The Wave-Function as a Multi-Field
TThe Wave-Function as a Multi-Field
Mario Hubert ∗ Davide Romano † January 22, 2018
Forthcoming in the European Journal for Philosophy of Science
It is generally argued that if the wave-function in the de Broglie–Bohm theory is aphysical field, it must be a field in configuration space. Nevertheless, it is possibleto interpret the wave-function as a multi-field in three-dimensional space. This ap-proach hasn’t received the attention yet it really deserves. The aim of this paper isthreefold: first, we show that the wave-function is naturally and straightforwardlyconstrued as a multi-field; second, we show why this interpretation is superior toother interpretations discussed in the literature; third, we clarify common miscon-ceptions.
Contents ∗ Université de Lausanne, Faculté des lettres, Section de philosophie, 1015 Lausanne, Switzerland. E-mail: [email protected] † E-mail: [email protected] a r X i v : . [ phy s i c s . h i s t - ph ] J a n Can the Wave-Function Be a Field?
The wave-function is a peculiar object: it is mathematically defined in configurationspace, and yet it is supposed to determine the motion of particles in three dimensions.We need to translate this mathematical picture into a coherent ontological story. Sincethe particles and the wave-function are defined in too different mathematical spaces,it has been generally agreed that the wave-function cannot be interpreted as a field inthree-dimensional space. Whereas the electromagnetic field assigns to every point inthree-dimensional space a unique field value, the wave-function doesn’t do so.One strategy to save a field interpretation would be to declare configuration space asthe fundamental space of the world. In this space, the wave-function indeed assigns toevery point a unique value. The following passage by John Bell is usually interpreted tosuggest such that:
No one can understand [the de Broglie–Bohm] theory until he is willing to think of ψ as a real objective field rather than just a ‘probability amplitude’. Even thoughit propagates not in 3-space but in 3 N -space. (1987, p. 128) The most developed interpretation along these lines is the marvelous point interpretationby David Albert (1994).Another strategy for a field interpretation of the wave-function was pursued by TravisNorsen (2010). In fact, this is rather a new theory than just a mere interpretationbecause the wave-function is reduced to local fields in three-dimensional space governedby a revised Schrödinger equation.In this paper, we steer a middle course between these two proposals. We aim atshowing that the wave-function can be indeed construed as a new kind of field in three-dimensional space, namely, as a multi-field . We will defend this interpretation againstAlbert’s interpretation and Norsen’s theory. For the multi-field incorporates the best ofboth worlds: an ontology in three-dimensional space without changing the mathematicalformalism.
The multi-field view starts from the idea to generalize a classical field, which specifiesa definite field value for each location of three-dimensional space. A charged particlethat is posited at a given location will feel the force generated by the value of the fieldat this location. The multi-field generalizes this concept to N -tuples. Given an N -particle system, a multi-field specifies a precise value for the entire N -tuple of points inthree-dimensional space, thus determining, given the actual positions of N particles, themotion of all particles.We suggest that the wave-function is the mathematical representation of such a multi-field. In the first-order formulation of the de Broglie–Bohm theory, the multi-field spec-ifies the velocity of the particles according to the guiding equation, whereas in thesecond-order formulation it specifies the acceleration of the particles, according to the2uantum Newtonian law: F C + F Q = ma . In other words, the multi-field assigns a“multi-velocity” to the configuration of particles in the first-order theory and a “multi-acceleration” induced by a multi-quantum-force to the configuration of particles in thesecond-order theory. The multi-velocity and multi-acceleration then generate the veloc-ities and accelerations for each single particle of the configuration.There is an important difference between the classical field and the multi-field. Inthe classical case, the field of N particles can always be decomposed into a sum ofthe single-particle fields, because every particle produces its own field. As the wave-function is not produced by “quantum sources”, the multi-field of an N -particle systemis not decomposable into a sum of single-particle quantum fields. If the wave-functionis factorizable, it may be decomposable into single-particle wave-functions, but each ofthese wave-functions is not generated by its corresponding particle. Thus, the multi-fieldis rather a holistic or relational field assigned to sets of N particles.The idea of the multi-field is not novel. It dates back to Forrest (1988, Ch. 5), wherehe introduced the concept of polywaves as a generalization of classical mono-waves : I posit polywaves, which are disturbances to polyadic fields. The familiar monowaves(monadic waves) are assignments to each location of some member of the set of pos-sible field-values for that location. Likewise an [ N ]-adic polywave is an assignmentto each ordered n-tuple of locations of a member of the set of possible field-valuesfor that [ N ]-tuple of locations. The integer [ N ] is just the “number of particles”.And the possible field-values are [ N ]-adic relations. (1988, p. 155) In the original proposal by Forrest, the concept of polywaves is defined in the frame-work of standard quantum mechanics. The problem with this approach is that theinteger N in the definition above cannot be straightforwardly understood as the numberof particles, since in standard quantum mechanics a system is described only by thewave-function. This is the reason why Forrest’s metaphysics seems to be constitutedby pure relations between (empty) points of space, which seems to be a rather peculiarontology. This problem is solved in the de Broglie–Bohm theory, where the integer N refers to the number of real particles, and the relations between particles are easily ex-plained by the dynamical correlations induced by the multi-field as we have describedabove.Belot (2012) was then the first to apply Forrest’s polywaves to the de Broglie–Bohmtheory and to dub this approach the multi-field interpretation. After a brief sketch ofthe multi-field view, Belot dismisses it for the following four reasons (pp. 72-3):1. The multi-field doesn’t have sources.2. The multi-field violates the action-reaction principle.3. The multi-field doesn’t restore energy-momentum conservation.4. The multi-field transforms under Galilean boosts differently from the electromag-netic field. 3bjections 1, 2, and 4 arise from the requirement that the multi-field must have essen-tially the same features as classical fields. We expect the multi-field to have differentphysical features than its classical counterpart. This is not problematic for the existenceof the multi-field, but also desirable: if a multi-field behaved like a classical field, therewould be no point of introducing a novel entity in the ontology of the physical world.A crucial question is to understand which features are essential to fields in general andwhich are only essential to classical fields. We want to make this distinction starting froma definition of a classical field, and then giving a generalization of it for the multi-field. We can think of a classical field to be defined by the following features:(a) it is an assignment of intrinsic properties to the points of space it is defined on,and(b) it ensures energy and momentum conservation.Now, in the case of the multi-field, we must substitute (a) with(c) it is an assignment of intrinsic properties to particular N -tuples of points of three-dimensional space.In sum, we suggest that the definition of a multi-field is captured by statements (b) and(c), and that only classical fields are required to obey (a) and (b). So, energy conservationis an important factor to reify a mathematical function into a physical field, and we willsee indeed that the multi-field permits to restore energy-momentum conservation in thede Broglie-Bohm theory. Now, we are ready to answer Belot’s objections against themulti-field. No Sources and no Action-Reaction
We think that these two features are intrinsically connected with each other: a particlemust act back on a field if it has generated the field itself. In the de Broglie-Bohmtheory, the multi-field is not produced by particles, and therefore it is plausible thatthe action-reaction principle is violated. However, this is not problematic for the theory(since it is physically coherent) nor for considering the wave function as a field, since theaction-reaction principle is not included in the defining statements (b) and (c).
Energy-Momentum Conservation
In the de Broglie-Bohm theory, there is energy-momentum conservation for closed sys-tems, and this is assured by the wave-function. This can be easily shown within theHamilton-Jacobi formulation of the theory, where the classical potential V and the quan-tum potential Q = − (cid:126) m ∇ ΨΨ together contribute to energy-momentum conservation. In However, we will not enter here in the metaphysical issue concerning the nature of fields. As thenotion of a classical field can be framed in different metaphysical views (for example, Humean view,dispositional view, etc.), the same procedure is in principle applicable to the multi-field. V and Q are time-independent (that is, if the system is closed), the totalenergy of particles along their trajectories is conserved (Holland, 1993, p. 285): N X i =1 mv i + V + Q = const . A similar relation holds for momentum conservation, in the absence of external (classicaland quantum) forces. If P Ni =1 ∇ i ( V + Q ) = 0 then N X i =1 p i = const.A concrete example may help to illustrate the situation. Consider a one-particle systemin an infinite potential well. The wave-function of the system inside the well isΨ( x, t ) = A sin( kx ) e i (cid:126) Et , with total energy E tot = k (cid:126) m . Yet, the kinetic energy of the particle is zero ( v = ∂S∂x = 0, since the wave functionis real), and the classical potential energy V is zero (by definition inside the potentialwell). So where does the energy of the system come from? It comes from the quantumpotential. For we have E tot = E kin + V + Q = 0 + 0 − (cid:126) m ∇ ΨΨ = k (cid:126) m . In sum, the total energy of the system is absorbed by the quantum potential, which isproduced by the wave-function. Interpreting the wave-function as a multi-field permitsto account for energy conservation in a natural way.
Galilean Transformation
Regarding the transformation properties of the multi-field, it is a mathematical fact thatthe wave-function transforms differently under Galilean boosts than the electromagneticfield. The correct transformation involves a multiplication of the boosted wave-functionby a plane wave with the same velocity. We can accept this transformation behavior ofthe multi-field as a non-classical feature arising from a non-classical law, the Schrödingerequation. Furthermore, a plane wave can be split up into two parts, one dependent onspace and the other independent of space. The space-independent part can be interpretedas a phase of the wave-function. But this phase factor doesn’t change the motion ofparticles. Therefore, this would indicate that the mathematical representation of themulti-field is just determined up to a phase factor. Moreover, this objection, like theone of the action-reaction principle, does not undermine the possibility to regard the5ave-function as a field, since the ordinary Galilean transformation is not included inthe statements (b) and (c).We think that the the multi-field account naturally explains the non-local behaviorof Bohmian particles: since the value of the multi-field depends on N -tuples of pointsand not on single points, the behavior of a given configuration of particles is intrinsicallynon-local, as it were a single structure moving in three-dimensional space. The multi-field is thus a new type of field mathematically represented by the wave-function. Itfills the physical space with precise values for every N -tuple of points. Particles willfeel a certain velocity and a certain acceleration depending on the actual configuration( x , . . . , x N ) and depending on the precise value of the quantum field Ψ( x , . . . , x N ).Although the multi-field is a physical field in three-dimensional space, its mathematicalrepresentation is given by the usual wave-function in configuration space. Therefore,the multi-field interpretation does not have to be confused with Norsen’s theory of localfields, which aims at defining the wave-function itself in three dimensions. We show howthis idea differs from the multi-field in the next section.
Travis Norsen (2010) proposed a Bohmian quantum theory, in which there is no longera wave-function in configuration space (see also Norsen et al., 2015). Instead, the maindynamical entity is the conditional wave-function associated to every particle in three-dimensional space. The one-particle conditional wave-functions, however, don’t sufficeto recover all the predictions of the de Broglie–Bohm theory, since they cannot describeentangled states between particles. Norsen presents a nice example.Imagine two particles that are about to collide. We can prepare the system in twodifferent ways. In the first case, we start with a non-entangled wave-function (see Fig.1), and in the second case, we prepare the system to be entangled (see Fig. 2). Theinitial particle positions are the same. And, more importantly, the initial conditionalwave-functions of both particles are the same, too. Yet, we can prepare each system insuch a way that the particles move differently after collision.This shows that conditional wave-functions cannot do the job alone in retrieving allBohmian trajectories. While conditional wave-functions can render the correct trajec-tories in the first example, they cannot do so for the entangled state. The informationabout entanglement gets lost in the definition of conditional wave-functions—that’s thesame for the reduced density matrix in an EPR experiment, where it merely gives usthe statistics for one particle irrespective of what happens to the other particle.Norsen’s idea is therefore to add additional local fields to the conditional wave-functions to recover quantum entanglement. The task of these new fields is to changethe conditional wave-functions of each particle in such a way that they render the correct Chen (2017b) challenges the view of defining the wave-function in mathematical terms; he insteadproposes a nominalistic approach. The conditional wave-function ψ t ( x ) of a particle is defined by the universal wave-function Ψ, oncethe positions of all the other particles in the universe Y ( t ) are fixed: ψ t ( x ) := Ψ( x, Y ( t )). x -axis approaching a particlethat sits at x = 0. The potential of the resting particle is marked as a lightgrey diagonal line. Their initial wave-function is marked in dark grey. Afterscattering, the first moving particle stops, and the other particle moves up-wards. The wave-function is then in a superposition indicated by two lightgrey wave-functions. (Picture from Norsen, 2010, p. 1867)trajectories even if the system is entangled; in fact, these fields are non-zero only if thereis entanglement.Norsen’s theory of exclusively local beables makes the very same empirical predictionsas the de Broglie–Bohm theory—it even predicts the very same trajectories—, but theprice to be paid is a more contrived law for the evolution of all those local fields. First,it turns out that there are infinitely many such interacting fields since the evolution ofthe interaction fields requires further interaction fields. . . a never ending recursion. Andit’s not clear yet that one can get satisfactory results with only a finite set of these fields.Second, each conditional wave-function follows a modified Schrödinger equation, in whichthe other interaction fields are included. And these interaction fields themselves havetheir own evolution equation. This makes the theory mathematically very complicatedand almost impractical for calculating empirical predictions.We now realize that Norsen does not present a theory of a multi-field. There areinfinitely many ordinary local fields in physical space, which interact. In fact, this isinstead a theory of a multitude of ordinary fields!Still, Norsen’s theory of local beables is sometimes confused with the multi-field: We may consider the multi-field option—this postulates a multitude of fields in 3dspace, corresponding to each physical particle. Each of these fields is determined bythe wavefunction in configuration space, [. . . ] Yet, at each instant in time, a field is x = 0. Their entangled initial wave-function is depictedin dark grey. After scattering, the resting particle starts moving to the rightparallel to the x -axis, while the other particle stops at x = 0. The post–scattering wave-function is drawn in light grey. (Picture from Norsen, 2010, p.1868) defined in 3-D space corresponding to each particle. (Suárez, 2015, p. 3211) It is apparent that this is not the multi-field account that we have presented in theprevious section. Indeed, in the multi-field account, there is no “multitude of fields” onphysical space; rather, there is just one field that assigns a value to a set of N particles.You cannot decompose these values corresponding to each fields assigned to each particlebecause, for entangled systems, one cannot attribute a wave-function to a single particle. David Albert (1996) argued that if the wave-function is a field it has to be a field inconfiguration space. In doing so, he takes this space as the fundamental space of physics,in which only one single universal particle exists:
On Bohm’s theory, for example, the world will consist of exactly two physical objects.One of those is the universal wave-function and the other is the universal particle .And the story of the world consists, in its entirety, of a continuous succession ofchanges of the shape of the former and a continuous succession of changes in the position of the latter.And the dynamical laws that govern all those changes – that is: the Schrödingerequation and the Bohmian guidance condition – are completely deterministic, and in the high-dimensional space in which these objects live) completely local . (Albert,1996, p. 278) The major reason for Albert to develop an ontology in configuration space is to havethe wave-function as a local beable : the wave-function is determined by the local valuesit assigns to every point of configuration space. Hence, the motion of the universalparticle is completely determined by the field value at its location, exactly as in classicalelectrodynamics, where the motion of a charged particle is determined by the value ofthe electromagnetic field at its location. We doubt, however, that this kind of localityis a universal principle for physical theories, especially in the quantum domain. In our opinion, the major drawback of this interpretation is that it doesn’t distinguishbetween the mathematical space and physical space. The fact that a physical objectis mathematically defined in configuration space, does not necessarily imply that theobject itself has to exist in configuration space. Nor does it imply that the world is3 N dimensional. We encounter a similar issue in the meaning of dimensions in classicalmechanics. The number of dimensions to mathematically describe a physical objectis given by the number of degrees of freedom that we need to describe the state andthe motion of that object. For example, a rigid body that translates and rotates isrepresented in a six-dimensional space. But, of course, this does not mean that therigid body is really a six-dimensional object: in fact, it exists in a three-dimensionalspace, but we need six degrees of freedom to fully describe its state of motion. Here, thedistinction between the mathematical representation and the actual physics is obvious,since everybody agrees on the ontology, on what a rigid body is.Albert takes a different approach in quantum mechanics, as well as in classical mechan-ics. He starts with the mathematical formalism and searches for criteria in how to distillan ontology. His primary principle is locality: find the mathematical space in which thephysical entities are locally defined. You may also call this principle separability , sinceit is about the ontology of objects and not about their dynamical behavior. For thede Broglie–Bohm theory this principle distinguishes configuration space as the space inwhich the wave-function is locally defined. So, he takes this space as fundamental.In classical mechanics, the ontology in three-dimensional space and in phase space areboth separable. So here the locality principle does not single out one space over theother. Therefore, Albert invokes a second principle: take the local space that has thelowest dimension. Applying this principle, we get that the fundamental space of classicalmechanics is three dimensional.We think that Albert’s approach to extract an ontology from the mathematical for-malism of physics relies on a heterodox understanding of what physical theories are.Physical theories are not bare mathematics (for this critique, see also Maudlin, 2013). Many arguments were given to criticize Albert’s ontology (for instance, Chen, 2017a; Maudlin, 2013;Monton, 2013). We will focus on one argument that we think is the strongest with respect to themulti-field interpretation. More explicitly, there are two types of locality: ontological locality and dynamical locality. The formercoincides with separability and is about local beables, while the latter is about the behavior of physicalobjects, for instance, Bell’s notion of local causality or Einstein’s locality. in the real world . This distinguishesphysics from mathematics. In particular, the standard commentary of the formalism ofthe de Broglie–Bohm theory relates the mathematics to N -particles in three-dimensionalspace. We agree that mathematical equations do not uniquely determine what they referto, but we disagree that the principle of separability singles out a preferred ontology.The primary task of physics is to explain what we observe, to explain our manifestimage (to use an expression of Sellars, 1963). A theory does so by proposing a scientificimage, namely an ontology with corresponding laws of nature. We think that Albert’sseparability criterion may lead to ontologies that hardly fill the gap between the scientificand manifest image. Albert’s scientific image consists of a point in configuration spacethat is locally guided by the wave-function. He gets the manifest image by a functionalanalysis of the Hamiltonian in the Schrödinger equation. The Hamiltonian has thestructure of giving rise to a three-dimensional world, with tables and chairs, on a coarse-grained level.In the multi-field view, the scientific image consists of many particles in three dimen-sions, guided by a non-local field in this very space. The advantage of this view is thatthe scientific image and the manifest image are situated in the very same space. So themacroscopic objects in the manifest image are mereologically composed of microscopicparticles from the scientific image. This mereological composition is lacking in Albert’sontology. Although he introduces particles in three-dimensions, they are mere shadows of the marvelous point (Albert, 2015, p. 130). Indeed, it is not sufficient to explainthe manifest image just by the dynamical laws but also by how these laws give rise tothe manifest image from the fundamental ontology. That real particles exist in three-dimensional space composing tables and chairs is much more plausible and straightfor-ward than a marvelous point in a very high dimensional space explaining the behaviorof tables and chairs by a functional analysis of the Hamiltonian (see also Emery, 2017,for an argument along these lines). We think that we should prefer ontologies that arecloser to the manifest image of the physical world over those representing a significantdeparture from it. In the following, we list the advantages of the multi-field interpretation in more sys-tematic order. Some of the points have been mentioned above, but here we make themexplicit.
General Physical Interpretation
The multi-field interpretation is largely independent of the concrete formulation of the deBroglie–Bohm theory in terms of a first-order formulation (Dürr et al., 2013; Valentini,2010), second-order formulation (Bohm and Hiley, 1993), or quantum Hamilton-Jacobiformulation (Holland, 1993). In the first-order formulation the multi-field specifies thevelocities of each particle. In the second-order formulation, the acceleration is specified10y means of the quantum potential. We think, however, that the multi-field interpre-tation is particularly useful for the second-order theory, because it allows to retrievethe entire classical scheme of how motion is generated: field → potential → force → acceleration.Recent interpretations, like the nomological interpretation (Goldstein and Zanghì,2013), the Humean interpretation (Bhogal and Perry, 2017; Callender, 2015; Esfeld et al.,2014; Miller, 2014), or the dispositional interpretation (Esfeld et al., 2014; Suárez, 2015)require a commitment to what laws of nature are. The multi-field interpretation, on theother hand, shows that one can have an ontological interpretation of the wave-functionwithout a commitment to the status of laws of nature. We think that the wave-functionis best regarded as an objective physical field than a nomological entity, since the wavefunction is in general time-dependent. Even if the universal wave-function turned out tobe time-independent, it will still be a solution of a dynamical equation, like the Wheeler-DeWitt equation. Having a law for a nomological entity seems to us not in the spirit ofa nomological entity. This is different for the Hamiltonian in classical mechanics, whichdoes not arise from a law, and therefore may be interpreted as a nomological entity.These two arguments suggests to regard the wave-function as a field, similarly to theelectromagnetic field (being a solution of the Maxwell equations). The Entire Ontology in Three Dimensions
That Bohmian particles are guided by the wave-function is often taken merely as aheuristic metaphor. As the wave is defined in configuration space, it cannot directlyinfluence the motion of particles. Having the wave-function as a multi-field, however,gives this intuition an ontological underpinning. The de Broglie–Bohm theory is hencea pilot-multi-wave theory, where the wave directly guides particles in three-dimensionalspace. This has also the advantage that there is an ontological continuity betweenone-particle and many-particle scenarios. A one-particle wave-function cannot be onlyvisualized in three-dimensional space because this space mathematically coincides withconfiguration space, but there is indeed a wave in three-dimensional space. When thisone-particle wave-function gets entangled with an external N -particle system, the new( N + 1)-particle system is still guided by a wave in three dimensions. The double-slitexperiment provides a vivid example of the explanatory advantage of this view. Whilethe particle goes through one of the slit, the wave literally enters both slits, therebydetermining the motion of the particle and accounting for the characteristic interferencepattern on the screen. Simplicity
The multi-field view is different from Norsen’s approach, which reduces the universalwave-function to a set of one-particle wave-functions. There, each particle has an associ-ated local “pilot-wave” in three-dimensional space, so that an N -particle wave-functionreduces to a set of local beables. But in order to make the theory empirically ade-quate and to account for entanglement, Norsen has to introduce an infinite number of11nteracting fields for the guiding fields. Thus, the formalism of the theory becomes ex-traordinarily complicated. This is due to the recursive structure of the local Schrödingerequation, leading to a kind of Taylor series for every local guiding field comprising aparticle. This would be a totally new structure for a fundamental law of nature. Sincethe total infinite series of fields cannot be calculated, it is an open question at what orderone can truncated this series to have approximately good empirical results. We thinkthat Norsen’s attempt to write down a theory of exclusively local beables shows howunnatural it is to mathematically embed the wave-function in three-dimensional space.The multi-field view, on the other hand, is much simpler: it is an interpretation of thewave-function as a new type of field in three-dimensional space. In particular, it doesnot require us to modify the definition of the wave-function, and so it does not require tomodify the mathematical formalism of the theory. Quantum non-locality is encoded byhaving the mathematics in an abstract high-dimensional space. This explains quantumnon-locality in the simplest way. Instantiating a Non-Local Beable
For Maudlin (2013), the wave-function in the de Broglie–Bohm theory refers to a realphysical entity that determines the behavior of particles. The wave-function is accordingto him best regarded as the mathematical representation of a new physical object: the quantum state . Yet, contrary to Albert, Maudlin suggests that we should not extractthe ontology of the quantum state directly from its mathematical representation for thefollowing reasons:1. The wave-function contains some degrees of freedom that are merely gauge, sincethey do not lead to different quantum states. One example is the overall phase:wave-functions with different overall phases represent the same quantum state,since their empirical predictions are exactly the same.2. The wave-function (for an N -particle system) is naturally expressed in configura-tion space because it refers to the positions of a real configuration of N particlesin three-dimensional physical space.Nevertheless, Maudlin does not specify which sort of physical entity the quantum stateis. He merely describes it as a non-local beable : From the magisterial perspective of fundamental metaphysics, then, our precisequantum theories have a tripartite ontology: a space-time structure that assumes afamiliar approximate form at mesoscopic scale; some sort of local beables (particles,fields, matter densities, strings, flashes) in that space-time; and a single universalnon-local beable represented by the universal wave-function Ψ. (Maudlin, 2015, p.356)
The multi-field idea, on the other hand, goes beyond Maudlin’s characterization of thewave-function: it is a genuine physical field. Because of its properties, the multi-field isa non-local beable. It is a beable since it is a real physical object. And it is non-local12ince the value of the multi-field is specified not for one point but only for N -tuplesof points in three-dimensional space. The multi-field thus explains what the quantumstate is, and it instantiates a non-local beable. This relation of instantiation between anon-local beable and the multi-field could be also analyzed in terms of grounding in thesense of Schaffer (2009). A thorough discussion of this issue, however, would go beyondthe scope of this paper. In the course of writing the paper, we have received some critical remarks about themulti-field approach. We now mention the most important ones followed by our reply.1.
The multi-field doesn’t give us a new ontological understanding of the wave-function.
We regard the multi-field interpretation as a new ontological interpretation of thewave-function that has been so far ignored. This approach is incompatible withthe nomological interpretation of the wave-function, and its metaphysical charac-terizations in terms of dispositionalism and Humeanism. But one may still groundthe multi-field on an extended Humean mosaic that also comprises contingentnon-reducible relations (Darby, 2012). In this way, we would have a Humeaninterpretation, in which the wave-function is no longer a nomological entity be-cause it is not part of the best system. The mosaic would be comprised of pointscomposing three dimensional space, a particle configuration, and the multi-field.Similarly, Loewer (1996) embedded Albert’s marvelous point interpretation in aHumean framework. The difference is, however, that Loewer’s mosaic is separablein configuration space, while Darby’s mosaic is non-separable in three-dimensionalspace.2.
You don’t change the formalism. How can the wave-function be something in three-dimensional space, if it is defined on configuration space?
There is a difference between the mathematical structure that we use to define aphysical object and the ontology of this object. The multi-field interpretation isbased on this distinction. The configuration space is the mathematical space thatwe need to describe a function which generally depends on 3 N degrees of freedom(where N is the number of particles of the system); three-dimensional space is thephysical space in which the object represented by that function is defined.A simple example can be of some help here. Historically, the first formulationof classical mechanics was due to Newton. Newton’s theory of mechanics was anontological theory, that is, a theory with clear ontological commitments: particlesmoving in three-dimensional space accelerated by force acting on them. The sametheory can be casted in the Hamiltonian formulation. Here, the system is rep-resented by a particle moving in phase space with a trajectory described by theHamiltonian function. Nevertheless, it is understood that the Hamiltonian formu-lation is just a mathematical representation of the ontological picture of classical13echanics given by Newton’s theory. What we usually do in practice is to use bothformulations simultaneously: the physical ontology of Newton’s theory (that’s whatthe world is built of) and the mathematics of the Hamiltonian formulation (whichis often more convenient in doing calculations).Quantum mechanics has followed an inverse path: the formalism of the theory is anextension of the Hamiltonian formulation of classical mechanics. And the historicalmistake was to try to extract from this formulation the physical interpretation ofthe theory. We regard the de Broglie–Bohm theory as the ontological theory ofquantum mechanics (the analogue of Newton’s theory for the classical case), whichis a theory of particles moving in three-dimensional physical space acted upon bya multi-field. In a nutshell, the multi-field interpretation bears the advantagesof Norsen’s ontology (that is, having the wave-function physically as a field onthree-dimensional space) and the simplicity of the standard formalism (that is, thewave-function as a mathematical object in configuration space).3. The multi-field runs into the problem of communication.
Suárez claims the following:
According to this view, the multi-fields are defined at each instant by the wave-function in configuration space, and the question is how the wave-function“communicates” to physical three-dimensional space in order to fix each of thefields and the positions of the particles for any system of N particles. [. . . ] Also,note that the communication is curiously one way: while the wave-function fixesthe physical properties, including positions, of particles in three-dimensionalspace, these have no effect back onto the wave-function, which essentially ig-nores which are the actual particle trajectories amongst all the possible trajec-tories compatible with the dynamics. (2015, p. 3212)
It is difficult to see how the problem of communication arises in the multi-fieldapproach. Indeed, there is a problem of communication when different physicalentities live in different physical spaces and yet influence each other. But, in thecase of the multi-field approach, both the guiding wave and the particles live inthe very same space, namely, three-dimensional space. So, there is no problem ofcommunication in the multi-field account.Moreover, the fact that the communication is “one way” is independent of theproblem of communication; it ought to be an objection to regarding the wave-function as a field in the first place. This can become a problem if we want thewave-function to behave like a classical field, where particles and field mutuallyaffect each other. But there are good reasons to think that, if the wave-function isa field, it must certainly be a new type of field, possessing completely new physicalfeatures. The action-reaction principle may thus be thought of as a characteristicsof classical fields, and it is better to abandon this principle in the ontology ofquantum theory. After all, there is no logical inconsistency in thinking of particlesnot acting back on the field. This could just confirm that the type of physical14nteraction between the (Bohmian) particle and the multi-field is after all not aclassical interaction.4.
Fields have to be local and propagate with finite speed.
We disagree that these features are essential to fields, although they are crucial toclassical fields. In section 2, we proposed that the multi-field is specified by thestatements (b) and (c), that is, by an assignment of properties to N -tuples of pointsof three-dimensional space and momentum-energy conservation for closed systems.All other features about locality and dynamics are derived from the equations thatimplement these fields in the theory.5. Physicists have always understood the wave-function as a kind of multi-field.
We could not find historical evidence that the multi-field used to be the default viewamong physicists, not even the default view among Bohmians. Since de Broglie’sand Bohm’s work, physicists were struggling to understand the ontological statusof the wave-function because it is mathematically defined in configuration space,and yet it communicates with particle configurations in three-dimensional space.This tension has lead people to claim that we live in configuration space, or thatwe must give up the idea that the wave-function literally guides the particles.We think that both scenarios are problematic. We argue, instead, that the wave-function can exist as a certain kind of physical field on three-dimensional space,its mathematical definition notwithstanding.6.
How can the multi-field fill up physical space? For example, if I point to a locationof physical space, what part of the multi-field is there?
The multi-field is a non-local beable; therefore, it does not assign any value tosingle points of space according to Bell’s definition of local beables. The multi-field exists as a physical field in three-dimensional space by assigning propertiesto all N -tuples of points of space. The ontology of particles, on the other hand, islocal because particles occupy single points in space. The connection between thenon-local beable and the local beables is accomplished by the guidance equation,which projects the non-local property assigned to the N -tuple into local propertiesof particles.One may still object that if the multi-field doesn’t assign a value to any pointof three-dimensional space, the multi-field cannot exist on this space. But thisobjection presupposes that the multi-field is a local beable. The multi-field existsat each point of space as a a non-separable or relational field, which needs N spatialpoints to deliver a field value. Without specifying these N points the multi-fielddoes not give you a precise value. The field value assigned to an N -tuple hasmetaphysical priority, since this property cannot be built up from local properties.Nevertheless, for an actual configuration of particles the multi-field gives everyparticle a local property in form of a specific velocity (first-order formulation) ora specific acceleration (second-order formulation).15on-local beables are unfamiliar in physics and apart from the wave-function hardto find. But one can define a non-local quantity in General Relativity that sharessimilar features with the multi-field. This quantity goes by the name of ADMenergy (Arnowitt, Deser, and Misner). In General Relativity, it’s not possible todefine a local energy density for the gravitational field, unless one invokes additional(unjustified) structures on space-time, like a preferred coordinate system (Wald,1984, section 11.2). Still, it’s possible to define a global energy on an asymptoticallyflat space-times by means of an integral over an infinitely large 2-sphere. Thisintegral is the ADM energy. In this case, too, it would be meaningless to askwhat the energy is at a single point of space-time, since the energy is only globallydefined. There are no local values for the ADM energy, like there are no localvalues for the multi-field.
We have argued that the multi-field interpretation has been dismissed for the wrongreasons owing to prejudices from classical fields. Construing the Bohmian wave-functionas a multi-field is actually the most conservative physical interpretation compared toAlbert’s marvelous point interpretation and Norsen’s local fields theory. For it describesthe entire ontology of the theory in three-dimensions without changing the mathematicalformalism, thus solving the problem of communication and the problem of perception,and it provides a natural metaphysical explanation for the non-local behavior of particles.Tim Maudlin regards the wave-function to be a non-local beable building on the workof John Bell. The multi-field interpretation starts from a completely different route,namely, by an analysis and extension of the classical field concept. In our view, thewave-function is also a non-local beable because it is, indeed, a multi-field.
Acknowledgements
We wish to thank David Albert, Guido Bacciagaluppi, Michael Esfeld, Dustin Lazarovici,Tim Maudlin, Matteo Morganti, Travis Norsen, Andrea Oldofredi, Charles Sebens, andTiziano Ferrando for many helpful comments on previous drafts of this paper. We alsothank the audience of the 3 rd Annual Conference of the Society for the Metaphysicsof Science (SMS) and especially Lucas Dunlap for commenting on our paper at thisevent. We also thank two anonymous referees for their very detailed reviews. DavideRomano’s research was funded by the Swiss National Science Foundation (grant no.105212_149650).
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