, is given by (anti)-symmetric wave functions, symmetric potentials, and equal parameters, as we shall see in Sects. 3 and 4. NB: this is not the only way to define identical particles in BQM. As claimed by Belousek (2000), the (anti)-symmetric state is only a special case of a more general criterion for identical particles that generates a permutation-invariant dynamics. However, this discussion lies beyond the scope of this paper.
In this paper, a system of particles with different mass is considered. The general case for a system of particles with charge, spin, and angular momentum is given in Holland (1995).
\(L^{2}(\mathbb {R}^{3n},{\mathbb {C}})\) is the set of square-integrable functions in configuration space with 1 complex components. Interpretational issues with respect to the ontological status of these functions and the space in which they are defined are beyond the scope of this paper.
Note that if a Bohmian property is state-independent then it is intrinsic, whereas Bohmian intrinsic properties are not, in general, state-independent (e.g., the trajectories of the particles). As mentioned in a Sect. 1 footnote, I shall assume the broader definition of intrinsic property, via the ‘non-involvement of other objects’ and not via ‘state-independence’.
Given this, there is no need to appeal to this principle again, when referring to distinguishability as the guarantor of individuality.
I shall define relationals in Sect. 5. However, the introduction of relationals to the PII means that I aim to establish the appropriate notion of a distinguishable object (as opposed to the notion of individuality).
As we shall see in Sect. 4, identical Bohmian particles (as opposed to identical particles in standard QM) possess a well-defined, distinguishing position extended along trajectories. That this does not hold for the case of non-identical particles is precisely the task to be elaborated in this section.
In more precise terms, identical particles are statistically indistinguishable if we can associate a label with each particle with no effects of any observable quantity. Similarly, we can say that identical particles are statistically indistinguishable if any observable quantity is invariant under permutations of the labels associated with any pair of such particles. Since observable quantities in BQM are ultimately averages over ensembles of particle trajectories (i.e., expectation values for particle positions), this notion is not a feature of the behaviour of each individual particle but of the observational behaviour of the system as a whole.
Contrary to what happens in a classical mechanical system of particles, the trajectory of each Bohmian particle is fully specified by the initial configuration and the wave function of the whole system, as described by the Bohmian state \((\Psi _{t},\mathbf{Q})\) and the deterministic nature of the equations of motion. Note that this specification presupposes that the initial velocity of the particles is not a degree of freedom of the system, something which ultimately determines which particle is which (and which trajectory corresponds to which particle) in the classical regime. In fact, any set of two or more particles in classical mechanics can be distinguished via the trajectories they follow thanks to the possibility of determining the trajectory associated with each particle according to its initial velocity, which is chosen independently of its initial position and the initial conditions of the other particles. Thus, the second order nature of classical mechanics enables the reference-correspondence between each particle and its own trajectory, whereas in BQM each particle cannot refer to its corresponding trajectory unless it refers to some intrinsic property instantiated in its initial position at the start.
According to Brown et al. (1996), experimental results from the above strongly suggest that the Bohmian parameters should be considered actual properties of either the wave function alone (what has been called the ‘principle of parsimony’) or of both the wave function and the particles (the ‘principle of generosity’). (Valentini 1992; Holland 1995; Brown et al. 1996, 1999) argue that the principle of generosity may seem to be more compelling because the guiding equation (1) describes the evolution of the particles by explicitly appealing to these parameters whereas advocates of wave-function realism would endorse something like the principle of parsimony. My hunch is that the interpretation of these parameters as features of the wave function must be elaborated in depth (as these parameters are classically conceived as being properties of localized objects, either by point particles or fields). However, since I am not concerned with the metaphysics of the wave function—I am neutral with respect to the interpretation of this entity—I shall not address this interpretation here.
Haecceity is a scholastic term characterized in contemporary metaphysics as a property of primitive thisness, according to Adams (1979), but also as transcendental individuality, according to Post (1963). The metaphysical nature of haecceity and the metaphysical differences between these characterizations lie beyond the scope of this paper.
The notion of haecceity is literally indescribable in terms of any epistemically accessible property. Some contemporary philosophers, such as French (1998) and French and Krause (2006), have used names or labels to express haecceity in formal terms via the notion of self-identity \((a=a)\). They formally extend Adams’ definition, according to which haecceity is “the property of being identical with a certain individual—not the property that we all share, of being identical with some individual or other, but my property of being identical with me, your property of being identical with you, etc. (Adams 1979, p. 6).” Furthermore, they have argued that haecceity lies behind the idea of countable pluralities. Interpreting countable systems of particles as ordinality, one may count how many Bohmian particles are in a given state by assigning a rigid label to each element of the system. Of course, the worry with this definition is that haecceity cannot be expressed by labels as their descriptive reference, which designate the individuals with predicates representing properties. As one reviewer of this paper has helpfully emphasized, self-identity is a monadic predicate that holds for every particle and does not set one particle apart from another. Similar complaints have been made by Auyang (1995, p. 125) in regard to its universal applicability: “Identity does not say anything beyond one thing; rather, it discloses the meaning of being an entity, and the disclosure signifies our primordial understanding.”
That ‘superfluous’ metaphysics is something undesirable rests upon the following assumption of ontological parsimony: as far as the notion of individuality may be grounded in terms of the physics, there is no need to expand the metaphysical baggage of the theory (Ladyman et al. 2007; Ladyman 2007). As such, haecceity would be superfluous if there were other ‘theoretically admissible’ accounts that could be ‘read off’ directly from the theory regardless of whether or not this metaphysical notion is considered.
Note that the counterfactual situation in which the labels of the particles permute is a physical possibility that belongs to the spectra of nomic states. Since our ontological commitments are limited to a partial set of nomic states which are realized in the actual world (i.e., the equivalence class of states that differ up to permutations), some would say that this interpretation is coherent through the lens of a non-Humean view of laws. This view admits modal connections that constrain what happens in the universe, as opposed to the Humean view, according to which laws cannot ground physical possibilities that do not (and will never) occur in the actual world. What the laws of nature are, according to the Humean, depends upon what there is in the counterfactual situation of any given local matter of particular fact, instead of that situation depending on the laws of nature. However, one could argue that there is an alternative Humean way to prevent grounding (metaphysically speaking) this kind of counterfactual in BQM. We can suppose that each single trajectory associated with the equivalence class of trajectories differing up to permutations forms the supervenience basis of the laws in the actual world, meaning that Bohmian laws are a systematized summary of all trajectories, each one being associated with this class and being constitutive of the actual world. From this perspective, the counterfactual situation of permuting the labels of the particles does not belong to the space of nomic states. Thus, we are not forced to conceive of this alternative interpretation as non-Humean with respect to laws. A Humean view of laws can also do this job.
Although in the majority of cases the underdetermination between different ways of interpreting the entities in question may seem uninteresting, the question regarding the mere existence of counterfactual nomic states associated with such entities should not be ignored. If we assume that we acquire knowledge of the world via BQM, it is tolerable to be sceptic about the deep metaphysical nature of objects, once the existence of these objects has been assumed, e.g., whether they are tropes, substances, etc., but it is intolerable to ignore what does (and could) exist and what does not (and will never) exist as regards the unobservable. Furthermore, along similar lines to Chakravartty (2003) one could argue that this form of metaphysical underdetermination can be tolerated as it is not an actual challenge for attaining knowledge of the world via BQM because, as long as the theory explains the relevant predicted phenomena, the Bohmian should not be concerned with the realizability of certain nomic states in the actual world in the same way that she is unconcerned with whether everyday macroscopic objects and properties are conceived in terms of substances, tropes, etc. However, we should note that there is an essential difference between discussing underdetermined ways to conceive of everyday macroscopic objects and properties, and being vague about the mere actuality of certain trajectories predicted by BQM. The underdetermination at the level of everyday objects can be interpreted as uninteresting because in this case we could have different ways to conceive of individual objects without the problematic decision of whether to draw a line between nomic states that are capable of being realized and those that are just surplus theoretical structure. Since observable objects are manifested entities, it makes no sense to stipulate that only some part of the behaviour associated with observable objects does not (and will never) exist in the actual world. In this case, we can directly know through observation whether any kind of behaviour is actually realized. However, the unobservable condition of Bohmian particles leaves open the possibility of deciding whether or not a nomic state does (or could) exist in the actual world, to the extent that the remaining theoretical structure explains the observed phenomena.
In this section, I shall denote BQM-(in)distinguishability as the term used in BQM, whereas (in)distinguishability as the term used in standard QM.
The reduced configuration space \(\mathbb {R}^{3N}{/}S^{n}\) is the space obtained by associating the equivalence class of points differing under permutations in the standard configuration space with a single point, as if part of the configuration space were ‘shrinking’ into a single point.
Considering this physical equivalence, Dürr et al. (2006) have argued that although a system of identical particles can be formulated in the standard configuration space, the reduced configuration space forms its natural mathematical basis as there is no surplus structure in this formulation, known as the standard formulation for identical particles. At this point of the discussion the question of ‘naturalness’ becomes apparent. A natural formulation is that which can be constructed with the resources of the relevant structure of a physical system (the state space) without relevant constraints that must be entered by hand. In the case in question, Dürr et al. (2006) may seem to conform to a natural formulation for a system of identical Bohmian particles because there is no need to introduce by hand the state constraints imposed by the standard configuration space formulation. However, since our discussion is purely metaphysical, and as these formulations represent the same real situation, irrespective of whether or not they are natural, questions of naturalness lie beyond the scope of this paper.
Please refer to Goldstein et al. (2005) for details.
Although, both BQM* and Dürr et al. (2006) can be indistinctly used to describe a system of identical particles, for general systems including non-identical particles it turns out that BQM* is the most appropriate symmetrized formulation, for in BQM* we can treat all particles as BQM-indistinguishable while still accounting for the different species of elementary particles varying in mass, charge or both. In this sense, there is no fact in the BQM* world about what sort of particle exists, such as electron points as distinct from muon points, only that a particle exists. Furthermore, note that this alternative formulation is different from the metaphysical view illustrated in the last part of Sect. 3. Although both are about non-identical particles, BQM* does not need to confine our ontological commitments to certain (not all) theoretical possibilities by mere stipulation. On the contrary, BQM* is rather a theory in which a physically admissible interpretation of BQM-indistinguishable particles can be conceived with labels referring only to their corresponding trajectories.
Galilean space-time is the space of classical mechanics, namely, a four-dimensional Euclidean space (three of which correspond to spatial coordinates and one to temporal coordinates) with no single privileged inertial frame of reference.
NB: do not confuse the relational ontological view of space-time (as opposed to the substantivalist view) with the view of matter points distinguished by relations holding between space-time points. As we shall see below, these views are different and they can only be associated on the basis of additional arguments. For example, according to Esfeld and Deckert (2017, p. 28), this association holds because if one were to be a substantivalist about space, then the theory would contain more ontological structure than what is required to account for the empirical evidence, which consists of relative particle positions and changes to these positions.
See, for example, Huggett and Norton (2013). They provide an updated account of relational properties as ‘extrinsic’.
So, two fermions are weakly discernible only when there is a two-place, non-reflexive but symmetric permutation-invariant relation R that holds between them. In the particular case of a singlet state of two fermions, no spin direction can be specified to either particle, but a relation can be ascribed to them in terms of which their spins in any given direction are opposite to each other (R reads as “the spin of x is opposite to the spin of y’). A rigorous generalization for elementary particles has also been given in Muller and Seevinck (2009).
According to the standard notion of individuality within standard QM, the distinction between physical and statistical indistinguishability is not well defined. Particles are indistinguishable because the observational quantities of the system are insensitive to permutations of the labels of these particles. There is no sense in introducing any statistical notion if there is no well-defined formal notion of the individual particle to begin with. However, under a relational interpretation of the standard theory, we can differentiate between statistical and physical distinguishability because we can talk of the behaviour of particles in relational terms.
The philosophical stance known as OOR is basically introduced in French and Krause (2006) and French (2011) to embrace those metaphysical accounts that are opposed to the thesis of structural realism. As such, although it incorporates STI, it embraces other object-oriented metaphysical accounts, such as that of non-individuality within the formal framework of quasi-set theory.
OSR is a philosophical thesis according to which all things in the world constitute a physical structure (i.e., a nexus of relations holding between relata) and denies the existence of non-structural objects and properties (i.e., the relata upon which these relations take place) (Ladyman et al. 2007; French 2014).
Contrary to Esfeld et al., my revisionary methodology allows us to critically investigate any logically possible metaphysical candidate that may seem to establish a notion of a distinguishable object in Bohmian theory. So, instead of justifying what has been initially preferred on an a priori basis, I am encouraged to explain on a neutral basis the virtues or failures of different metaphysical views, and after a deliberative critical evaluation will conclusively choose which alternative best suits my principal aim.
This group is a ten-parameter simply-connected group that accepts true (non-projective) representations in the Hilbert space and the physical space (such as standard spatial translations, rotations and transformations between different coordinate frames).
See my work (Manero 2019) for details.
References
Adams, R. (1979). Primitive thisness and primitive identity. Journal of Philosophy, 76(1), 5–26.
Aharonov, Y., & Bohm, D. (1959). Significance of electromagnetic potentials in the quantum theory. Physical Review, 115(3), 485–491.
Anderson, J. (1967). Principles of relativity physics. New York: Academic Press.
Auyang, S. (1995). How is quantum field theory possible? Oxford: Oxford University Press.
Bell, J. S. (1971a). Introduction to the hidden-variable question. In A. Aspect (Ed.), Speakable and unspeakable in quantum mechanics (pp. 29–39). Cambridge: Cambridge University Press.
Bell, J. S. (1971b). Against measurement. In A. Aspect (Ed.), Speakable and unspeakable in quantum mechanics (pp. 213–231). Cambridge: Cambridge University Press.
Belousek, D. W. (2000). Statistics, symmetry, and (in)distinguishability in Bohmian mechanics. Foundations of Physics, 30(1), 153–164.
Blackburn, S. (1990). Filling in space. Analysis, 50, 62–65.
Bohm, D. (1952a). A suggested interpretation of the quantum theory in terms of “hidden’’ variables. I. Physical Review, 85(2), 166–179.
Bohm, D. (1952b). A suggested interpretation of the quantum theory in terms of “hidden’’ variables. II. Physical Review, 85(2), 180–193.
Bohm, D., & Hiley, B. J. (1993). The undivided universe: An ontological interpretation of quantum theory. London: Routledge.
Brown, H. R., Dewney, C., & Horton, G. (1995). Bohm particles and their detection in the light of neutron interferometry. Foundations of Physics, 25(2), 329–347.
Brown, H. R., Elby, A., & Weingard, R. (1996). Cause and effect in the pilot-wave interpretation of quantum mechanics. In J. T. Cushing, A. Fine, & S. Goldstein (Eds.), Bohmian mechanics and quantum theory: An appraisal (pp. 309–319). Dordrecht: Boston Studies in the Philosophy of Science.
Brown, H., Sjöqvist, E., & Bacciagaluppi, G. (1999). Remarks on identical particles in the de Broglie–Bohm theory. Physics Letters A, 251(4), 229–235.
Castellani, E. (1998). Galilean particles: An example of constitution of objects. In E. Castellani (Ed.), Interpreting bodies. Classical and quantum objects in modern physics (pp. 181–194). Princeton: Princeton University Press.
Chakravartty, A. (2003). The structuralist conception of objects. Philosophy of Science, 70, 867–878.
De Gosson, M. (2001). The principles of Newtonian and quantum mechanics: The need for Planck’s constant, h. London: Imperial College Press.
De Gosson, M., & Hiley, B. (2011). Imprints of the quantum world in classical mechanics. Foundations of Physics, 41(9), 1415–1436.
Dürr, D., Goldstein, S., & Zanghi, N. (1992). Quantum equilibrium and the origin of absolute uncertainty. Journal of Statistical Physics, 67(5), 843–907.
Dürr, D., Goldstein, S., & Zanghi, N. (1995). Bohmian mechanics and the meaning of the wave function. In A. Shimony, R. S. Cohen, M. Horne, & J. Stachel (Eds.), Experimental metaphysics: Quantum mechanical studies for Abner Shimony (pp. 25–38). Dordrecht: Boston Studies in the Philosophy of Science.
Dürr, D., Goldstein, S., & Zanghi, N. (2004). Quantum equilibrium and the role of operators as observables in quantum theory. Journal of Statistical Physics, 116(1–4), 959–1055.
Dürr, D., Goldstein, S., & Zanghi, N. (2006). Topological factors derived from Bohmian mechanics. In D. Dürr, S. Goldstein, & N. Zanghi (Eds.), Quantum physics without quantum philosophy (pp. 205–220). Berlin: Springer-Verlag.
Esfeld, M., & Lam, V. (2008). Moderate structural realism about space-time. Synthese, 160(1), 27–46.
Esfeld, M., Lazarovici, D., Lam, V., & Hubert, M. (2017). The physics and metaphysics of primitive stuff. The British Journal for the Philosophy of Science, 68(1), 133–161.
Esfeld, M., & Deckert, D. (2017). A minimalist ontology of the natural world. New York: Routledge.
French, S. (1998). On the withering away of physical objects. In E. Castellani (Ed.), Interpreting bodies: Classical and quantum objects in modern physics (pp. 93–113). Princeton: Princeton University Press.
French, S. (2011). Metaphysical underdetermination: Why worry? Synthese, 180(2), 205–221.
French, S. (2014). The structure of the world: Metaphysics and representation. Oxford: Clarendon Press.
French, S., & Krause, D. (2006). Identity in physics: A historical, philosophical, and formal analysis. Oxford: Oxford University Press.
Goldstein, S., Taylor, J., Tumulka, R., & Zanghi, N. (2005). Are all particles identical? Journal of Physics A: Mathematical and General, 38(7), 1567–1576.
Goldstein, S., & Zanghi, N. (2013). Reality and the role of the wave function in quantum theory. In A. Ney & D. Albert (Eds.), The wave function: Essays on the metaphysics of quantum mechanics (pp. 263–278). Oxford: Oxford University Press.
Gracia, J. J. E. (1988). Individuality: An essay on the foundations of metaphysics. New York: State University of New York Press.
Greenberger, D. M. (2001). Inadequacy of the usual Galilean transformation in quantum mechanics. Physical Review Letters, 87(10), 1004051–1004054.
Holland, P. R. (1995). The quantum theory of motion: An account of the de Broglie–Bohm causal interpretation of quantum mechanics. Cambridge: Cambridge University Press.
Huggett, N., & Norton, J. (2013). Weak discernibility for quanta, the right way. British Journal for the Philosophy of Science, 65(1), 39–58.
Ladyman, J. (2007). Scientific structuralism: On the identity and diversity of objects in a structure. The Aristotelian Society, 81(1), 23–43.
Ladyman, J., Ross, D., Spurret, D., & Collier, J. (2007). Every thing must go: Metaphysics naturalized. Oxford: Oxford University Press.
Laidlaw, M. G. G., & DeWitt, C. M. (1971). Feynman functional integrals for systems of indistinguishable particles. Physical Review D, 3(6), 1375–1378.
Leinaas, J. M., & Myrheim, J. (1977). On the theory of identical particles. Il Nuovo Cimento B, 37(1), 1–23.
Manero, J. (2019). Imprints of the underlying structure of physical theories. Studies in History and Philosophy of Modern Physics, 68, 71–89.
Muller, F. A. (2011). Withering away, weakly. Synthese, 180(2), 223–233.
Muller, F. A., & Saunders, S. (2008). Discerning fermions. The British Journal for the Philosophy of Science, 59(3), 499–548.
Muller, F. A., & Seevinck, M. P. (2009). Discerning elementary particles. Philosophy of Science, 76(2), 179–200.
Post, H. (1963). Individuality and physics. The Listener, 70, 534–537.
Saunders, S. (2003). Physics and Leibniz’s principles. In K. Brading & E. Castellani (Eds.), Symmetries in physics: Philosophical reflections (pp. 287–307). Cambridge: Cambridge University Press.
Saunders, S. (2006). Are quantum particles objects? Analysis, 66(289), 52–63.
Sjöqvist, E., & Carlsen, H. (1995). Fractional statistics in Bohm’s theory. Physics Letters A, 202(2–3), 160–166.
Stachel, J. (2005). Structural realism and contextual individuality. In Y. Ben-Menahem (Ed.), Hilary Putnam (pp. 203–219). Cambridge: Cambridge University Press.
Valentini, A. (1992). On the pilot-wave theory of classical, quantum and subquantum physics. Dissertation, Scuola Internazionale Superiore di Studi Avanzati, Trieste.
Acknowledgements
I am grateful to Olimpia Lombardi and Kerry Mckenzie for their careful reading of earlier drafts. Thanks to anonymous referees for comments, to the editors, and to Michael Pockley for linguistic advice. The work on this paper was supported by the Formal Epistemology—the Future Synthesis grant, in the framework of the Praemium Academicum programme of the Czech Academy of Sciences.
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I confirm that the work on this paper was supported by the Formal Epistemology—the Future Synthesis grant, in the framework of the Praemium Academicum programme of the Czech Academy of Sciences. These funding sources have no involvement in study design; in the collection, analysis and interpretation of data; in the writing of the report; and in the decision to submit the article for publication.
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Manero, J. Quantum pointillism with relational identity. Synthese 199, 10639–10666 (2021). https://doi.org/10.1007/s11229-021-03262-w
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DOI: https://doi.org/10.1007/s11229-021-03262-w