Nonseparability, Classical and Quantum Wayne C. Myrvold Department of Philosophy University of Western Ontario wmyrvold@uwo.ca Abstract This paper examines the implications of the holonomy interpretation of classical electromagnetism. As has been argued by Richard Healey and Gordon Belot, classical electromagnetism on this interpretation evinces a form of nonseparability, something that otherwise might have been thought of as confined to non-classical physics. Consideration of the differences between this classical nonseparability and quantum nonseparability shows that the nonseparability exhibited by the classical electromagnetism on the holonomy interpretation is closer to separability than might at first appear. 1 Introduction How are we to think of classical electromagnetic fields? There is a view, which has its roots in the work of Wu and Yang ([1975]), and which has been advanced in the philosophical literature by Richard Healey ([1997],[ 2004],[ 2007]) and Gor- don Belot ([1998]), according to which the traditional representation of classical electromagnetism, in terms of electric and magnetic fields (or, in Lorentz covari- ant formulation, in terms of the Faraday tensor), is not the most perspicuous representation of the physical content of the theory. In its place is offered what Healey ([2007]) calls the holonomy interpretation. This rests on a representation of electromagnetism that takes as its basic elements assignments of unit complex numbers to loops in spacetime. As both Healey and Belot have pointed out, the holonomy interpretation has consequences for the metaphysics of classical elec- tromagnetism; electromagnetism on the holonomy interpretation evinces a form of nonseparability, which one might otherwise have thought was a hallmark of non-classical physics. This, in turn has consequences for the view that Lewis calls Humean Supervenience (Healey [2007], pp. 124–27). These points are well taken. Nevertheless, there remain important differences between quantum nonseparability and the nonseparability exhibited by the holon- omy interpretation of classical electromagnetism. An examination of these differ- ences, I shall argue below, shows that the nonseparability exhibited by the classi- cal electromagnetism on the holonomy interpretation is closer to separability than might at first appear. Furthermore, the consequences for Humean Supervenience are not as dire as would seem. 2 Interpretations of Classical Electromagnetism In his book, Healey ([2007]) distinguishes three main lines of approach to classical gauge theories, such as electromagnetism. One is the “no new gauge potential properties view.” On this view, the electromagnetic potentials represent no new properties; an electromagnetic potential represents physical reality only insofar as it encodes the electromagnetic field, represented by the Faraday tensor. This runs into trouble when we consider the manner in which electromagnetic fields couple to a quantum wave function. As the Aharonov-Bohm effect dramatically illus- trates (though the issue is, of course, not confined to that particular set-up), the coupling between electric and magnetic fields and the wave-function of a charged particle cannot be a local coupling. We can achieve local coupling by express- ing the electromagnetic field in terms of the electromagnetic potential Aµ. This has been taken as an argument for the reality of the potential (notably, by Feyn- man [1964], §15-5). However, a representation of electromagnetism in terms of the electromagnetic potential has the disadvantage that it is not gauge-invariant. Any two electromagnetic potentials that are related by a gauge transformation are regarded as representing the same physical state of affairs. The integral of the potential around a closed curve is gauge invariant, as is the Dirac phase factor exp[− ie h̄ ∮ C Aµdx µ]. Wu and Yang ([1975]) argued that the set of all such phase factors—that is, a map- ping from closed curves in spacetime to complex numbers of unit modulus—yields a representation that captures all of the physical content of the theory without superfluous structure. Such a representation, according to Wu and Yang, “con- stitutes an intrinsic and complete description of electromagnetism” (p. 3845). Healey ([1997],[ 2004],[ 2007]) and Belot ([1998]) have examined the consequences of taking this representation of classical electromagnetism, the holonomy represen- tation, as basic, and in particular, its consequences for locality and separability. Locality and separability are conditions on physical theories, both found in the 2 writings of Einstein on quantum theory (though not always clearly distinguished by Einstein; it is due to the work of Don Howard ([1985]) that these have widely come to be recognized as distinct principles). Separability, as Einstein puts it, is the condition that “things claim an existence independent of one another, insofar as these things ‘lie in different parts of space’” (Einstein [1948], p. 321; tr. Howard [1985], p. 188). Locality, or the Prinzip der Nahewirkung, is the condition that, if A and B are spatially separated systems, “an external influence on A has no immediate effect on B” (Einstein [1948], pp. 321–22; tr. Howard [1985], p. 188).1 For Einstein, (classical) field theories were the epitomes of theories satisfying these conditions. Speaking of the separability condition, he wrote, Field theory has carried out this principle to the extreme, in that it localizes within infinitely small (four-dimensional) space-elements the elementary things existing independently of one another that it takes as basic, as well as the elementary laws it postulates for them (Einstein [1948], p. 320; tr. Howard [1985], p. 188). And the locality condition, according to Einstein, “is applied consistently only in field theory” (Einstein [1948], p. 322; tr. Howard [1985], p. 188). It has become commonplace to say that classical state descriptions are sepa- rable, quantum state descriptions, nonseparable. The holonomy view of classical electromagnetism has introduced a complication into this neat dichotomy; on this view the state descriptions of classical electromagnetism exhibit a form of non- separability (see §5, below). The argument for the holonomy view, to be sure, rests on quantum considerations, but it is a classical field, nonetheless, that is at issue. If we take it as a principle that we should let our best physical theories be our guides in our views on matters ontological, then we should take Belot’s and Healey’s arguments seriously in considering how to think about classical electro- magnetic fields.2 If we accept the holonomy view of the classical electromagnetic field, and accept that, on this view, classical electromagnetism yields nonseparable state descriptions, does this obliterate any distinction between quantum nonseparability and the sort of nonseparability exhibited by classical electromagnetism? It would be rash to say so, and neither Healey nor Belot do.3 It is, therefore useful to examine the differences between quantum nonseparability and the nonseparability of classical electromagnetism. 3 3 Quantum nonseparability As a warm-up, it is useful to rehearse some familiar facts about quantum nonsep- arability. If two quantum systems A, B, are in an entangled state, the reduced states of A and B—that is, restriction of attention to what the state says about mea- surements performed on A, and what it says about measurements performed on B—do not suffice to determine the state of the combined system, as the reduced states underdetermine the correlations between measurement results on the two systems. This is a form of holism that does not exist in classical mechanics, as, in classical mechanics, a complete specification of the states of the parts that make up a whole determines the state of the whole. Comparison of the classical and quantum case is facilitated by the use of a framework that subsumes both theories. Both classical and quantum systems can be represented as theories whose observables are the self-adjoint part of a C∗-algebra (see Clifton, Bub, and Halvorson [2003]). The key difference is that classical observables can be thought of as functions on an underlying phase space; this entails that the algebra of observables is abelian. A consequence of this is that pure states are dispersion-free (that is, they assign definite values to all observables), and this in turn entails that the state space of a composite system contains no entangled states. A little more formally: let A, B be two physical systems, with associated C∗- algebras A, B. We associate with the composite system consisting of A and B the algebra C = A∨B, that is, the smallest C∗-algebra containing A and B.4 Let ω be a state5 of C, and let ωA, ωB be the restrictions of ω to A, B, respectively. If ω is a pure state, and either A or B is abelian, then ωA and ωB are also pure, and ω is uniquely determined by specification of ωA and ωB (that is: there is only one state of C that is compatible with ωA and ωB being the restrictions of ω to their respective subalgebras). If, on the other hand, both A and B are non-abelian, as will be the case when both systems are treated as quantum systems, there will be pure states ω that are not product states. Such a state is not uniquely determined by the reduced states ωA and ωB. This is true even if A and B are systems located some spatial distance apart. Thus, quantum states violate what Healey ([1997], p. 26) has called Spatial Separability. Spatial Separability. The qualitative intrinsic physical properties of a compound system are supervenient on those of its spatially separated component systems together with the spatial relations among these component systems. 4 4 The holonomy representation of the classical electromagnetic field. Although in this paper we are primarily interested in classical electromagnetism, other gauge theories also admit of a holonomy representation. To set up the holonomy representation of electromagnetism and other Yang-Mills theories, we must first introduce some terminology. We begin with a spacetime manifold M. A curve in M is a continuous, piecewise smooth mapping C : [0, 1] → M. The image of a curve is the set of points in M that it traces out. The inverse of a curve C, which we will denote by C−1, traverses the same set of points in the opposite direction, and is defined by C−1(x) = C(1 −x). A closed curve is a curve whose beginning point and endpoints are the same point, called its base point. If the endpoint of curve C1 is the beginning point of curve C2, we define the composition C2 ◦ C1 as a curve that traces out first C1, then C2.6 A closed curve is thin if it is possible to shrink it down to a point while remaining within its image. A loop is the oriented image of a non-self-intersecting closed curve. Closed curves C1,C2 are thinly equivalent iff C1 ◦ C−12 is thin. A hoop is an equivalence class of thinly equivalent closed curves.7 We will denote by [C] the hoop containing a closed curve C. Pick some point o ∈ M, and consider the set of closed curves with base-point o. The associated set of hoops can be given a natural group structure. For any two hoops α, β, take curves C1 ∈ α, C2 ∈ β, with common base point o. Define the composition β ◦ α as [C2 ◦ C1]. The identity element is the class of curves equivalent to the trivial curve that begins at o and remains there. For any hoop α, we can define its inverse α−1 as [C−1], where C is any element of α. Clearly, α−1 ◦α is [C−1 ◦C], which is the identity element. Let Lo be this group of hoops. We can define a holonomy as a smooth8 homomorphism from Lo into a suitable Lie group; this gives rise to a representation of these as holonomies of a connection on a principal fiber bundle with that Lie group as structure group. In the case of electromagnetism, things are particularly simple. The structure group is just U(1), the set of rotations of a circle. This group has a natural representation as complex numbers of unit modulus; thus, in electromagnetism, a holonomy assigns to each hoop in Lo a complex number of unit modulus. Written in terms of the electromagnetic potential, the value a holonomy H assigns to a hoop γ is, H(γ) = exp[ ie h̄ ∮ C Aµdx µ], where C is any closed curve in the equivalence-class γ. Since the group U(1) is abelian, holonomies are independent of base point. 5 Any closed curve is composed of non-intersecting curves, whose oriented im- ages are loops. To specify an electromagnetic holonomy on the group of hoops, therefore, it suffices to specify a value for each loop (since these values are inde- pendent of base point). At the price of some abuse of notation, we will write H(l) for the value of H on a hoop γl containing a curve whose oriented image is the loop l. The holonomy interpretation of electromagnetism takes as its basic description of the electromagnetic state of the world an assignment of holonomy values to loops in spacetime. The electromagnetic state of a region R of spacetime is an assignment of holonomy values to loops in R. These values will not be independent of each other. Suppose loops l1 and l2 are the images of closed curves C1, C2, with common base point. It may happen that there is a curve C3, thinly equivalent to C2 ◦C1, whose oriented image is a loop, which we will denote l1 ⊕ l2. Because a holonomy must respect the group structure of the group of hoops, we must have, for any holonomy H, H(l1 ⊕ l2) = H(l2)H(l1). Call this the loop composition condition. Though doing so is a bit more complicated than in the traditional represen- tation, it is possible to write the classical action for electromagnetism in terms of loop-dependent quantities (see Ugon et al. [1994], and Gambini and Pullin [1996], §4.6). It is also possible to write Maxwell’s equations in terms of loop- dependent quantities. These preserve the expected relations of dependence. Let Σ1, Σ2 be two maximal spacelike hypersurfaces in Minkowski spacetime, with Σ2 to the future of Σ1. Let R2 be some subset of Σ2, and let R1 be the past domain of dependence of R2 in Σ1. Then the holonomy-state of R2 is determined by the holonomy-state of R1. Thus, as Healey ([2007], p. 127) notes, there is no threat to relativistic locality from the holonomy interpretation. We can also construct a holonomy interpretation of a classical non-abelian Yang-Mills theory. The holonomy is now a homomorphism from the group of hoops into a non-abelian Lie group. These holonomies will not, in the non-abelian case, be independent of the base point o, and will not be gauge-invariant. What will be gauge-invariant and independent of base point will be the Wilson loops. WA(γ) = Tr [ P exp ( i ∮ C Aµdx µ )] , where P denotes the path-ordered product. It can be shown (see Gambini and Pullin [1996], §4.3.2) that all the gauge-invariant content of the theory can be reconstructed from the these Wilson loops. 6 5 Holonomy and nonseparability: the classical electromagnetic field Suppose we represent a classical electromagnetic field in this way: as a smooth assignment of elements of U(1) to loops in spacetime. Any particular holonomy H will represent a state of the electromagnetic field, and the restriction of a state to the set of observables associated with a spacetime region R will be given by {H(l)}, where l ranges over loops in R. This immediately eliminates any version of separability that requires physical states to supervene on properties assigned to spacetime points. Call this Pointil- liste Separability.9 Pointilliste Separability. All physical processes occupying a region R of space-time supervene on an assignment of qualitative intrinsic physical properties at spacetime points in R. But perhaps we have reasons to reject pointillisme, independent of the issues we’re discussing here.10 On the traditional view, electromagnetic fields are specified by specifying fields B, E, or, in covariant form, by specifying the electromagnetic field tensor field Fµν. At first glance, we have here a perfect instance of pointil- liste separability: the electromagnetic tensor field is an assignment of a tensor to each point of spacetime. The question is whether these can be regarded as prop- erties intrinsic to the spacetime points. Jeremy Butterfield ([2006]) has argued that instantaneous velocities of a classical point particle are to be regarded as ex- trinsic but local to the spacetime points at which they obtain. Extrinsic because possession of a velocity at an instant has implications for matters of fact at other instants, and local because the velocity of a particle at time t is determined by the particle’s trajectory in any neighborhood of t, no matter how small. A similar argument can be given for the vectors (or tensors) assigned to points by an electromagnetic field. Let us note, first of all, that a vector (or tensor) field is usually taken to be a smooth assignment of a vector (or tensor) to each point of the spacetime manifold. The requirement of smoothness means that assignment of a field value to a point x has implications for assignments at points other than x, so, if an electromagnetic field must be smooth, the values by a field to a spacetime points is an extrinsic, though local, property of the point. Even if we relax the requirement that an electromagnetic field be a smooth assignment of field values to points, perhaps considering as a metaphysical possi- bility a field that is discontinuous or not everywhere defined (whether or not this is regarded as physically possible), a case can be made for rejecting pointillisme about EM fields. 7 Suppose a spacetime point p in world w has associated with it a nonzero electromagnetic field tensor Fµν(p). Let there be a duplicate of p—that is, a point q possessing all of p’s intrinsic properties—in a world w′ that differs from w at points other than q. Since the intrinsic properties of q place no constraints on properties of other points, we may suppose that the electromagnetic field is zero at all points other than q in some neighbourhood of q. If Fµν represents an intrinsic property of p, then its duplicate q must have the same property, and we will have an electromagnetic field that takes on the value Fµν at q, but is zero everywhere in some neighbourhood of q. This is problematic. The property that Fµν(p) represents is the existence of a certain electromagnetic field at p, not merely an assignment of some tensor to p that is devoid of physical significance. In order for Fµν to count as an electromag- netic field tensor, the physical laws of w′ must include some that we are willing to countenance as electromagnetic laws, laws in which Fµν performs something like the function it does according to w’s laws. It is probably not easy to be precise about what conditions are necessary in order for something to count as an electromagnetic field, but it seems that such a field must be implicated in the physically possible trajectories of charged particles via something like the Lorentz force law, which determines the acceleration of a trajectory (or rather, the com- ponent of acceleration attributable to electromagnetic forces) passing through a point in terms of the electromagnetic field tensor and the four-velocity of the tra- jectory at that point.11 The possibility that the field correctly yield accelerations of possible test particles places a constraint on field values. The acceleration at a point on a trajectory is a derivative. Though derivatives need not be continuous, they must, by Darboux’s theorem, have the intermediate value property.12 Thus, the acceleration at a point on a particle’s trajectory carries implications about accelerations at other points. If field values are to yield possible accelerations, the field value assigned to a point must carry implications about field values assigned to other points, and cannot take on finite nonzero values at isolated points. Therefore, the electromagnetic field value assigned to a point carries impli- cations about its value at other points; it is an extrinsic property of the point. Extrinsic, but local: the field value assigned to a point p can be regarded as an intrinsic property of any neighborhood N of p, no matter how small, as the value of the field at p is compatible with any field outside of N. We want a separability condition that does not carry with it a commitment to pointillisme. Versions of this are offered by both Healey ([2007], p. 46), who calls it Weak Separability, and Belot ([1998], p. 540), whose term is Synchronic Locality. The idea is that the state of a region supervenes on assignments of intrinsic properties to patches of the region, where the patches may be taken to be arbitrarily small. If it is possible to do so, it is preferable to define our 8 separability condition in terms that don’t invoke a notion of the size of patches, as it is preferable to have a condition that applies even when no spacetime metric is available. This can be done by speaking of arbitrarily fine coverings of a spacetime region. A covering N is a refinement of a covering M if every element of N is contained in some element of M; if N is a refinement of M, we say that N is finer than M (note that this is reflexive). Let us take, as a notion that captures the idea of separability without a commitment to pointillisme: Patchy Separability. For any spacetime region R, there are arbitrarily fine open coverings N of R such that the state of R supervenes on an assignment of qualitative intrinsic physical properties to elements of N . This means: for any open covering M of R, there is a covering N that is a refinement of M, such that the state of R supervenes on assignments of intrinsic physical properties to elements of N .13 As Belot ([1998], p. 544) and Healey ([2007], p. 125) point out, Patchy Separa- bility, too, fails, on the holonomy view. Consider the set-up of the Aharonov-Bohm effect, and let R be a ring that circles the solenoid. If we cover this ring with small patches, the intrinsic properties assigned to those patches will be holonomies on loops within the patches. If we choose our cover so that none of the patches con- tain loops that circle the solenoid, the holonomy assigned to any loop contained in a patch will be 1. This fact, that all loops in elements of our covering are assigned the value 1, leaves undetermined the value assigned to a loop in R that circles the solenoid. To tie this in with the considerations raised in §3, we can associate a C∗- algebra with a spacetime region R region by forming a suitable set of bounded functions of the holonomy values in R; since holonomy values are just complex numbers, this will be an abelian algebra. A holonomy-state of R will assign holonomy-values to all loops in R, which will determine values for all functions in the algebra we associate with R. Take regions A, B, such that R = A∪B, and consider a holonomy-state of R. If R is not simply connected, but A and B are, a holonomy state of R will not be uniquely determined by specification of values of the holonomy on all loops in A and all loops in B. Thus, there might seem to be a tension here with what was said in §3, about absence of nonseparability in the state of a composite system whose observable-algebra is abelian. A crucial assumption there, however, was that, if algebras A, B are associated with systems A, B, the algebra associated with the composite of A and B is A∨B, the smallest algebra containing both A and B.14 Since there will be loops associated with a composite system that are not composed of loops belonging to the parts, this assumption fails for the holonomy interpretation of electromagnetism. Classical 9 electromagnetism on the holonomy interpretation achieves nonseparability, not via entanglement, but because there are observables associated with a composite system whose values are underdetermined even by an assignment of dispersion-free states to the component systems. Patchy Separability fails because it contains a quantification over all spacetime regions R. However, if we consider a simply connected region R, then it is true that, for any open covering N of R, the value of a holonomy H will be determined by its values on loops contained in elements of N . The idea is this. Given a loop σ, take two points x,y on σ, and join them by a path g. Let c1 be a loop that starts at x, traverses σ until it reaches y, and then takes g−1 back to x. Let c2 be a loop that starts at x, traverses g to y, and then traverses σ back to x. Let σ1, σ2 be the segments of σ traversed by c1, c2, respectively. Then we have c1 = g −1 ◦σ1 c2 = σ2 ◦g σ = σ2 ◦σ1. Then, for any holonomy H H(c2 ◦ c1) = H(σ2 ◦g ◦g−1 ◦σ1) = H(σ2 ◦σ1) = H(σ), and so the value of H on σ is determined by its values on c1 and c2. H(σ) = H(c1 ◦ c2) = H(c1)H(c2). We can further subdivide c1 and c2, and so on, and, as long as R is simply connected, we will be able to continue this process until each of the component loops lies within an element of our open cover. Thus, the following condition, weaker than Patchy Separability, will be satis- fied. Patchy Separability for Simply-Connected Regions. For any simply- connected spacetime region R, there are arbitrarily fine open cover- ings N of R such that the state of R supervenes on an assignment of qualitative intrinsic physical properties to elements of N . In particular, the state of the whole world (that is, either the state of M, or the state of some maximal spacelike surface in M, depending on whether one is in a 4-dimensionalist or 3-dimensionalist mood) will supervene on assignments of states to arbitrarily small regions, if spacetime (or, for the 3-dimensionalist, space) is simply connected. Separability fails only when we consider subregions of the world, and ones with special properties, at that. We have, when M is simply connected, 10 Global Patchy Separability. All physical processes supervene on an assignment of qualitative intrinsic physical properties to elements of arbitrarily fine coverings of M. Let us consider again spatial separability. Spatial Separability. The qualitative intrinsic physical properties of a compound system are supervenient on those of its spatially separated component systems together with the spatial relations among these component systems. If A and B are spatially separated regions, then there are no loops in A∪B that are not in either A or B. It follows that a holonomy assigning elements of U(1) to loops in A∪B supervenes on assignments to loops in A and B. Classical electromagnetism, on the holonomy view, satisfies spatial separability. If we ask what the holonomy view requires a conservative classical metaphysi- cian (ccm) to abandon, then certainly, pointillisme is something that has to go. But pointillisme should be rejected even on the traditional field interpretation of electromagnetism. Patchy Separability, with its quantification over all spacetime regions whatsoever, though satisfied by the field interpretation, is violated by the holonomy interpretation of electromagnetism.15 But a ccm who rejects pointil- lisme will be able to retain Spatial Separability and, provided her spacetime is simply connected, Global Patchy Separability. Thus, there is a marked difference between the form of non-separability exhibited by classical electromagnetism on the holonomy interpretation, and quantum non-separability, as the latter requires rejection of both Spatial Separability and Global Patchy Separability. 6 On Humean Supervenience In the introduction to Volume II of his Philosophical Papers, Lewis formulates the thesis of Humean Supervenience (HS). Humean supervenience ... is the doctrine that all there is to the world is a vast mosaic of local matters of particular fact, just one little thing and then another ... We have geometry: a system of external relations of spatiotemporal points. Maybe points of spacetime itself, maybe point-sized bits of matter or aether or fields, maybe both. And at those points we have local quantities: perfectly natural intrinsic properties which need nothing bigger than a point at which to be instantiated (Lewis [1986], ix–x). 11 Later, in “Humean Supervenience Debugged,” we have, Humean Supervenience ... says that in a world like ours, the funda- mental relations are exactly the spacetime relations: distance relations, both spacelike and timelike, and perhaps also occupancy relations be- tween point-sized things and spacetime points. And it says that in a world like ours, the fundamental properties are local quantities: per- fectly natural intrinsic properties of points, or of point-sized occupants of points. Therefore it says that all else supervenes on the spatiotem- poral arrangement of local quantities throughout all of history, past and present and future. The picture is inspired by classical physics (Lewis [1994], p. 474). The primary motivation behind Humean Supervenience is the idea that laws of nature are compact summaries of patterns in non-nomic facts. Lewis’ statement of HS commits him to pointillisme, but it is not clear that this is essential to the core idea. A Lewisian metaphysician might be satisfied with properties that are extrinsic but local: note that in the first quotation Lewis first refers to “local matters of particular fact,” and then proceeds to put a pointilliste slant on this, and in the second quotation “local quantities” is glossed as assignments of intrinsic properties to points. It is not clear that Lewis is distinguishing between local matters of particular fact and matters of particular fact intrinsic to points. One could, perhaps, without doing too much violence to Lewis’ intentions, advance a version of Humean supervenience that has all else supervene on local matters of fact, construed as assignments of intrinsic properties to elements of an arbitrarily fine covering of the world.16 This version of Humean Supervenience would be committed only to some version of Patchy Separability. Some version, but which? Patchy Separability, which requires the state of any part of the world to supervene on intrinsic properties of elements of an arbitrarily fine covering, or Global Patchy Separability, which only requires this to be true of the whole world? The language “all there is to the world” in the first quotation suggests the global version. If we take it this way, and replace talk of properties intrinsic to points with talk of local properties, then, in spite of the nonsepara- bility evinced by classical electromagnetism in the holonomy representation, this nonseparability is compatible with Humean supervenience—as long as spacetime is simply connected. That it matters which of these two versions of separability we consider is perhaps surprising. There is a temptation to think that, if the state of the whole world supervenes on assignments of intrinsic properties to arbitrarily small patches of it, the same must be true of any subregion of the world. One lesson from the 12 holonomy interpretation of electromagnetism is that this need not be the case. Global Patchy Separability does not entail Patchy Separability. It seems that Global Patchy Separability ought to be enough to satisfy the intuitions that underwrite the thesis of Humean supervenience. If Global Patchy Separability is satisfied, then the state of the world can be specified by an as- signment of properties to elements of an arbitrarily fine covering, and laws can be taken to be patterns in such assignments. Thus, it seems that Healey over- states matters when he says that any kind of non-separability violates Humean supervenience (Healey [2007], p. 124).17 There is a complication, however.18 Note that, in the passage quoted from “Humean Supervenience Debugged,” Humean Supervenience is a thesis about worlds like ours, not just about our world.19 If we take this to be integral to the thesis, then, if Humean Supervenience is to hold at a world w, then it must also hold in some “neighbourhood” of w, consisting of the worlds that are like w. Suppose we take a classical world w with a simply connected spacetime, and consider the class of worlds that are just like w, except that an infinite open cylinder has been removed from space, and holonomy values other than 1 are permitted for loops that circle the missing cylinder. Do these count as worlds like w, or not? Lewis’ “inner sphere” of possibility consists of those worlds that contain no natural properties or relations that are alien to this world (Lewis [1986], p. x). A change in the topology of spacetime, therefore, does not entail a move out of the inner sphere. But not all worlds in the inner sphere are “worlds like ours” (Lewis [1994], pp. 474–45). The question, then, is: should a Lewisian metaphysician count worlds with distinct spacetime topologies as unlike each other? Prima facie, the answer would seem to be no; removal of a chunk of w’s spacetime produces a world that is still like w. If that is right, and if HS is taken to hold at a world w only if it also holds at worlds like w, then classical electromagnetism, on the holonomy interpretation, violates Humean Supervenience, whether or not spacetime is simply connected. But perhaps a classical world with a multiply-connected spacetime should be counted as relevantly unlike a classical world with a simply connected space- time. When it comes to doing physics—and, in particular, when it comes to doing electromagnetism— the difference is not a minor one. In a simply-connected space- time, specifying the electromagnetic field tensor field Fµν suffices to determine the holonomy associated with any loop; this is not the case for a multiply-connected spacetime. This also has consequences for the quantum theory of a charged par- ticle; there are inequivalent quantizations corresponding to different holonomies on loops that circle the missing cylinder (see Belot [1998], §4.1 for discussion). So perhaps a world that differs from w in having a cylinder removed is not a world 13 like w, after all. 7 Conclusion There can be nonseparability in classical physics. This is the surprising lesson of the holonomy interpretation of classical electromagnetism. However, this non- separability has nothing to do with entanglement, which remains a non-classical phenomenon, and is decidedly weaker than quantum nonseparability. The classi- cal nonseparability we have found in electromagnetism is compatible with suitably restricted separability theses, such as Patchy Separability for Simply Connected Regions, and, in a simply connected world, Global Patchy Separability. Fur- thermore, though Lewis included a commitment to separability, and, indeed, to Pointilliste Separability in his statement of Humean Supervenience, it seems that the holonomy interpretation of classical electromagnetism can still respect the ba- sic intuitions underlying HS. I conclude that, though the nonseparability exhibited by classical electromagnetism on the holonomy interpretation is something that should give the conservative classical metaphysician pause, it’s nothing that the ccm should lose sleep over. 8 Acknowledgments This paper has its origin in my comments at the Author Meets Critics session for Gauging What’s Real at the 83rd Annual Meeting of the Pacific APA, Vancouver, April 9, 2009. I extend thanks to Craig Callender for organizing the session, to Larry Sklar for chairing it, and to Richard Healey, Bill Unruh, Gordon Belot, and the audience for their comments. Thanks also to the Foundations of Physics discussion group at the University of Western Ontario, and in particular Chris Smeenk, Bob Batterman, and Sona Ghosh. 14 Notes 1I take Belot’s condition of synchronic locality to be essentially equivalent to separability, and his diachronic locality, to what we are here calling locality. See Belot ([1998]), p. 540. 2Belot ([1998]) has expressed the point dramatically: “until the discovery of the Aharonov-Bohm effect, we misunderstood what electromagnetism was telling us about the world ” (p. 532). 3In his reply ([1999]) to Maudlin ([1998]), who emphasized the disanalogies be- tween the case of classical electromagnetism and quantum nonseparability, Healey made it clear that he was not denying the reality of these disanalogies. 4We assume that AB = BA for all A ∈A, B ∈B. 5That is, a normalized, positive linear functional. 6“Concatenation” might be a more appropriate term, but we stick with stan- dard terminology. 7The terminology adopted here is that used in Healey ([2007]). In Healey ([2004]), hoops are called loops (following Gambini and Pullin [1996]) and loops are called rings. 8For the meaning of “smooth” that is relevant here, see condition H3 of Barrett ([1991]). 9The negation of this is called Non-separability in Healey ([1991],[ 1994],[ 2007]). 10Pointillisme is defined by Butterfield as “the doctrine that a physical the- ory’s fundamental quantities are defined at points of space or of spacetime, and represent intrinsic properties of such points or point-sized objects located there” (Butterfield [2006], p. 709). He credits Lewis with appropriating the art move- ment’s name for this doctrine. I take this to be essentially the same as the view that Hawthorne calls “Pointy Fact Fundamentality,” defined as the view that “[f]acts about points and the relations between them are more fundamental than facts about extended regions” (Hawthorne [2008], p. 264). 11Some readers may question this. But whatever conditions are required for a tensor assigned to a point to count as an electromagnetic field tensor, these conditions must involve some sort of participation in electromagnetic laws. It seems plausible that, for any reasonable choice of such conditions, it will not be true that an assignment of an electromagnetic field at a point p is compatible with absolutely arbitrary states of affairs everywhere else. 12This means: if any component of the acceleration takes on distinct values at two points on the trajectory, it must take on every intermediate value at points in between. This is satisfied by continuously varying quantities, but is a weaker condition than continuity. 13It won’t do to simply say that the state of a region supervenes on assignments 15 of intrinsic properties to elements of an arbitrary open covering. Suppose R is some open set; one covering of R consists simply of R itself. Though the assignment of a field value to a point in an open region can be taken to be an intrinsic property of that region, it is less clear that the totality of field values assigned to points in the region is intrinsic: consider, for example, a region R consisting of two disconnected parts, and take the field to be a scalar field that necessarily has the intermediate value property. The definition of Patchy Separability is phrased to sidestep this complication. We can always assign to each element of R a neighborhood np, with np 6= nq for distinct p, q, and take as our covering the set {np |p ∈ R}. Then the field on R supervenes on a assignment of a field value to a single point in each element of our covering. 14This assumption is introduced without comment by Clifton, Bub, and Halvor- son ([2003]). It is interesting to find a setting in which it fails in a natural way. 15Unless the ccm is willing to make a move to a claim that subsets of spacetime that are not simply connected don’t count as bona fide regions. 16It is worth noting in connection with this that Lewis intends a vector field to be an arrangement of local quantities (Lewis [1994], p. 474; [1999], p. 209). And so it is, but, as argued above, the vector assigned to a point by a vector field should be taken to be a local but extrinsic property of that point. 17One might also consider a version of Humean Supervenience (or, rather, a thesis like Humean Supervenience) that captures the basic intuition that laws are no more than patterns in non-nomic facts, without Lewis’ commitment to separability, of any sort. This has been suggested by Chris Smeenk ([2009]) in a review of Healey’s book, citing Earman and Roberts ([2005]) for the idea. 18This point was raised by Richard Healey in his remarks at the Author Meets Critics session at the 83rd Annual Meeting of the Pacific APA, Vancouver, April 9, 2009. 19 It might sound odd to some readers—I hope it sounds odd to some readers— that a thesis “inspired by classical physics” might be thought to hold of worlds like ours, in spite of evidence that classical physics is not fundamental. Lewis in 1986 declared himself “not ready to take lessons in ontology from quantum physics as it now is” (Lewis [1986], p. xi). This doesn’t affect the project of this paper, whose concern is how we should think of the classical theories that apply, in some approximation, to our world. 16 References Barrett, J. W. [1991]: ‘Holonomy and Path Structures in General Relativity and Yang-Mills Theory’, International Journal of Theoretical Physics, 30, pp. 1171– 1215. 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