The physics and metaphysics of primitive stuff
Michael Esfeld, Dustin Lazarovici, Vincent Lam, Mario Hubert
aa r X i v : . [ phy s i c s . h i s t - ph ] N ov The Physics and Metaphysics of PrimitiveStuff forthcoming in the British Journal for the Philosophy of Science
Michael Esfeld ∗ , Dustin Lazarovici † , Vincent Lam ‡ , Mario Hubert § December 1, 2014
The paper sets out a primitive ontology of the natural world in terms of primitivestuff, that is, stuff that has as such no physical properties at all, but that is nota bare substratum either, being individuated by metrical relations. We focus onquantum physics and employ identity-based Bohmian mechanics to illustrate thisview, but point out that it applies all over physics. Properties then enter into thepicture exclusively through the role that they play for the dynamics of the primitivestuff. We show that such properties can be local (classical mechanics), as well asholistic (quantum mechanics), and discuss two metaphysical options to conceivethem, namely Humeanism and modal realism in the guise of dispositionalism.
Keywords : primitive ontology, primitive stuff, ontic structural realism, identicalparticles, Bohmian mechanics, Humeanism, modal realism, dispositionalism.
There are two main options pursued in current research on the ontology of quantumphysics. One option is to take the formalism of the quantum theory that one adopts to ∗ Université de Lausanne, Faculté des lettres, Section de philosophie, 1015 Lausanne, Switzerland. E-mail: [email protected] † LMU Munich, Mathematical Institute, Theresienstr. 39, 80333 Munich, Germany. E-mail:[email protected] ‡ Université de Lausanne, Faculté des lettres, Section de philosophie, 1015 Lausanne, Switzerland.School of History, Philosophy, Religion and Classics, The University of Queensland, St Lucia QLD4072, Australia. E-mail: [email protected] § Université de Lausanne, Faculté des lettres, Section de philosophie, 1015 Lausanne, Switzerland. E-mail: [email protected] primitive ontology as regards that matter. Therole of the quantum state, represented by the universal wave-function, then is limitedto the dynamics, that is, its role is to guide or govern the temporal development of thedistribution of matter in physical space (see Allori et al. 2008). In brief, the motivationfor the primitive ontology option is to uphold the commitment to physics being aboutmatter in ordinary space also when it comes to quantum physics, although the quantumstate is defined on a very high-dimensional space (see e.g. Monton 2006, Maudlin 2010,Belot 2012).It is obvious that a dualism consisting in a conjunction of these two options is notan attractive position: if one takes the quantum state as represented by the universalwave-function to be the physical object of quantum physics, then that state as it exists inthe very high-dimensional space on which the universal wave-function is defined, is thephysical reality. There then is no point to take that state to be also the state of matterdistributed in three-dimensional space. The task rather is to show how the dynamics ofthe quantum state existing in the configuration space of the universe can be such thatthis state develops in that space – e.g. through decoherence – into something that canaccount for our experience of matter being distributed in a three-dimensional space.By the same token, if one commits oneself to a primitive ontology of matter distributedin three-dimensional space being the referent of the formalism of quantum physics, thenthere is no point in adding to that commitment a commitment to the quantum stateexisting in the high-dimensional configuration space of the universe. The reason is, inbrief, that it is not intelligible how the quantum state could fulfill the role that it has inthe primitive ontology theories of quantum physics, namely to guide the temporal devel-opment of the primitive ontology, if it were a physical object on a par with the primitiveontology, but existing in another space; it would, for instance, be unclear how a fieldexisting in the very high-dimensional configuration space of the universe, represented2y the universal wave-function, could guide the motion of matter in three-dimensionalspace. Since the quantum state enters the primitive ontology theories through the rolethat it plays for the temporal development of the primitive ontology, it is reasonable toregard it as nomological, by contrast to a physical entity on a par with the primitiveontology (cf. Dürr, Goldstein, and Zanghì 2013, chs. 11.5 and 12). We will consider insections 4 and 5 two proposals to spell out what it means that the quantum state isnomological, namely Humeanism and dispositionalism.This paper is concerned with the second option. Its aim is to push the idea of aprimitive ontology of quantum physics to its ultimate consequence and to show thatthe primitive ontology option applies throughout physics. The ultimate consequence isto maintain that matter is primitive stuff, materia prima , having as such no physicalproperties at all. What is usually regarded as physical properties enters into the theorythrough its role for the dynamics of the primitive stuff, that is, through its nomologicalrole. In other words, the way in which the primitive ontology theories of quantum physicsare often presented, namely in terms of introducing the elements of the primitive ontologyas being characterized by classical properties such as mass and charge, is incoherent, asis the dualism of a primitive ontology existing in three-dimensional space and a quantumstate existing in configuration space. The reason for this incoherence is that the dynamicsof classical physics is fundamentally different: in classical physics, dynamical variablessuch as mass and charge are attributed to point particles taken individually. Given thelaws of classical physics, the distribution of mass and charge in the universe fixes howthe particles move.In quantum physics, by contrast, it is in general not possible to attribute a wave-function to the particles taken individually, but in the last resort only to the whole con-figuration of matter in the universe at a given time. In the primitive ontology theoriesof quantum physics, the wave-function then has the job to fix the temporal developmentof the configuration of matter (in a deterministic or probabilistic manner). It is inco-herent to assume that the determination of the dynamics encoded in the wave-functionis superimposed on a determination of the dynamics through the classical properties ofthe particles taken individually, that is, their mass and their charge. In brief, either thedynamics is determined from above so to speak, namely by variables that apply only tothe primitive ontology as a whole, or the dynamics is determined from below, namely byvariables belonging to the elements of the primitive ontology taken individually. Sincethe latter option is excluded for quantum physics, it is reasonable to pursue the formerone. This implies taking the primitive ontology to be primitive stuff, instead of particlesthat are equipped with intrinsic properties each, and conceiving the dynamics as being3etermined by variables belonging to the whole configuration of the primitive stuff.In the next two sections, we first elaborate on the metaphysics and then on the physicsof matter as primitive stuff, using the formalism of Bohmian mechanics for identicalparticles. In section 4, we show how a recent proposal for a Humean conception of thedynamical variables can shed light on this view of matter. In section 5, we apply thisproposal to the more ambitious metaphysical stance according to which the dynamicalvariables literally determine the temporal development of an initial configuration ofprimitive stuff.
The term primitive ontology goes back to Dürr, Goldstein, and Zanghì (2013, ch. 2, seeend of section 2.2, paper originally published 1992). They write:
What we regard as the obvious choice of primitive ontology—the basic kinds ofentities that are to be the building blocks of everything else (except, of course, thewave function)—should by now be clear: Particles, described by their positions inspace, changing with time—some of which, owing to the dynamical laws governingtheir evolution, perhaps combine to form the familiar macroscopic objects of dailyexperience. (Quoted from the reprint in Dürr, Goldstein, and Zanghì 2013, p. 29; aforerunner of this notion can be found in Mundy 1989, p. 46)
This term has been created in the context of quantum mechanics in order to remindus of the fact that the formalism of quantum mechanics is supposed to represent some-thing, namely matter in space, and is supposed to describe its behavior, for instance inmeasurement situations. The first sense in which the ontology of matter distributed inphysical space is primitive is that this ontology cannot be inferred from the formalismof textbook quantum mechanics, but has to be put in as the referent of that formalism.According to the proposal pursued in this paper, that ontology is furthermore primitivein the sense that it consists in primitive stuff, that is, stuff that has as such no physicalproperties. Dürr, Goldstein, and Zanghì allude to this meaning of “primitive” in thequotation above when they say that the particles are described only by their position inspace. That is to say, a particle being located at a point of space merely signifies thatthe point in question is occupied instead of being empty. But as far as the primitiveontology is concerned, there are no physical properties – such as a mass or a charge –instantiated by the particle.The de Broglie-Bohm theory, going back to de Broglie (1928) and Bohm (1952) andknown today as Bohmian mechanics (see Dürr, Goldstein, and Zanghì 2013) is the oldest4rimitive ontology theory of quantum mechanics. Bohmian mechanics puts forward adiscrete primitive ontology of point particles, whereby, as mentioned above, a particlebeing located at a point of three-dimensional space means that the point in questionis occupied by primitive stuff instead of being empty. What accounts for the primitivestuff occupying points being particles is that, according to Bohmian mechanics, thereare continuous lines of occupation of points in space-time, so that there are worldlinesconstituting particle trajectories. In Bohmian mechanics, the role of the wave-function,developing according to the Schrödinger equation, is to determine, via what is known asthe guiding equation, the velocity of each particle at any time t given the position of allthe particles at t . We will go into the physics of Bohmian mechanics in the next section.For present purposes, it is only important to note that velocity is not a property thatthe particles have over and above being located in space, but simply the first temporalderivative of position.The view according to which all the physical properties, including mass and charge, arebest understood at the level of the wave-function rather than at the level of the Bohmianparticles has been suggested in the literature on the basis of experimental considerationsinvolving interference phenomena, for instance in the context of the Aharonov-Bohmeffect and of certain interferometry experiments (see e.g. Brown et al. 1995 and refer-ences therein; cf. also most recently Pylkkänen et al. 2014). Brown et al. (1996) explic-itly discuss this view – which they call the parsimonious view –, but only within theframework of a dualistic ontology that recognizes both the Bohminan particles and thewave-function as genuine ontological entities on their own right. However, as we haveargued above in section 1, there is no point in doing so; in particular, it remains entirelymysterious how the wave-function understood as a physical object on configuration spacecould guide the Bohmian particles. Indeed Brown et al. (1996, § 4) acknowledge thisfact when they concede that their parsimonious view faces what they call the problem ofrecognition , namely to explain how the wave-function of a given particle “knows” whichparticle to guide when there are several particle species in a region of overlap of therespective wave-functions, assuming a factorizable total wave-function for simplicity.Furthermore, there are two primitive ontology theories of quantum mechanics usingthe dynamics proposed by Ghirardi, Rimini, and Weber (GRW) (1986), which seeks toinclude the textbooks’ postulate of the collapse of the wave-function upon measurementinto a modified Schrödinger equation. Bell (1987, ch. 22) suggests that whenever there isa spontaneous localization of the wave-function in configuration space, this developmentof the wave-function in configuration space represents an event occurring at a point inphysical space. These point-events are today known as flashes ; that term was introduced5y Tumulka (2006, p. 826). According to the GRW flash theory (GRWf), the flashesare all there is in space-time. As far as the primitive ontology is concerned, the GRWflash theory is the Bohmian particle ontology without the trajectories: instead of particletrajectories – that is, continuous lines of occupation of points in space-time –, there areonly isolated points being occupied by primitive stuff.Bohmian mechanics and the GRW flash theory both propose a primitive ontology ofprimitive stuff that is discrete : particles or flash-events at space-time points. By contrast,Ghirardi, Grassi, and Benatti (1995) develop an ontology of a continuous matter densitydistribution in physical space (GRWm). The wave-function in configuration space andits temporal development as described by the GRW equation represent at any timethe density of matter in physical space, and the spontaneous localization of the wave-function in configuration space (its “collapse”) represents a spontaneous contraction ofthe matter density in physical space, thus accounting for measurement outcomes andwell localized macrophysical objects in general (see also Monton 2004). Again, matteris primitive stuff, as pointed out by Allori et al. (2014): Moreover, the matter that we postulate in GRWm and whose density is given bythe m function does not ipso facto have any such properties as mass or charge; itcan only assume various levels of density. (Allori et al. 2014, pp. 331–332) Matter thus is gunk, filling all of space. This, however, implies that the primitive stuffadmits of degrees, as expressed by the m function in the GRWm formalism: there is morestuff at some points of space than at others, with the density of matter at the pointsof space changing in time; otherwise, the theory would not be able to accommodatevariation. But it remains unclear what could constitute the difference in degrees ofstuff at points of space, if matter just is primitive stuff. The GRWm theory hence iscommitted to the view of matter being a bare substratum with its being a primitivefact that this substratum has various degrees of density at points of space or space-time. In other words, there is a primitive stuff-essence of matter that admits differentdegrees of density. On Bohmian mechanics and the GRW flash ontology, by contrast, theonly variation consists in some points of space being occupied while others are empty,with there being a change in time in which points of space are occupied. This ontologycan then easily account for the concentration of matter in certain regions of space bymaintaining that in some regions of space, more points are occupied than in other regionsof space.Nonetheless, Bohmian mechanics and the GRW flash theory face the following ques-tion: What is it that occupies points of space? In other words: What accounts forthe difference between a point of space being occupied and its being empty? There are6o intrinsic properties such as mass or charge available that could make up for thatdifference. That is to say, Bohmian particles or GRW flashes do not have an intrinsicessence constituted by intrinsic properties. Even if there are no intrinsic properties, onecould still maintain that Bohmian particles or GRW flashes have a primitive thisness(haecceity). However, haecceitism is a very controversial metaphysical stance. In anycase, it is a purely metaphysical view that is always available if one is willing to pay theprice, physics be as it may. In other words, there is no motivation for haecceitism fromphysics (especially given the explicit and generalized permutation invariance that we willexplain below in section 3). It seems hence that also in the case of a primitive ontologyof discrete objects (particles, flashes), we have to fall back into admitting a primitivestuff-essence of matter that accounts for the difference between a point of space beingempty and its being occupied. The only difference between a primitive ontology of dis-crete objects and a primitive ontology of gunk would then be that in the latter casethat primitive stuff-essence also has to include different degrees of density at points ofspace. In a nutshell, it seems that the primitive ontology theories of quantum physicsare committed to conceiving matter as a Lockean bare substratum. This consequence puts these theories in an uncomfortable position: the commitment toa bare substratum is a controversial metaphysical stance. One may motivate this stanceby claiming that there has to be a primitive stuff-essence at the bedrock of matter. Butone can also with reason object that a primitive stuff-essence in the guise of a baresubstratum is mysterious. In any case, again, the view that there is a primitive ontologyof physics is well-motivated – since physics, including quantum physics, can with goodreason be taken to be about matter in space-time –, but there is no motivation fromphysics to conceive the primitive ontology in terms of a primitive stuff-essence of matter(cf. the objection that Ladyman and Ross 2007, p. 136 note 15, raise against Bohmianparticles). The upshot of these considerations hence is that if one admits an essence ofmatter, that essence be better constituted by properties – or relations, as we shall argue–, but never be primitive.The impasse into which the question of what accounts for the difference between apoint of space being occupied and its being empty runs is a consequence of conceivingthe primitive ontology theories in terms of a commitment to absolute space into whichmatter is inserted. Only in the case of a dualism of there being points of space andmatter occupying these points does that question arise. However, whereas workingwith an absolute background space certainly is an elegant manner of presenting thesetheories (at least as long as the issue of including gravity is left out), there is no reason We are grateful to one of the referees for raising this objection. matter is primitive stuff. It is discrete, consisting in matter points. These are matterpoints, because there is a non-vanishing three-dimensional distance between any two suchpoints.
In other words, they are matter points in virtue of being connected by metricalrelations. In that way, switching from absolutism to relationalism about space removesthe commitment to a bare substratum or a primitive stuff-essence because it opens upthe possibility to conceive the primitive stuff in terms of standing in metrical relationsthat are its essence.A primitive ontology theory that treats matter as primitive stuff, but seeks to avoida commitment to a primitive stuff-essence or bare substratum cannot but adopt theCartesian characterization of matter in terms of spatial extension. In a nutshell, whatdistinguishes points of a primitive matter stuff from points of a hypothetical primitivemental stuff only is that the former, by contrast to the latter, are connected by metricalrelations. Nonetheless, there are no space-time points. There are substances that arenot extended in themselves (points). These are material, because they are connected byspatial relations and move, so that there is change in their spatial relations and thus atemporal development of the spatial configuration of these point-substances. If they werenot connected by spatial relations, but by hypothetical fundamental mental relations,they would not be primitive matter stuff (matter points) and not be physical entities,but primitive mental stuff.Hence, what makes it that a point is a matter point is nothing intrinsic of that point– no intrinsic properties, no primitive thisness, no bare substratum or primitive stuff-essence –, but the fact that it stands in spatial relations. The view of matter as primitivestuff thereby joins the stance known as (moderate) ontic structural realism in claimingthat the identity of the fundamental physical objects, namely the matter points in thiscase, is provided by certain relations, namely metrical relations (see Esfeld and Lam2008). Nonetheless, these relations are strong enough to allow the matter points tofulfill Leibniz’ principle of the identity of indiscernibles in that they can be absolutelydiscernible: it is possible that each matter point is distinct from all the other ones bysome of the distance relations that it bears to other matter points. If they are absolutely Recall that two entities are absolutely discernible if and only if there is a physically meaningful monadicpredicate or, more generally, a physically meaningful formula with one free variable that applies toone but not to the other. res extensa only.9
The Physics of Matter as Primitive Stuff
Let us turn to Bohmian mechanics in order to illustrate the physics of matter as primi-tive stuff, since Bohmian mechanics is the best known example of a primitive ontologyformulation of non-relativistic quantum mechanics and since it is the only primitive on-tology theory for which there is a version worked out in terms of permutation invarianceavailable. Although, if spelled out in a consequent manner, the primitive ontology ofmatter as primitive stuff should go with relationalism about space, we will use for thesake of this illustration the formulation in terms of an absolute background space withsome points of that space being occupied, whereby these occupied points make up theconfiguration of matter in the universe. Our primary aim in this section is to show whatthe physics of primitive stuff in contrast to the physics of material objects with intrinsicessences looks like. Casting that physics at the same time in relationalist terms aboutspace and time would by far go beyond a single paper – the main challenge in this respectis to investigate whether Bohmian mechanics admits a universal wave-function that hasall the right symmetries to depend only on the metrical relations between the particles.Bohmian mechanics, as commonly presented (see the papers in Dürr, Goldstein andZanghì 2013 and the textbook Dürr and Teufel 2009), is a theory about point particlesmoving in three-dimensional space, whereby the quantum wave-function figures in a non-local law of motion for the configuration of particles. Usually, the theory is introduced byformulating the laws of motion on the configuration space R N , where N is the numberof particles and Q ( t ) = ( Q ( t ) , . . . , Q N ( t )) ∈ R N represents their positions at time t .The configuration then evolves according to the guiding equationd Q k d t = ~ m k ψ ∗ ∇ ψψ ∗ ψ ( Q , . . . , Q N ) , (1)where ψ ( q , . . . , q n ) is the wave-function representing the quantum state of the system.The time-evolution of this wave-function, in turn, is given by the Schrödinger equation ı ~ ∂ψ∂t = (cid:16) − N X j =1 ~ m j ∆ j + V ( q , . . . , q n ) (cid:17) ψ, (2)familiar from standard quantum mechanics. The non-local character of the law is man-ifested in the fact that the velocity of any particle at time t depends on the position ofevery other particle at time t ; the law of motion, in other words, describes the evolutionof the particle configuration as a whole . This is necessary in order to take quantumnon-locality – as illustrated for instance by Bell’s theorem (Bell 1987, ch. 2) – into10ccount.The parameters m k appearing in equation (1) and (2) correspond to the mass of the k -th particle. Furthermore, we observe that for (static) electromagnetic interactions, thecharges e k of the particles enter the Schrödinger equation via the Coulomb potential V ( q , . . . , q n ) = X i 12y the electromagnetic interactions and deflected towards A , if it has opposite charge, oraway from A if it has equal charge as particle one. However, if that second particle werepassing near region B , it would be affected in the very same way , no matter how faraway that is from the actual position of the other particle. This scenario demonstrates,firstly, the explicitly non-local character of Bohmian mechanics. It also shows that itwould thus be wrong to think of charge in the familiar way as something localized at theposition of the particles. A similar reasoning would apply to the particle mass, in so faras gravitational interactions play a role in quantum mechanics.A common reply to this issue is that mass and charge should be regarded not as prop-erties of the particle, but as properties of the wave-function, the intuition (presumably)being that the absolute value of ψ – or rather | ψ | – can represent (something akin to) acharge distribution. But this view is untenable for a variety of reasons. To begin with,we have already seen that the view of (effective) wave-functions as physical entities overand above the particles is unwarranted. If the wave-function associated with a particleis not a physical entity, it cannot carry physical properties. Moreover, as soon as weconsider an entangled wave-function of two or more particles, it will correspond to a (nonfactorizing) function on a high-dimensional configuration space and cannot be taken torepresent a distribution of physical quantitates in three-dimensional space.So what attitude shall we adopt vis-à-vis mass and charge in Bohmian mechanics?First and foremost, one should take the theory seriously in its own right and acknowledgethat all the (classical) intuitions that we associate with mass and charge are a priori questionable. In the first instance, m , . . . , m N and e , . . . , e N are merely numericalparameters that appear in the formulation of the Bohmian laws of motion. To illustratethis point, let us assume for the moment that there exists but a single species of particles,i.e. that m k = m l = m and e k = e l = e for all k, l ∈ { , . . . , N } . Hence, we see fromequations (1) and (2) that we are left with the same three constants ~ , m , and e appearingin the equation of motion for any of the N particles. There is then plainly no reason totreat m and e differently from Planck’s constant ~ . In particular, there is no justificationto attach m or e to the particles or to interpret them as localized physical quantities,any more than we would do for ~ . Rather we would regard m and e as nothing morethan two additional constants of nature, numerical parameters entering the equations ofmotion without referring to anything in the physical ontology.However, the commitment to a single type of particle, that is, a single elementarymass and charge, is clearly unsustainable from a physical point of view. Modern particlephysics introduces an entire zoo of elementary particles varying in mass or charge or both:electrons, positrons, muons, anti-muons, the nucleons, respectively their constituent13uarks, and so on. Hence, there must be something in the world which makes it the casethat certain terms (respectively certain coordinates) in the equations of motion refer to,say, an electron rather than a muon.Goldstein et al. (2005b,a) demonstrated the possibility to account for the differentspecies of elementary particles in modern particle physics by reformulating Bohmianmechanics in a way that reflects the ontological commitment to propertyless particlesas primitive stuff, treating all particles as identical . To appreciate what this means andhow the reformulation is carried out, let us begin with the following observation. If weinsist that particles are distinguished only by their position, that is, spatial relationsinstead of intrinsic properties, we note that the configuration space R N has too muchmathematical structure in that it “cares” about permutations of the particle labels. Thatis to say the following: the nature of the Bohmian law of motion (being a first-orderdifferential equation on configuration space) is such that it determines at every time t the change of the system’s spatial configuration depending on the current configuration Q ( t ). However, unless one presupposes a primitive identity or haecceity of the particles,the instantaneous configuration of an N -particle system is completely characterized bya set of N points in physical space that are designated as being occupied by matter.There are no intrinsic properties, nor internal or external relations distinguishing theconfiguration represented by the tuple ( Q , Q , . . . , Q N ) from, let’s say, the configurationrepresented by the tuple ( Q , Q , . . . , Q N ) with the particles 1 and 2 interchanged. It isthus understood that – for so-called identical or indistinguishable particles – the naturalconfiguration space of an N -particle system is not R N , but N R := n S ⊆ R | ♯S = N o , (5)which is the set of all subsets of R containing exactly N elements. Note that thisspace lacks the mathematical structure to represent permutations of the particle labelsin contrast to R N : a point Q ( t ) = { Q , Q , . . . , Q N } ∈ N R – in contrast to the orderedN-tupel ( Q , Q , . . . , Q N ) – describes the fact that at time t there is a particle occupyingspace-point Q , a particle occupying space-point Q , and so on; it does not state thatparticle 1 occupies Q , particle 2 occupies Q , etc.Consequently, the wave-function of the system should now be defined on the config-uration space N R as well, which in fact can be done (Goldstein et al. 2005b, section4). Nevertheless, it is still more convenient, in general, to represent the quantum stateas a function on R N (which can be regarded, mathematically, as the universal coveringspace of N R ). As long as we consider a system in which all particles are associatedwith the same mass and charge, the demand of consistency then leads immediately to14 wave-function that is symmetric or anti-symmetric under permutations of the particlecoordinates and hence to the famous boson/fermion alternative . In Dürr et al. (2006), N R was thus already introduced as the configuration space of identical or indistinguish-able particles, referring to a single species of particles, and it is shown how the quantumstatistics of identical particles thus arise in the Bohmian theory (see also Dürr and Teufel2009, ch. 8.5).However, we now note that as soon as we have to admit more than one value for theparameters m k , the standard formulation of Bohmian mechanics breaks down. That isbecause equation (1) no longer defines a law of motion on N R , since it discriminatesdifferent particles by their associated mass, while configurations represented on N R donot do so. The basic idea of Goldstein et al. (2005b,a) is thus to symmetrize equation(1) in order to get a permutation invariant equation, because any permutation invariantequation on R N defines, in a canonical way, a law of motion on N R , the configurationspace of identical particles. In this way, they show that we can treat all particles asidentical, while still accounting for the empirical data that, as usual, are explained interms of a particle “zoo”.To preserve equivariance of the law, i.e. the conservation of total probability by theBohmian flow, the symmetrization has to be done in the following way. The standardguiding equation (1) can be written in the formd Q d t = j ( Q ( t )) ρ ( Q ( t )) , (6)where ρ = ψ ∗ ψ is the probability density and j = ( j , . . . , j N ) with j i = ~ m i Im ψ ∗ ∇ i ψ the probability current corresponding to the system’s wave-function ψ . In equation (6),numerator and denominator have to be symmetrized independently by summing over allpossible permutations of the particle labels 1 , . . . , N . Hence, we get a new, permutation-invariant guiding equation, which readsd Q k d t = P σ ∈ S N j σ ( k ) ◦ σ P σ ∈ S N ρ ◦ σ ( Q ( t )) . (7)15ere, the sum goes over all elements of the permutation group S N , and σQ := (cid:16) Q σ − (1) , . . . , Q σ − ( N ) (cid:17) means that every coordinate Q i is assigned a new index Q σ − ( i ) , changing the order inthe N -tupel.In this theory, which Goldstein et al. (2005b,a) dubbed identity-based Bohmian me-chanics, we do not attribute a priori any mass to any specific particle. The law of motionmerely determines N trajectories for N particles, and it is a characteristic of this law that one of those trajectories happens to behave – at least in the relevant circumstances– like the trajectory of a particle with mass m , another like the trajectory of a particlewith mass m , and so on, depending only on the (contingent) initial conditions of thesystem, respectively the universe.To illustrate how this works, let us discuss an example given in Goldstein et al.(2005b, section 3) that compares the standard formulation of Bohmian mechanics withthe identity-based version. Consider a two-particle universe consisting of an electronwith mass m e and a muon with mass m µ . Suppose, for simplicity, that they are in anon-entangled state Ψ( q , q ) = φ ( q ) χ ( q ) (note that we could symmetrize this wave-function, though this would be redundant when plugged into the symmetrized guiding-equation). Then, the standard guiding law (1) leads to the following equations of motion:d Q d t = ~ m e Im ∇ φ ( Q ) φ ( Q ) , d Q d t = ~ m µ Im ∇ χ ( Q ) χ ( Q ) . (8)In contrast, the symmetrized guiding equation (7) readsd Q d t = ~ m e | χ ( Q ) | Im ( φ ∗ ( Q ) ∇ φ ( Q )) + ~ m µ | φ ( Q ) | Im ( χ ∗ ( Q ) ∇ χ ( Q )) | φ ( Q ) | | χ ( Q ) | + | φ ( Q ) | | χ ( Q ) | d Q d t = ~ m µ | φ ( Q ) | Im ( χ ∗ ( Q ) ∇ χ ( Q )) + ~ m e | χ ( Q ) | Im ( φ ∗ ( Q ) ∇ φ ( Q )) | φ ( Q ) | | χ ( Q ) | + | φ ( Q ) | | χ ( Q ) | . (9)We see that equation (8) ascribes – or presupposes – an intrinsic mass and thus a distincttype to every particle: particle 1, described by the coordinates Q , is the electron withmass m e , while particle 2, described by the coordinates Q , is the muon with mass m µ .In equation (9), by contrast, neither Q nor Q is designated as the position of theelectron, respectively the muon. A priori , the two particles are distinguished only by16he position that they occupy at time t . However, if we consider a situation in which φ and χ have disjoint support, say, when one wave-packet is propagating to the left andthe other one to the right, one of the two sums in the nominators and denominatorswill be zero, so that the equation of motion effectively reduces to equation (8) (possiblywith the indices 1 and 2 interchanged). This is to say, in particular, that in situationswhere the two-particle wave-function is suitably decohered, one of the particles will playthe role of the electron – being effectively described by equations (1) and (2) with theparameter m e – while the other one will play the role of the muon – being effectivelydescribed by equations (1) and (2) with parameter m µ .Which trajectory turns out to be guided by which part of the wave-function therebydepends only on the law of motion and the (contingent) initial conditions of the system,rather than on intrinsic properties of the particles. In fact, if both parts of the wave-function were brought back together and then separated again, one and the same particlecould switch its role from being the electron to being the muon, and vice versa. Hence,like a particle’s spin, we must conclude that to be an electron, a muon, or a positron,etc. is nothing more and nothing less than to move – in the relevant circumstances –electronwise, muonwise, or positronwise, and so forth. There are no properties in thistheory defining different species of particles, but only primitive stuff , following a law ofmotion that accounts for the phenomena conventionally attributed to a multiplicity ofparticle-types.Apart from such circumstances in which the different parts of the wave-function arewell separated, one could say that the particles in the previous example are guidedby a superposition of (what one would usually call) an electron wave-function and amuon wave-function. However, it would be misleading to claim that this amounts toa superposition of being an electron and being a muon. Ontologically, there are nosuperpositions of anything, only propertyless particles moving on definite trajectories.Rather, the labels “electron”, “muon”, etc. are meaningless in the general case.One obvious objection to the move proposed by Goldstein et. al. is that the guidinglaw (7) is much more contrived than the one in standard Bohmian mechanics. To someextent, this is a correct observation and we are indeed trading a sparse ontology for amore complicated mathematical formalism by endorsing the symmetrized theory. Thatnotwithstanding, a few things can be said to address this worry. First, one should notethat the apparent complexity of equation (7) is really just the price for expressing a lawof motion for configurations in N R on the coordinate space R N and doesn’t necessarilyamount to more complicated physics. Second, it should be noted that (modulo somesubtleties discussed by Goldstein et al. 2005b,a) the symmetrized theory will give rise17o the familiar statistical description of subsystems in terms of effective wave-functions,which is really all that matters for most practical purposes.In this context, it should also be noted that, given the universal wave-function, the“right” statistical description of subsystems – that is, the one agreeing with the predic-tions of standard quantum mechanics, arises for typical initial conditions in terms of theparticle configuration, that is, in quantum equilibrium (see Dürr, Goldstein, and Zanghì2013, ch. 2). Hence, the emergence of different particle types as empirically observed innature is not attributed to special initial conditions (quite the opposite), though it isascribed to the particular form of the universal wave-function, i.e. to the physical law, ifthe latter is understood as nomological (we will expand on the nomological view of thewave-function in the upcoming sections).Finally, concerning the (empirical) content of the proposed theory, it should be empha-sized that the trajectories described by identity-based Bohmian mechanics will in generaldiffer from those obtained from standard Bohmian mechanics, but that the statisticalpredictions for experimental outcomes are the same. In this sense, the symmetrizedtheory is empirically equivalent to Bohmian mechanics and hence empirically equivalentto standard quantum mechanics. This shows, once more, that the physical ontology canneither be empirically determined, nor read off from the measurement-formalism of stan-dard quantum mechanics, while, on the other hand, the choice of a primitive ontology can supplement or enlighten the structure and formulation of the theory.In particular, if the physical ontology is one of propertyless particles, this stronglysuggests permutation invariant laws of motion in which all particles are treated as iden-tical – to borrow once more the terminology commonly employed in physics. Of course,this terminology is misleading in the sense that there is obviously a plurality of parti-cles instead of just one particle. The meaning of identity-based Bohmian mechanics –and more particularly the meaning of permutation invariance within this framework – israther that we are committed to an ontology of primitive stuff in the sense of particlesthat do not possess any intrinsic properties nor any intrinsic identity. Permutation in-variance thus means precisely that there is nothing to the particles beyond their positionin the total configuration, in particular nothing that the laws of motion could refer toin order to establish a different dynamical role for different particles depending on someintrinsic characteristics.To sum up, identity-based Bohmian mechanics provides for a clear ontological meaningof permutation invariance: it encodes a primitive stuff ontology of individuals withoutany intrinsic identity and properties, though (absolutely) discernible in virtue of theirposition in the total configuration. Furthermore, permutation invariance applies here to18ll the particles, since there is only primitive stuff and no different species of particles,by contrast to concerning only the particles of the same species as in standard Bohmianmechanics or the wave-function supposedly corresponding to particles of the same speciesas in textbook quantum mechanics. Staying within the framework of a primitive ontology of particles as in Bohmian me-chanics, how are we to conceive dynamical variables such as mass or charge that areattributed to the particles taken individually without making up for intrinsic essencesthat constitute different species of particles? Moreover, as mentioned in section 1, inthe primitive ontology approach, it is reasonable to conceive the quantum state as anomological entity by contrast to a physical entity on a par with the primitive ontol-ogy. But what does this mean? In this section and the next one, we will show that themain philosophical views about laws of nature can be employed in order to answer thesequestions. We will focus on Humeanism on the one hand and dispositionalism on theother.Let us start with Humeanism. According to this view, the world is a vast mosaic oflocal matters of particular fact, such as point particles being connected only by relationsof spatio-temporal distance. Given an initial configuration of such point particles, thereis nothing about that configuration that puts a constraint on its temporal development.A certain temporal development just happens to occur; there is nothing that guides,governs, or determines it. Nevertheless, considering that temporal development as awhole – that is, the distribution of the point particles throughout the whole of space-time –, that distribution exhibits certain patterns or regularities. Consequently, if onesets out to put forward a description of the distribution of the point particles in space-time, one can do better than dressing a very long list that registers each particle position.According to what is known as the Humean best system account, the laws of nature arethe axioms of the system that achieves the best balance between being simple and beinginformative in describing the distribution of matter throughout the whole of space andtime. In brief, laws of nature both simplify and are informative, striking the best balancebetween these two virtues (see notably Lewis 1973, ch. 3.3, pp. 72–75, and 1994, section3, as well as Cohen and Callender 2009; there is no space here and it is not the aim ofthis paper to consider the internal problems of Humeanism).Hall (2009, § 5.2), in particular, has put forward a version of Humeanism that regards19he vast mosaic of local matters of particular fact as consisting only in point particlesstanding in relations of spatio-temporal distance. These particles are just primitive stuff.Their distribution throughout space-time – that is, the development of the metricalrelations among the particles – exhibits certain regularities. Suppose that the laws ofclassical mechanics and electrodynamics figure in the Humean best system that capturesthese regularities. Then dynamical variables such as mass and charge appear in theselaws. On the basis of these laws being part of the Humean best system, one can thenattribute properties like mass and charge to the particles. That is to say: predicates suchas “mass” and “charge” apply to the particles. However, these predicates do not representproperties that the particles have per se , as something essential or intrinsic to them.They apply to the particles in virtue of the contingent fact that their motion throughoutthe whole of space-time happens to manifest certain regularities. Hence, what makesthe application of these predicates true is nothing over and above the distribution ofprimitive stuff throughout space and time. Nonetheless, this is not instrumentalism:Humeanism, applied to the primitive ontology approach in physics, is the view that theprimitive ontology is the entire ontology. However, the primitive ontology is an ontology that stands on its own feet: it consists in theoretical entities such as point particles thatexist in the world independently of observers and their beliefs.This idea can also be applied to the wave-function in any of the primitive ontologytheories of quantum physics, notably Bohmian mechanics (see Miller 2014, Esfeld 2014,Callender 2014; see also already Dickson 2000). It can thus be employed to spell out whatit means that the wave-function is nomological by contrast to being a physical entity ona par with the primitive ontology. Again, the Humean mosaic consists in the distributionof primitive stuff throughout the whole of space-time – such as particle trajectories, flash-events, or a matter density field. That distribution exhibits certain regularities. Supposethat the laws of quantum mechanics figure in the Humean best system that captures theseregularities, and let these laws be the Bohmian guiding equation and the Schrödingerequation, or a GRW-type equation and a law establishing a link with the primitiveontology. Then a universal wave-function describing the quantum state of the primitivestuff appears in these laws, and the quantum state includes parameters such as mass andcharge. However, as these latter parameters do not require an ontological commitmentto anything more than the distribution of primitive stuff throughout the whole of space-time in classical mechanics, so the quantum state is no addition to being: given the wholedistribution of the primitive stuff throughout space-time, a law describing the temporaldevelopment of a universal wave-function enters into the Humean best system as a meansto achieve a description of the distribution of the primitive stuff that strikes the best20alance between being simple and being informative about how the stuff is distributed.This law simplifies and is informative in any case, since in a deterministic theory such asBohmian mechanics, specifying the particle configuration and the wave-function at anygiven time is sufficient to capture the particle configuration at any other time. Giventhe law in which the wave-function figures, one can then attribute a quantum state asrepresented by the universal wave-function to the particle configuration, in the sensethat the propositions doing so are true; but their truth-maker is the distribution of theprimitive stuff throughout the whole of space-time, and not a quantum state that existsover and above the particle configuration.Finally, coming back to the link between the primitive ontology approach to physicsand relationalism about space-time, one can apply the Humean view of the primitiveontology being the entire ontology to space-time itself. Suppose that there is an initialconfiguration of matter points that are primitive stuff and that are connected by metricalrelations. The matter points move so that there is change in their spatial relationsand thus a temporal development of the initial configuration of matter points. OnHumeanism, there is nothing that puts a constraint on how that change has to occur.Some such change just happens. If one combines Humeanism with relationalism aboutspace and time, there is furthermore nothing about that initial configuration that singlesout a particular motion as inertial motion and a particular system of matter points asan inertial system. However, as Huggett (2006) has shown, given the whole motion ofthe matter points, there are some patterns or regularities in this motion that make itpossible to conceive a Humean best system achieving a good balance between beingsimple and being informative in describing that motion. Based on this best system, onecan then single out a certain motion as inertial and certain systems of matter pointsas inertial systems. One can thus account for absolute quantities such as accelerationin an ontology of a Humean space-time relationalism applied to the laws of Newtonianmechanics.It is evident that this strategy can be put to work for any space-time, not only aNewtonian one, as Humeanism is applicable to any primitive ontology theory of matter.One has to assume an initial configuration of extended stuff – such as matter pointsbeing connected by metrical relations –, that configuration happens to develop in a cer-tain matter. Considering that development as a whole, it exhibits certain patterns orregularities. Based on these patterns or regularities, there is a Humean best systemincluding the laws of both matter and space-time. Given that system, dynamical vari-ables can be attributed to the matter points, some systems of them can be singled outas inertial systems, etc. On Humeanism, whatever properties are attributed to matter21r space-time come all in one package, figuring in the Humean best system and beingdefined by their role in that system, instead of being properties that belong to matteror space-time as such. Although Humeanism is a coherent philosophical way to conceive matter as primitivestuff, showing how all the dynamical variables that are commonly attributed to materialobjects can be derived from the Humean best system, the physics of matter as primitivestuff is not committed to the metaphysics of Humeanism. In other words, if the universalwave-function is nomological rather than a physical object on a par with the primitiveontology, its nomological character does not have to be spelled out in the frameworkof Humeanism about laws of nature. Indeed, there are many well known philosophicalreservations against Humeanism in general. In particular, Humeanism cannot but regardit as a brute fact that the regularities on which we rely in science as well as in everydaylife always turn out to be well-confirmed. There is no constraint at all on which localmatters of particular fact can and which ones cannot occur in the future of any givenlocal matter of particular fact, since what the laws of nature are depends on whatthere will be in the future of any given local matter of particular fact, instead of thatfuture depending on the laws of nature. Hence, the laws of nature cannot be invokedto answer the question of why certain regularities – such as e.g. those ones experiencedas gravitation, or those exhibited in the EPR-experiment – always turn out to be well-confirmed. There simply is no answer to that question in Humeanism. Again, thereis nothing incoherent about this position. Yet the desire to obtain an answer to thatquestion motivates the search for a more ambitious metaphysical framework, that is, onethat admits modal connections in nature which put a constraint on what can and whatcannot happen in the universe given an arbitrary initial configuration of matter.The central anti-Humean answer to this question consists in anchoring the laws ofnature in properties that are attributed to the physical systems (see notably Bird 2007).These properties are such that it is essential for them to exercise a certain dynamicalrole for the temporal development of the physical systems. The laws, in turn, expressthat dynamical role. These properties hence are dispositions or powers . Thus, on thisview, mass and charge in classical mechanics are local properties of the particles whosefunction is to accelerate the particles as described by the laws of Newtonian mechanicsand classical electromagnetism. By way of consequence, on dispositionalism combinedwith a primitive ontology view of classical mechanics, the primitive stuff particles do22ndeed obtain properties each over and above standing in metrical relations. But theseproperties are not essential to the particles, and their role is not to provide an intrin-sic identity of the particles; their job exclusively is a dynamical one, namely to put aconstraint on how the particles move, given an initial configuration of particles whoseidentity is provided by the metrical relations in which they stand.When it comes to quantum physics, it is no longer possible to conceive mass and chargeas local dispositional properties or powers that belong to the particles taken individually,as various thought experiments such as the ones mentioned in section 3 make clear (andsee Brown et al. 1995 for more such experiments). Against this background, we haveargued in section 3 that what stands for mass and charge in the equation of motionare mere parameters without any direct ontological correlate. In brief, there are nomass and charge distributions influencing the motion of the particles. There only is theuniversal wave-function representing the quantum state. However, the quantum state isdefined on configuration space. Hence, if one intends to attribute to the quantum statean ontological weight as a dynamical variable – by contrast to regarding it simply asa convenient means to capture the salient regularities in the motion that the particleshappen to take –, one faces the difficulty of having to avoid the incoherent dualismmentioned at the beginning of this paper, namely the dualism of being committed toparticles existing in physical space and a quantum state existing in configuration space.Dispositionalism avoids this pitfall in the following manner: as in classical physicsthe particles taken individually instantiate dynamical, dispositional properties that de-termine their motion and that are represented by the mass and the charge variablesin the laws of motion, so in quantum physics, the particle configuration as a whole in-stantiates a dynamical, dispositional property that determines its temporal developmentand that is represented by the universal wave-function figuring in the laws of (identity-based) Bohmian mechanics or the GRW theory. In brief, according to dispositionalism,the universal wave-function represents the common disposition of motion of the particleconfiguration (see Belot 2012, pp. 77-80, and Esfeld et al. 2014, sections 4 and 5). Themotion of the particles is then such that, in certain specific circumstances, it is possibleto consider them as if they were carrying some local properties such as mass and chargethat influence their motion, even though, from an ontological point of view, there is nosuch thing. Behaving like a “massive particle” or a “charged particle” is only the resultof the particular particle motion (rather than its determinant) and contingent on theuniversal wave-function and the initial conditions.In the framework of dispositionalism, the shift from classical to quantum mechanicshence amounts to a shift from local dynamical properties determining the motion of23he particles to one holistic property of the particle configuration (the quantum state,represented by the universal wave-function) doing so. It is then more appropriate tocharacterize this property as a power than as a disposition, since if there is one holisticproperty of the particle configuration determining its temporal development, there is noquestion of an external stimulus or triggering condition for its manifestation (as a massor a charge qua local property of a particle requires another massy or charged particleto manifest itself in the acceleration of the particles). In brief, on Bohmian disposi-tionalism, the primitive stuff particles collectively instantiate one power (represented bythe universal wave-function) that determines their motion by determining their velocity.Consequently, this collective power relates strictly speaking all the particles with one an-other, determining their motion in tandem so to speak and thereby explaining quantumentanglement and the EPR correlations.In general, if dispositions or powers instantiated by the particles taken individuallycan influence their motion, so can a collective power instantiated by the particle config-uration. In both cases, the particle positions, consisting in the metrical relations thatindividuate them, by no means fix the disposition or power that determines the mo-tion and thus the temporal development of the particle positions (i.e., their metricalrelations). Such a disposition or power is in any case a modal property that has to beadmitted in addition to the primitive ontology, but instantiated by the elements of theprimitive ontology, as that what fixes what is possible and what is not possible abouttheir motion. The metaphysical conception of dispositionalism applied to the primi-tive ontology of physics – that is, the commitment to properties as that what puts aconstraint on the temporal development of the primitive stuff – is the same in bothcases. A holistic property or collective power doing so is no less intelligible and no moremysterious than local properties or powers doing so. The latter just are more familiarto us given our familiarity with classical physics and our unfamiliarity with quantumphysics. In other words, the shift from local properties to holistic or collective propertiesis imposed upon us by the transition from classical to quantum physics. Any philosoph-ical theory of properties has to adapt itself to this shift. Dispositionalism does so bycountenancing dynamical properties (dispositions, powers) that are instantiated by theparticle configuration as a whole instead of by the particles taken individually.One can further illustrate this conception by linking it up with ontic structural real-ism. Since this collective power relates all the particles with one another, one can alsoconceive it as a structure defined on the configuration of the particles. This again isan ontic structure since, according to dispositionalism, it exists in the world over andabove the primitive stuff (the particle configuration), albeit instantiated by it. However,24ne has to be careful not to confuse this ontic structure with the relational or structuralindividuation of the primitive stuff explained at the end of section 2: quantum entangle-ment conceived as an ontic structure in the framework of the physics of primitive stuffhas nothing to do with the individuation or the discernibility of the physical entities;that individuation and discernibility, both at a time and in time, is obtained throughthe metrical relations in which the matter points stand at any time. It is not touchedby the issue of Humeanism vs. modal realism (dispositionalism) as regards the laws ofnature. The entanglement structure, by contrast, concerns only the dynamics of thematter points. If this structure exists over and above the matter points, it is a modalstructure, putting a constraint on the temporal development of the matter points (theirmotion), whereas there is nothing modal about the metrical relations or structure insofaras they individuate the matter points, accounting for these points being matter pointsand being absolutely discernible. By way of consequence, only the modal realist but notthe Humean is committed to the entanglement structure. This paper started from recalling the two principled options for an ontology of quantumphysics: (1) quantum state realism, according to which the quantum state as definedby the universal wave-function on configuration space is the physical reality, and (2)a primitive ontology theory, according to which the physical reality consists in matterexisting in three-dimensional space or four-dimensional space-time. The aim of thispaper was to push the primitive ontology option to its ultimate consequence, which is toregard matter as primitive stuff, namely as points that are matter points only in virtueof the metrical relations in which they stand; in particular, these matter points do notcarry any intrinsic properties and do not possess any intrinsic identity. The metricalrelations individuate them, making them (absolutely) discernible. These matter pointsare particles, if they persist and if their motion traces out continuous lines in space(worldlines that can be conceived as particle trajectories). Consequently, there are nodifferent particle species in the fundamental ontology and “permuting” the particlesobviously does not lead to any new physical situation. There just are propertylessparticles qua matter points. We have shown how this view is naturally encoded inidentity-based Bohmian mechanics.We then elaborated on two principled options for introducing physical propertiesthrough the role that they play for the dynamics of the primitive stuff. Accordingto Humeanism, there is nothing over and above the primitive stuff throughout space25nd time. Given an initial configuration of primitive stuff, a certain temporal develop-ment of that configuration happens to occur. But there is nothing in nature that puts aconstraint on which temporal development can happen and which one cannot happen.Given the distribution of primitive stuff throughout the whole of space–time, that dis-tribution happens to exhibit certain patterns or regularities, which make it possible toformulate a Humean best system. The variables figuring in the Humean best systemcan then be attributed to the primitive stuff, but they do not represent an ontologicalcommitment to anything over and above spatio-temporally extended primitive stuff.According to modal realism, by contrast, there is something in nature over and abovethe primitive stuff that puts a constraint on its temporal development, fixing what canand what cannot happen given an initial configuration of primitive stuff. Disposition-alism spells this idea out in terms of dispositions or powers that the primitive stuffinstantiates over and above being individuated by the metrical relations in which thematter points stand. These dispositions or powers enter the ontology only through therole that they play in determining a certain temporal development of the primitive stuff.They thereby ground the laws of nature. In classical physics, these are dispositions orpowers that are instantiated by the matter points (the particles) taken individually; inquantum physics, there is in the fundamental ontology only one collective power instan-tiated by the configuration of the matter points (the particles) as a whole, tying theirtemporal development together.To sum up, the primitive ontology option applies to both classical and quantumphysics. If one endorses a primitive ontology of particles, the primitive ontology isthe same in classical and quantum physics: particles as primitive stuff, individuated bythe metrical relations in which they stand. The difference between classical and quantumphysics concerns only the dynamics, namely the dynamical properties (dispositionalism)or predicates (Humeanism) attributed to the particles in order to account for the changein their metrical relations (that is, their motion): local properties in classical physics, acollective one in quantum physics.Against the background of what has been achieved in this paper, we regard it as theforemost task for metaphysics to put the arguments for and against Humeanism andmodal realism (dispositionalism) in the framework of a primitive ontology of primitivestuff shared by both these metaphysical stances. 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