It has sometimes been suggested that quantum phenomena exhibit acharacteristic holism or nonseparability, and that this distinguishesquantum from classical physics. One puzzling quantum phenomenon ariseswhen one performs measurements on certain separated quantum systems.The results of some such measurements regularly exhibit patterns ofstatistical correlation that resist traditional causal explanation.Some have held that it is possible to understand these patterns asinstances or consequences of quantum holism or nonseparability. Justwhat holism and nonseparability are supposed to be has not always beenmade clear, though, and each of these notions has been understood indifferent ways. Moreover, while some have taken holism andnonseparability to come to the same thing, others have thought itimportant to distinguish the two. Any evaluation of the significanceof quantum holism and/or nonseparability must rest on a carefulanalysis of these notions and their physical applications.

• 1. Introduction
• 2. Methodological Holism
• 3. Metaphysical Holism
• 4. Property/Relational Holism
• 5. State Nonseparability
• 6. Spatial and Spatiotemporal Nonseparability
• 7. Holism and Nonseparability in Classical Physics
• 8. The Quantum Physics of Entangled Systems
• 9. Ontological Holism in Quantum Mechanics?
• 10. The Aharonov-Bohm Effect and Field Holonomies
• 11. Alternative Approaches
• 12. Quantum Field Theory
• 13. String Theory
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction

Holism has often been taken as the thesis that the whole is more thanthe sum of its parts. Several different interpretations of thisepigram prove relevant to physics, as we shall see. Here is acorrespondingly vague initial statement of nonseparability: The stateof the whole is not constituted by states of its parts. It is alreadyapparent both that holism and nonseparability are related notions andthat their exact relation needs to be clarified.

In one interpretation, holism is a methodological thesis (Section 2), to the effect that the best way to study the behavior of a complexsystem is to treat it as a whole, and not merely to analyze thestructure and behavior of its component parts. Alternatively, holismmay be taken as a metaphysical thesis (Section 3): There are some wholes whose natures are simply not determined by thenature of their parts. Methodological holism stands opposed tomethodological reductionism, in physics as well as in other sciences.But it is a certain variety of metaphysical holism that is moreclosely related to nonseparability. What is at issue here is theextent to which the properties of the whole are determined by theproperties of its parts: property holism (Section 4) denies such determination, and thereby comes very close to a thesisof nonseparability. In turn, nonseparability can be analyzed either asstate nonseparability (Section 5), or as spatiotemporal nonseparability (Section 6). By and large, a system in classical physics can be analyzed intoparts, whose states and properties determine those of the whole theycompose (Section 7). But all hands agree that the state of a system in quantum theoryresists such analysis. The quantum state of a system specifies itschances of exhibiting various properties on measurement. In ordinaryquantum mechanics, the most complete such specification is given bywhat is called a pure state. Even when a compound system has a purestate, some of its subsystems may not have their own pure states.Emphasizing this characteristic of quantum mechanics, Schrödingerdescribed such component states as “entangled” (Section 8). Superficially, such entanglement of states already demonstratesnonseparability. At a deeper level, it has been maintained that thepuzzling statistics that arise from measurements on entangled quantumsystems either demonstrate, or are explicable in terms of, holism ornonseparability rather than any problematic action at a distance (Sections 8,9). The Aharonov-Bohm effect (Section 10) also appears to exhibit action at a distance, as the behavior ofelectrons is modified by a magnetic field they never experience. Butthis effect may be understood instead as a result of the local actionof nonseparable electromagnetism. And indeed, this type of holism isnot rooted in quantum mechanics or entanglement: though themeasurement of the Aharonov-Bohm effect uses quantum ingredients, itbuilds on the existence of non-separable quantities in classical gaugetheory (e.g. electromagnetism) in a vacuum. But even focussing ontheories without gauge symmetry, we can obtain puzzling correlationsbetween distant simultaneous measurements in the vacuum, according toquantum field theory (Section 12). A form of quantum theory used to study them represents systems byalgebras of operators with new kinds of states defined on them,thereby making room for failures of state and system separability withno analogs in ordinary quantum mechanics. String theory (Section 13) is an ambitious research program in the framework of quantum fieldtheory. According to string theory, all fundamental particles can beconsidered to be excitations of underlying non-pointlike entities in amulti-dimensional space. The particles’ intrinsic charge, massand spin may then arise as nonseparable features of the world at thedeepest level.

2. Methodological Holism

Methodologically, holism stands opposed to reductionism, somewhat asfollows.

Methodological Holism: An understanding of a certain kind ofcomplex system is best sought at the level of principles governing thebehavior of the whole system, and not at the level of the structureand behavior of its component parts.

Methodological Reductionism: An understanding of a complexsystem is best sought at the level of the structure and behavior ofits component parts.

This seems to capture much of what is at stake in debates about holismin social and biological science. In social science, societies are thecomplex systems, composed of individuals; while in biology, thecomplex systems are organisms, composed of cells, and ultimately ofproteins, DNA and other molecules. A methodological individualistmaintains that the right way to approach the study of a society is toinvestigate the behavior of the individual people that compose it. Amethodological holist, on the other hand, believes that such aninvestigation will fail to shed much light on the nature anddevelopment of society as a whole. There is a corresponding debatewithin physics. Methodological reductionists favor an approach to(say) condensed matter physics which seeks to understand the behaviorof a solid or liquid by applying quantum mechanics (say) to itscomponent molecules, atoms, ions or electrons. Methodological holiststhink this approach is misguided: As one condensed matter physicistput it “the most important advances in this area come about bythe emergence of qualitatively new concepts at the intermediate ormacroscopic levels—concepts which, one hopes, will be compatiblewith one’s information about the microscopic constituents, butwhich are in no sense logically dependent on it.” (Leggett 1987,p.113)

It is surprisingly difficult to find methodological reductionistsamong physicists. The elementary particle physicist Steven Weinberg,for example, is an avowed reductionist. He believes that by asking anysequence of deeper and deeper why-questions one will arrive ultimatelyat the same fundamental laws of physics. But this explanatoryreductionism is metaphysical in so far as he takes explanation to bean ontic rather than a pragmatic category. On this view, it is notphysicists but the fundamental laws themselves that explain why“higher level” scientific principles are the way they are.Weinberg (1992) explicitly distinguishes his view from methodologicalreductionism by saying that there is no reason to suppose that theconvergence of scientific explanations must lead to a convergence ofscientific methods.

3. Metaphysical Holism

The metaphysical holist believes that the nature of some wholes is notdetermined by that of their parts. One may distinguish three varietiesof metaphysical holism: ontological, property and nomologicalholism.

Ontological Holism: Some objects are not wholly composed of basic physical parts.

Property Holism: Some objects have properties that are notdetermined by physical properties of their basic physical parts.

Nomological Holism: Some objects obey laws that are notdetermined by fundamental physical laws governing the structure andbehavior of their basic physical parts.

All three theses require an adequate clarification of the notion of abasic physical part. One way to do this would be to consider objects as basic, relative toa given class of objects subjected only to a certain kind of process,just in case every object in that class continues to be whollycomposed of a fixed set of these (basic) objects. Thus, atoms wouldcount as basic parts of hydrogen if it is burnt to form water, but notif it is converted into helium by a thermonuclear reaction. But thisway excludes consideration of time-slices and point events (forexample) as basic (spatio)temporal parts of an object. What counts asa part, and what parts are basic, are matters best settled in aparticular context of enquiry.

Weinberg’s (1992) reductionism is opposed to nomological holismin science. He claims, in particular, that thermodynamics has beenexplained in terms of particles and forces, which could hardly be thecase if thermodynamic laws were autonomous. In fact thermodynamicspresents a fascinating but complex test case for the theses both ofproperty holism and of nomological holism. One source of complexity isthe variety of distinct concepts of both temperature and entropy thatfigure in both classical thermodynamics and statistical mechanics.Another is the large number of quite differently constituted systemsto which thermodynamics can be applied, including not just gases andelectromagnetic radiation but also magnets, chemical reactions, starclusters and black holes. Both sources of complexity require a carefulexamination of the extent to which thermodynamic properties aredetermined by the physical properties of the basic parts ofthermodynamic systems. A third difficulty stems from the problematicstatus of the probability assumptions that are required in addition tothe basic mechanical laws in order to recover thermodynamic principleswithin statistical mechanics. (An important example is the assumptionthat the micro-canonical ensemble is to be assigned the standard,invariant, probability distribution.) Since the basic laws ofmechanics do not determine the principles of thermodynamics withoutsome such assumptions (however weak), there may well be at least oneinteresting sense in which thermodynamics establishes nomologicalholism. The related entryphilosophy of statistical mechanics contains further discussion of these difficulties, especially inSection 6.

4. Property/Relational Holism

While some form ofontological holism has occasionally been considered, the variety of metaphysical holismmost clearly at issue in quantum mechanics and in gauge theory isproperty holism. But to see just what the issue is we need a morecareful formulation of that thesis.

First the thesis should be contextualized tophysicalproperties of compositephysical objects. We are interestedhere in how far a physical object’s properties are fixed bythose of its parts, not in some more general determinationistphysicalism. Next, to arrive at an interesting formulation of propertyholism we must accept that this thesis is not only concerned withproperties, and not concerned with all properties. The properties of awhole will typically depend uponrelations among its properparts as well as on properties of the individual parts. But if we arepermitted to considerall properties and relations among theparts, then these trivially determine the properties of the whole theycompose. For one relation among the parts is what we might call thecomplete composition relation—that relation among the partswhich holds just in case they compose this very whole with all itsproperties.

Let us call a canonical set of properties and relations of the partswhich may or may not determine the properties and relations of thewhole the supervenience basis. To avoid trivializing the theses we aretrying to formulate, only certain properties and relations can beallowed in the supervenience basis. The intuition as to which theseare is simple—the supervenience basis is to include just thequalitative intrinsic properties and relations of the parts, i.e., theproperties and relations which these bear in and of themselves,without regard to any other objects, and irrespective of any furtherconsequences of their bearing these properties for the properties ofany wholes they might compose. Unfortunately, this simple intuitionresists precise formulation. It is notoriously difficult to sayprecisely what is meant either by anintrinsic property or relation, or by a purely qualitative property or relation. And theother notions appealed to in expressing the simple intuition arehardly less problematic. But, imprecise as it is, this statementserves already to exclude certain unwanted properties and relations,including the complete composition relation, from the superveniencebasis.

Finally, we arrive at the following opposing theses:

Physical Property Determination: Every qualitative intrinsic physical property and relation of a setof physical objects from any domainD subject onlyto typeP processes supervenes on qualitativeintrinsic physical properties and relations in the supervenience basisof theirbasic physical parts relative toD andP.

Physical Property Holism: There is some set of physical objects from a domainDsubject only to typeP processes,not all of whose qualitative intrinsic physical properties andrelations supervene on qualitative intrinsic physical properties andrelations in the supervenience basis of theirbasic physical parts (relative toD and \(P)\).

If we take the real state of a set of physical objects to be given bytheir qualitative intrinsic physical properties and relations, thenphysical property determination says (while physical property holismdenies) that the real state of wholes is determined by the real stateof their parts.

There is some residual unclarity in the notion of supervenience thatfigures in these theses. The idea is familiar enough—that therecan be no relevant difference in objects inDwithout a relevant difference in their basic physical parts. I take itthat the modality involved here is not logical but broadly physical.One might try to explicate the notion of supervenience here in termsof models of a true, descriptively complete, physical theory. At issueis whether such a physical theory has two models which agree on thequalitative intrinsic physical properties and relations of the basicparts of one or more objects inD but disagree onsome qualitative intrinsic property or relation of these objects.

Teller (1989) has introduced the related idea of what he callsrelational holism.

Relational Holism: There are non-supervening relations—that is, relations that donot supervene on the nonrelational properties of the relata. (p. 214)

Within physics, this specializes to a close relative of physicalproperty holism, namely:

Physical Relational Holism: There are physical relationsbetween some physical objects that do not supervene on theirqualitative intrinsic physical properties.

Physical property holism entails physical relational holism, but notvice versa. For suppose thatF is some qualitativeintrinsic physical property or relation of one or more elements ofDthat fails to supervene on qualitative intrinsicphysical properties and relations in the supervenience basis of theirbasic physical parts. We may define a (non-intrinsic) physicalrelation \(R_{F}\) to hold of the basic physical parts of elements ofD if and only ifF holds ofthese elements. Clearly \(R_{F}\) does not supervene on thequalitative intrinsic physical properties of these parts. So physicalproperty holism entails physical relational holism. But the converseentailment fails. For let \(R_{G}\) be a physical relation that holdsbetween the basic parts of some elements inD whenand only when those elements are in the relation \(S_{G}\). \(R_{G}\)may fail to supervene on the qualitative intrinsic physical propertiesof these basic parts, even though all qualitative intrinsic physicalproperties and relations of elements ofD(including \(S_{G}\)) supervene on the qualitative intrinsic physicalpropertiesand relations of their basic parts.

Physical relational holism seems at first sight too weak to captureany distinctive feature of quantum phenomena: even in classicalphysics the spatiotemporal relations between physical objects seem notto supervene on their qualitative intrinsic physical properties. Butwhen he introduced relational holism Teller (1987) maintained a viewof spacetime as a quantity: On this view spatiotemporal relations doin fact supervene on qualitative intrinsic physical properties ofordinary physical objects, since these include their spatiotemporalproperties.

5. State Nonseparability

Physics treats systems by assigning them states. The thermodynamicstate of a gas specifies its pressure, volume and temperature. Thestate of a system of classical particles is represented as a point ina phase space coordinatized by their positions and momenta. Oneexpects that if a physical system is composed of physical subsystems,then both the composite system and its subsystems will be assignedstates by the relevant physical theory. One further expects that thestate of the whole will not be independent of those of its parts, andspecifically that if a system is composed of two subsystems,AandB, then it will satisfy aprinciple formulated by Einstein (1935). Howard (1985, p.180) givesthe following translation of this principle, which I’ll callthe

Real State Separability Principle: The real state of the pair \(AB\) consists precisely of the realstate ofA and the real state ofB,which states have nothing to do with one another.

But the assignment of states to systems in quantum mechanics seems notto conform to these expectations (see related entryquantum mechanics). Recall that the quantum state of a system specifies its chances ofexhibiting various properties on measurement. At least in ordinaryquantum mechanics, the mathematical representative of this state is anobject defined in a Hilbert space—a kind of vector space. Thisis analogous in some ways to the representation of the state of asystem of particles in classical mechanics in a phase space. Let usformulate a principle of

State Separability: The state assigned to a compound physical system at any time issupervenient on the states then assigned to its component subsystems.

This principle could fail in one of two ways: the subsystems maysimply not be assigned any states of their own, or else the statesthey are assigned may fail to determine the state of the system theycompose. Interestingly, state assignments in quantum mechanics havebeen taken to violate state separability in both ways.

The quantum state of a system may be either pure or mixed (see relatedentryquantum mechanics). In ordinary quantum mechanics a pure state is represented by a vectorin the system’s Hilbert space. On one common understanding, anyentangled quantum systems violate state separability in so far as a vector representing thestate of the system they compose does not factorize into a product ofvectors, one in the Hilbert space of each individual subsystem, thatcould be taken to represent their pure states. On the other hand, insuch a case each subsystem may be uniquely assigned what is called amixed state represented in its Hilbert space not by a vector but by amore general object—a so-called von Neumann density operator.But then state separability fails for a different reason: thesubsystem mixed states do not uniquely determine the compoundsystem’s state. A failure of state separability may not occasionmuch surprise if states are thought of merely in their role ofspecifying a system’s chances of exhibiting various possibleproperties on measurement. But it becomes more puzzling if onebelieves that a system’s quantum state also has a role inspecifying some or all of its categorical properties. For that rolemay connect a failure of state separability to metaphysical holism andnonseparability.

6. Spatial and Spatiotemporal Nonseparability

The idea is familiar (particularly to Lego enthusiasts!) that if oneconstructs a physical object by assembling its physical parts, thenthe physical properties of that object are wholly determined by theproperties of the parts and the way it is put together from them. Aprinciple of spatial separability tries to capture that idea.

Spatial Separability: The qualitative intrinsic physical properties of a compound systemare supervenient on those of its spatially separated component systemstogether with the spatial relations among these component systems.

If we identify the real state of a system with its qualitativeintrinsic physical properties, then spatial separability is related toa separability principle stated by Howard (1985, p. 173) to the effectthat any two spatially separated systems possess their own separatereal states. It is even more closely related to Einstein’s(1935)real state separability principle. Indeed, Einstein formulated this principle in the context of a pair\(A, B\) of spatially separated systems.

Spatial nonseparability —the denial of spatial separability—is also closelyrelated tophysical property holism. At least classically, spatial relations are the only clear examplesof qualitative intrinsic physical relations required in thesupervenience basis for physical property determination/holism: otherintrinsic physical relations seem to supervene on them, while anyinstance of physical property holism due to the spatial separation ofbasic physical parts would entail spatial nonseparability. But if onethought a spatially localized object has a determinate value for amagnitude like mass only by virtue of its mass relations to other suchobjects elsewhere then one might decide to include those relations inthe supervenience basis also (see Dasgupta (2013)).

If we take a spacetime perspective, then spatial separabilitynaturally generalizes to

Spatiotemporal Separability: Any physical process occupying spacetime regionRsupervenes upon an assignment of qualitative intrinsic physicalproperties at spacetime points inR.

Spatiotemporal separability is a natural restriction to physics ofDavid Lewis’s (1986, p. x) principle of Humean supervenience. Itis also closely related to another principle formulated by Einstein(1948, pp. 233–234 of Howard’s (1989) translation) in thefollowing words: “An essential aspect of [the] arrangement ofthings in physics is that they lay claim, at a certain time, to anexistence independent of one another, provided these objects‘are situated in different parts of space’” (thecontext of the quote suggests that Einstein intended his principle toapply to objects provided they occupy spacelike separated regionsof spacetime).

As Healey (1991, p. 411) shows, spatiotemporal separability entailsspatial separability, and so spatial nonseparability entailsspatiotemporal nonseparability. Because it is both more general andmore consonant with a geometric spacetime viewpoint, it seemsreasonable to consider spatiotemporal separability to be the primarynotion. Accordingly, separability without further qualification willmean spatiotemporal separability in what follows, and nonseparabilitywill be understood as its denial.

Nonseparability: Some physical process occupying a regionR ofspacetime is not supervenient upon an assignment of qualitativeintrinsic physical properties at spacetime points inR.

It is important to note that nonseparability entails neitherphysical property holism norspatial nonseparability: a process may be nonseparable even though it involves objects withoutproper parts. But this section has explained that either of the latterprinciples entails nonseparability under quite weak assumptions.

7. Holism and Nonseparability in Classical Physics

Classical physics presents no definitive examples of eitherphysical property holism ornonseparability. As section 6 explained, almost any instance of physical propertyholism would demonstrate nonseparability. This justifies restrictingattention to the latter notion. Now the assumption that all physicalprocesses are completely described by a local assignment of magnitudesforms part of the metaphysical background to classical physics. InNewtonian spacetime, the kinematical behavior of a system of pointparticles under the action of finite forces is supervenient uponascriptions of particular values of position and momentum to theparticles along their trajectories. This supervenience on localmagnitudes extends also to dynamics if the forces on the particlesarise from fields defined at each spacetime point.

The boiling of a kettle of water is an example of a more complexphysical process. It consists in the increased kinetic energy of itsconstituent molecules permitting each to overcome the short rangeattractive forces which otherwise hold it in the liquid. It thussupervenes on the assignment, at each spacetime point on thetrajectory of each molecule, of physical magnitudes to that molecule(such as its kinetic energy), as well as to the fields that give riseto the attractive force acting on the molecule at that point.

As an example of a process in Minkowski spacetime (the spacetimeframework for Einstein’s special theory of relativity), considerthe propagation of an electromagnetic wave through empty space. Thisis supervenient upon an ascription of the electromagnetic field tensorat each point in the spacetime.

But it does not follow that classical processes like these areseparable. For one may question whether an assignment of basicmagnitudes at spacetime points amounts to or results from anassignment of qualitative intrinsic properties at those points. Takeinstantaneous velocity, for example: this is usually defined as thelimit of average velocities over successively smaller temporalneighborhoods of that point. This provides a reason to deny that theinstantaneous velocity of a particle at a point supervenes onqualitative intrinsic properties assigned at that point. Similarskeptical doubts can be raised about the intrinsic character of other“local” magnitudes such as the density of a fluid, thevalue of an electromagnetic field, or the metric and curvature ofspacetime (see Butterfield (2006)).

One response to such doubts is to admit to a minor consequentviolation of separability while introducing a weaker notion,namely

Weak Separability: Any physical process occupying spacetimeregionR supervenes upon an assignment ofqualitative intrinsic physical properties at points ofRand/or in arbitrarily small neighborhoods of thosepoints.

Along with a correspondingly strengthened notion of

Strong Nonseparability: Some physical process occupying aregionR of spacetime is not supervenient upon anassignment of qualitative intrinsic physical properties at points ofR and/or in arbitrarily small neighborhoods ofthose points.

No holism need be involved in a process that is nonseparable, but notstrongly so, as long as the basic parts of the objects involved in itare themselves taken to be associated with arbitrarily smallneighborhoods rather than points. Versions ofWeakSeparability are offered by both Belot (1998, p. 540), whose termis Synchronic Locality, and by Myrvold (2011, p. 425), who calls itPatchy Separability. The idea is that the state of a region superveneson assignments of intrinsic properties to patches of the region, wherethe patches may be taken to be arbitrarily small.

Any physical process fully described by a local spacetime theory willbe at least weakly separable. For such a theory proceeds by assigninggeometric objects (such as vectors or tensors) at each point inspacetime to represent physical fields, and then requiring that thesesatisfy certain field equations. But processes fully described bytheories of other forms will also be separable. These include manytheories which assign magnitudes to particles at each point on theirtrajectories. Of familiar classical theories, it is only theoriesinvolving direct action between spatially separated particles whichinvolve nonseparability in their description of the dynamicalhistories of individual particles. But such processes are weaklyseparable within spacetime regions that are large enough to includeall sources of forces acting on these particles, so that theappearance of strong nonseparability may be attributed to a mistakenlynarrow understanding of the spacetime region these processes actuallyoccupy.

The propagation of gravitational energy according to generalrelativity apparently involves strongly nonseparable processes, sincegravitational energy cannot be localized (it does not contribute tothe stress-energy tensor defined at each point of spacetime as doother forms of energy). But even a non-locally-defined gravitationalenergy will still be supervenient upon the metric tensor defined ateach point of the spacetime, and so the process of its propagationwill be weakly separable.

The definition of nonseparability becomes problematic in generalrelativity, since its application requires that one identify the sameregionR in possible spacetimes with differentgeometries. But while there is no generally applicable algorithm formaking a uniquely appropriate identification, some identification mayappear salient in a particular case. For example, one can meaningfullydiscuss whether or not the field is the same everywhere in the regionoutside the solenoid in theAharonov-Bohm effect with an increased current flowing, even though the size of thecurrent will have a (tiny) influence on the geometry of that region.Note that the definition of nonseparability does not require that oneidentify the same point in spacetimes of distinct geometries.

While strictly outside the domain of classical physics, quantumphenomena such as theAharonov-Bohm effect may be considered manifestations of nonseparability and holism eveninclassical electromagnetism. Nonseparability would be atrivial notion if no qualitative intrinsic physical properties wereever assigned at spacetime points or in their neighborhoods. But thiswould require a thorough-going relationism that took not justgeometric but all local features to be irreducibly relational (cf.Esfeld (2004)). We will return to this point in our discussion of theAharonov-Bohm effect.

8. The Quantum Physics of Entangled Systems

Quantum entanglement is in the first instance a relation between notphysical but mathematical objects representing the states of quantumsystems. Different forms of quantum theory represent quantum states ofvarious systems by different kinds of mathematical object. So theconcept of quantum entanglement has been expressed by a family ofdefinitions, each appropriate to a specific form and application ofquantum theory (see Earman (2015)). The first definition(Schrödinger (1935)) was developed in the context of applicationsof ordinary non-relativistic quantum mechanics to pairs ofdistinguishable particles that have interacted, such as an electronand proton.

A hydrogen atom may be represented in ordinary non-relativisticquantum mechanics as a quantum system composed of two subsystems: anelectrone and a nuclear protonp.When isolated, its quantum state may be representedby a vector \(\Psi\) in a spaceH constructed as atensor product of spaces \(H_{p}\) and \(H_{e}\) used to representstates of \(e, p\) respectively. The states of \(e, p\) are thendefined as entangled if and only if

forevery pair of vectors \(\Psi_{p}, \Psi_{e}\) in\(H_{p}\), \(H_{e}\) respectively. This definition naturallygeneralizes to systems composed ofndistinguishable particles. But alternative definitions seem preferablefor a collection of indistinguishable particles—of electrons orof photons for example (see Ghirardiet al. (2002), Ladymanet al. (2013)).

It follows that the states of electron and proton in an isolatedhydrogen atom are entangled. But one may also represent the hydrogenatom as composed of a center-of-mass subsystemCand a relative subsystemR represented by vectorstates \(\Psi_{C}\), \(\Psi_{R}\) in \(H_{C}, H_{R}\) respectivelysuch that

If the state of the hydrogen atom is represented by \(\Psi\) then thestates of quantum subsystems \(C, R\) are not entangled but the statesof quantum subsystems \(p, e\) are entangled. This illustrates theimportant point that one cannot draw metaphysical conclusions from amathematical condition of quantum entanglement without first decidingwhich quantum systems are physical parts composing some physicalwhole. It may seem natural to take the physical parts of a hydrogenatom to be an electron and a proton. But note that the state of anisolated hydrogen atom is usually represented by \(\Psi_{R}\), and notby \(\Psi\) or \(\Psi_{e}\).

Viewed as basic physical parts of a hydrogen atom represented by state\(\Psi\), its electron and proton may be considered entangled physicalparts since \(\Psi\) cannot be expressed as a product of vectorsrepresenting the state of each. The electron and the proton may eachbe assigned a mixed state, but these do not uniquely determine thestate \(\Psi\): state separability is violated. This may occasion nosurprise if a system’s state merely specifies its chances ofexhibiting various possible properties on measurement. But it may havemetaphysical significance if a system’s quantum state plays arole in specifying its categorical properties—its real state, sothat thereal state separability principle is threatened. His commitment to this principle is one reason whyEinstein denied that a physical system’s real state is given byits quantum state (though it’s not clear what he thought itsreal state consisted in). But according to (one variant of) the rivalCopenhagen interpretation, the quantum state gives a physicalsystem’s real dynamical state by specifying that it containsjust those qualitative intrinsic quantum dynamical properties to whichit assigns probability 1. On this last interpretation, violation ofstate separability in quantum mechanics leads tophysical property holism: it implies, for example, that a pair of fundamental particles mayhave the intrinsic property of being spinless even though this is notdetermined by the intrinsic properties and relations of its componentparticles.

If an entangled pure vector state of a pair of quantum systemsviolates state separability then there are measurements of dynamicalvariables (one on each subsystem) whose joint quantum probabilitydistribution cannot be expressed as a product of probabilitydistributions for separate measurements of each variable. Quantumtheory predicts such a probability distribution for each of many typesof spatially separated measurements of variables including spin andpolarization components on a pair of entangled physical entitiesassigned such a state, and many of these distributions have beenexperimentally verified. If one thought that quantum theory treatseach dynamical variable by replacing a precise real value assignmentby a probability distribution for the results of measurements of thatdynamical variable, one might take this already to violate the realstate separability principle. But if one entertains a theory thatsupplements the quantum state by values of additional“hidden” variables, then the quantum probabilities wouldbe taken to arise from averaging over many distinct hidden states. Inthat case, it would rather be the probability distribution conditionalon a complete specification of the values of the hidden variables thatshould be taken to specify the underlying chances of system andsubsystems exhibiting various possible properties on measurement. Thereal state might then include all these conditional probabilitydistributions. The best known example of such a theory is the Bohmtheory (see the entry onBohmian mechanics), where the “hidden” variables are spatial positions. Ineach specific experimental context all conditional probabilities are 0or 1, so the joint conditional probability distributions indeedfactorize. But the outcome of a measurement of a chosen dynamicalvariable on one subsystem depends on what dynamical variable is chosenand measured on the other, no matter just when or how far apart thesemeasurements are selected and made.

Bell (1964, [2004]) reasoned that anylocal hidden variabletheory must yield conditional probabilities of 0 or 1 for each localoutcome in order to reproduce all quantum predictions but could notpermit these to depend on the choice of distant measurement. He thenproved that the probabilistic predictions of any local hidden variabletheory must satisfy particular inequalities violated by predictions ofquantum theory for certain entangled state assignments (see entryBell’s Theorem). In later work Bell (1990, [2004]) generalized this argument to applyto any theory of a certain type meeting a condition he called LocalCausality which, he claimed, quantum mechanics does not meet. Howard(1989, 1992) took outcome independence — the probabilisticindependence of the outcomes of a given pair of measurements, one oneach of a pair of entangled systems, conditional on definite valuesfor any assumed hidden variables on the joint system—as aseparability condition. Outcome independence may be contrasted withparameter independence—the condition that, given a definitehidden variable assignment, the outcome of a measurement on one of apair of entangled systems is probabilistically independent of whatmeasurement, if any, is made on the other system. Together withparameter independence, outcome independence implies the factorizationof conditional probabilities that leads to what are now called Bellinequalities. These inequalities constrain the patterns of statisticalcorrelations to be expected between the results of measurements ofvariables such as spin and polarization on a pair of entangled systemsin any quantum state. Quantum mechanics predicts, and experimentconfirms, that such Bell inequalities do not always hold. The Bohmtheory accommodates this fact by its violation of parameterdependence, and hence Local Causality. But Howard (1989), as well asTeller (1989), suggested that we appeal instead to a failure ofoutcome independence to understand why Bell inequalities do not alwayshold, and that this failure is associated rather with holism ornonseparability. Howard (1989) blamed the violation of Bellinequalities on the violation of his separability condition: Teller(1989) took it to be a manifestation ofrelational holism. They both acquit parameter independence of blame because they believethat (at least when the measurement events on the entangled systemsare spacelike separated) parameter independence (unlike outcomeindependence) is a consequence of relativity theory: (note that theBohm theory requires a preferred frame not provided by relativitytheory).

Henson (2013) and others have questioned this line of reasoning,including the conclusion that its appeal to holism or nonseparabilityhelps one to understand how these correlations involving entangledsystems come about without any action at a distance that violatesrelativity theory, Local Causality or Einstein’s (1948)

Principle of Local Action: IfA andB are spatiallydistant things, then an external influence onA hasnoimmediate effect onB.

Howard’s (1989,1992) identification of outcome independence witha separability condition has proved controversial, as hasTeller’s (1989) claim that violations of Bell inequalities areno longer puzzling if one embraces (physical) relational holism(Laudisa 1995; Berkowitz 1998; Henson 2013). Winsberg and Fine (2003)object that separability requires only that the conditional jointprobabilities be determined as some function of the marginalprobabilities, whereas outcome independence arbitrarily restricts thisto be the product function. By allowing other kinds of functionaldependence they are able to construct models of experiments whoseresults would exhibit violations of Bell inequalities. They claim thatthese models are both local and separable even though they violateoutcome independence. But Fogel (2007) presents alternativeformalizations of the separability condition, of which several indeedimply outcome independence. The view that violations of outcomeindependence are consistent with relativity theory, while violationsof parameter independence are not, has also been criticized (Jones& Clifton 1993; Maudlin 2011). But Myrvold (2016) has responded byarguing that locally-initiated state-vector collapse in violation ofoutcome dependence may be perfectly compatible with relativity.

While diverging from the Copenhagen prescription mentioned above, somemodal interpretations take real states of systems to be closely enough related to quantumstates that entangled systems’ violation of quantum stateseparability implies some kind of holism or nonseparability. VanFraassen (1991, p. 294), for example, sees his modal interpretation ascommitted to “a strange holism” because it entails that acompound system may fail to have a property corresponding to a tensorproduct projection operator \(P\otimes I\) even though its firstcomponent has a property corresponding toP. Infact, a clearer case of holism would arise in a modal interpretationthat implied that the component lackedP while thecompound had \(P\otimes I\):ceteris paribus, that wouldprovide an instance of physical property holism. Healey (1989,1994)offered a modal interpretation and used it to present a model accountof the puzzling correlations which portrays them as resulting from theoperation of a process that violates bothspatial andspatiotemporal separability. He argued that, on this interpretation, thenonseparability of the process is a consequence ofphysical property holism; and that the resulting account yields genuine understanding of howthe correlations come about without any violation of relativity theoryorLocal Action. But subsequent work by Clifton and Dickson (1998) and Myrvold (2001)cast doubt on whether the account can be squared with relativitytheory’s requirement of Lorentz invariance. More recently Healey(2016) has given a different account of how quantum theory may be usedto explain violations of Bell inequalities consistent with Lorentzinvariance and Local Action. This account involves no metaphysicalholism or nonseparability.

Esfeld (2001) takes holism, in the quantum domain and elsewhere, toinvolve more than just a failure of supervenience. He maintains that acompound system is holistic in that its subsystems themselves count asquantum systems only by virtue of their relations to other subsystemstogether with which they compose the whole.

9. Ontological Holism in Quantum Mechanics?

As applied to physics,ontological holism is the thesis that there are physical objects that are not whollycomposed of basic physical parts. Views of Bohr, Bohm and others maybe interpreted as endorsing some version of this thesis. In no case isit claimed that any physical object hasnonphysical parts.The idea is rather that some physical entities that we take to bewholly composed of a particular set of basic physical parts are infact not so composed.

It was Bohr’s (1934) view that one can meaningfully ascribeproperties such as position or momentum to a quantum system only inthe context of some well-defined experimental arrangement suitable formeasuring the corresponding property. He used the expression‘quantum phenomenon’ to describe what happens in such anarrangement. In his view, then, although a quantum phenomenon ispurely physical, it is not composed of distinct happenings involvingindependently characterizable physical objects—the quantumsystem on the one hand, and the classical apparatus on the other. Andeven if the quantum system may be taken to exist outside the contextof a quantum phenomenon, little or nothing can then be meaningfullysaid about its properties. It would therefore be a mistake to considera quantum object to be an independently existing component part of theapparatus-object whole.

Bohm’s (1980, 1993) reflections on quantum mechanics led him toadopt a more general holism. He believed that not just quantum objectand apparatus, but any collection of quantum objects by themselves,constitute an indivisible whole. This may be made precise in thecontext of Bohm’s (1952) interpretation of quantum mechanics bynoting that a complete specification of the state of the“undivided universe” requires not only a listing of allits constituent particles and their positions, but also of a fieldassociated with the wave-function that guides their trajectories. Ifone assumes that the basic physical parts of the universe are just theparticles it contains, then this establishes ontological holism in thecontext of Bohm’s interpretation. But there are alternativeviews of the ontology of the Bohm theory (see the entryBohmian mechanics).

Some (Howard 1989; Dickson 1998) have connected the failure of aprinciple of separability to ontological holism in the context ofviolations of Bell inequalities. Howard (1989) formulates thefollowing separability principle (pp. 225–6)

Howard Separability: The contents of any two regions of space-time separated by anonvanishing spatiotemporal interval constitute separable physicalsystems, in the sense that (1) each possesses its own, distinctphysical state, and (2) the joint state of the two systems is whollydetermined by these separate states.

He takes Einstein to defend this as a principle of individuation ofphysical systems, without which physical thought “in the sensefamiliar to us” would not be possible. Howard himselfcontemplates the possible failure of this principle forentangled quantum systems, with the consequence that these could no longer be taken to be whollycomposed of what are typically regarded as their subsystems. Dickson(1998), on the other hand, argues that such holism is not “atenable scientific doctrine, much less an explanatory one” (p.156).

One may try to avoid the conclusion that experimental violations ofBell inequalities manifest a failure ofLocal Action by invoking ontological holism for events. The idea would be to denythat these experiments involve distinct, spatiotemporally separate,measurement events, and to maintain instead that what we usuallydescribe as separate measurements involving an entangled system infact constitute one indivisible, spatiotemporally disconnected, eventwith no spatiotemporal parts. But such ontological holism conflictswith the criteria of individuation of events inherent in both quantumtheory and experimental practice.

10. The Aharonov-Bohm Effect and Field Holonomies

Aharonov and Bohm (1959) drew attention to the quantum mechanicalprediction that an interference pattern due to a beam of chargedparticles could be produced or altered by the presence of a constantmagnetic field in a region from which the particles were excluded.This effect has since been experimentally demonstrated. At firstsight, the Aharonov-Bohm effect seems to involve action at a distance.It seems clear that the (electro-)magnetic field acts on the particlessince it affects the interference pattern they produce; and this mustbe action at a distance since the particles pass through a region fromwhich that field is absent. But alternative accounts of the phenomenonare possible which portray it rather as a manifestation of (strong)nonseparability (Healey 1997). There need be no action at a distance if the behaviorboth of the charged particles and of electromagnetism are nonseparableprocesses. While such a treatment of electromagnetism (and other gaugetheories) is increasingly common in physics, to treat the motion ofthe charged particles as a nonseparable process is to endorse aparticular position on how quantum mechanics is to be interpreted.

An interpretation of quantum mechanics that ascribes a nonlocalizedposition to a charged particle on its way through the apparatus iscommitted to a violation ofspatiotemporal separability in the Aharonov-Bohm effect, since the particle’s passageconstitutes a nonseparable process. To see why the electromagnetismthat acts on the particles during their passage may also be taken tobe nonseparable it is necessary to consider contemporaryrepresentations of electromagnetism in terms of neither fields norvector potentials.

Following Wu and Yang’s (1975) analysis of the Aharonov-Bohmeffect, it has become common to consider vacuum electromagnetism to becompletely and nonredundantly described neither by the electromagneticfield, nor by its vector potential, but rather by the so-called Diracphase factor:

where \(A_{\mu}\) is the electromagnetic potential at spacetime point\(x^{\mu}\),e is the particles’ charge, andthe integral is taken over each closed loopC inspacetime. This can be seen as an instance of the more general notionof the holonomy of a closed curve, a notion that has come to the forein contemporary formulations of gauge theories includingelectromagnetism in terms of fiber bundles (Healey (2007)). Applied tothe Aharonov-Bohm case, this means that the constant magnetic field isaccompanied by an association of a phase factor \(S(C)\) with allclosed curvesC in space, where \(S(C)\) is definedby

(where \(\mathbf{A}(\mathbf{r})\) is the magnetic vector potential atpoint \(\mathbf{r}\) of space). This approach has the advantage thatsince \(S(C)\) is gauge-invariant, it may readily be considered aphysically real quantity. Moreover, the effects of electromagnetism inthe field-free region may be attributed to the fact that \(S(C)\) isnonvanishing for certain closed curvesC withinthat region. But it is significant that, unlike the magnetic field andits potential, \(S(C)\) is not defined at each point of space at eachmoment of time.

Can \(S(C)\) at some time be taken to represent an intrinsic propertyof a region of space corresponding to the curveC?There are two difficulties with this suggestion. The first is that thepresence of the quantitye in the definition of\(S(C)\) appears to indicate that \(S(C)\) rather codes the effect ofelectromagnetism on objects with that specific charge. If in factall charges are multiples of some minimal valuee,then this would no longer be a problem: the value of\(S(C)\) for this minimal charge could then be viewed as representingan intrinsic property of a region of space corresponding to the curveC. If not, one could rather take

to represent an intrinsic property ofC. The seconddifficulty is that closed curves do not correspond uniquely to regionsof space: e.g., circling the region in which there is a magnetic fieldtwice on the same circle will produce a different curve from circlingit once. But this does not prevent one from taking \(S(C)\) at sometime to represent an intrinsic property of a loop—the orientedregion of space traced out by a closed curveC thatintersects itself only at its end-point.

Once these difficulties have been handled, it is indeed possible toconsider electromagnetism in the Aharonov-Bohm effect as faithfullyrepresented at a time by a set of intrinsic properties of loops inspace (or more generally space-time). But if one does so, thenelectromagnetism itself manifests (strong)nonseparability. For these intrinsic properties do not supervene on any assignment ofqualitative intrinsic physical properties at spacetime points in theregion concerned, nor even in arbitrarily small neighborhoods of thosepoints. Whether the magnetic field remains constant or changes, theassociated electromagnetism constitutes a nonseparable process, and sothe Aharonov-Bohm effect violatesspatiotemporal separability. If the motion of the particles through the apparatus is anonseparable process, then it is possible to account for the \(AB\)effect in terms of a purely local interaction between electromagnetismand this process. For the particles effectively traverse loops tracedout by closed curvesC on their nonlocalized“trajectories”, and so they interact with electromagnetismprecisely where this is defined.

Assuming it exhibits nonseparability, does the Aharonov-Bohm effectinvolve some kind of holism? The states of the particles need not beentangled with each other. But the state of the field may be thoughtto be holistic, in so far as the electromagnetic properties of loopsdo not supervene on properties (such as electric and magnetic fieldstrengths) at the points that make up those loops. Since these areclassical fields, the Aharonov-Bohm effect may be taken to demonstrateholism as well as nonseparability even inclassical physics.However, a loop may also be thought of as traced out by a curvecomposed by “stringing together” a set of curves thattrace out smaller loops, and removing segments traversed by two suchcurves in opposite directions. In that case, in a simply-connectedregion, the holonomy properties of any loop will be determined bythose of an arbitrary set of smaller loops that compose it in thisway. But when spacetime is not simply connected, there are loops thatcannot be decomposed in this way: e.g. the loop that encompasses“the hole” in space cannot be further decomposed.

Myrvold (2011) interprets this result as saying that Weak Separability(what he calls Global Patchy Separability) fails for non-simplyconnected spacetimes, even though locally—over simply-connectedpatches—separability holds. Moreover, he here sees a markeddistinction between the holonomy and the field formalism for gaugetheories: in which supposedly separability always holds.

But Gomes (2021) effaces this distinction: he argues that a fieldformalism maintains separability, but only at the cost of dealing withgaugevariant field variables. Once we restrict attention tothe gauge-invariant quantities, the nonseparabilityillustrated above (using the holonomy variables) is restored.Moreover, he argues that only in a vacuum is nonseparabilityconditional on the topology of spacetime. In the presence of chargedmatter, there may be gauge-invariant quantities that cannot bedecomposed into smaller patches, even for simply-connected regions.And this is easy to illustrate: suppose that, instead of the phasefactor \(S(C)\), for a closed loop C, we write \(S(I)\) for an opensegmentI, that begins at pointx and endsat pointy. For a (Klein-Gordon) charged scalarfield \(\varphi\) that vanishes everywhere except at smallneighborhoods \(\nu_x, \nu_y\), ofx andy,the function:

is gauge-invariant, and it cannot be further decomposed intoquantities with support on patches that do not include the entirety of\(I. \)For instance, if there are two patches, each containing either\(\nu_x\) or \(\nu_y\), then that patch has insufficient charges tocomplete \(S(I)\) into a gauge-invariant function. This example is inline with the physics literature, which largely takes gauge-invariance(and in particular the existence of a generalized Gauss constraint),to signal some general type of nonlocality (cf. Strocchi (2013, Ch.7)).

11. Alternative Approaches

This entry has focused mainly onmetaphysical holism and its relation to nonseparability. That there is a variety ofalternative ways of understanding holism in physics is illustrated bya special issue of the journalStudies in the History andPhilosophy of Physics (2004) devoted to this topic.

Seevinck (2004) proposes an epistemological criterion of holism andillustrates its application to physical theories. A physical theorycounts as holistic by this criterion if and only if it is impossiblein principle to infer the global properties, as assigned in thetheory, by local resources available to an agent, where these include(at least) all local operations and classical communication. To applythis criterion it is necessary to specify how a theory assignsproperties, a matter on which different interpretations of the theorymay disagree. Seevinck (2004) argues that neither classical physicsnor Bohmian mechanics is holistic in this sense. Applying theeigenvalue-eigenstate link to a particular state of a bipartitequantum system he then shows that this manifests epistemologicalholism, even when the state of the system is notentangled.

Placek (2004) understands quantum state holism as involving a thesisabout probabilities: that the probability of a joint result of acombined measurement on a pair of entangled quantum systems is notdetermined by the probabilities of the two results. But he takes thisto be only one ingredient of a more complete conception, whoseformulation and analysis requires a modal framework combiningindeterminism, (rudiments of) relativistic space-time, andprobability—Belnap’s (1992) theory of branchingspace-times.

Esfeld (2004) argues for a metaphysics of relations based on acharacterization of quantum entanglement in terms of non-separability,thereby regarding entanglement as a sort of holism. He therecharacterizes non-separability as follows:

Non-separability: The states of two or more systems arenon-separable if and only if it is only the joint state of the wholethat completely determines the state-dependent properties of eachsystem and the correlations among these systems (to the extent thatthese are determined at all).

He takes this to imply that any case of quantum entanglement is a caseof non-separability, and non-separability is the reason why quantumentanglement is a sort of holism. (He discusses the relation betweennon-separability and holism in chapter 8 of Esfeld (2001).)

Lyre (2004) and Healey (2004) see electromagnetism and other gaugetheories as manifesting nonseparability for reasons different fromthose arising from quantum entanglement (cf.The Aharonov-Bohm Effect). Lyre takes this to be a variant of spatiotemporal holism, andconnects it to structural realism. Healey (2007) argues that generalrelativity does not manifest this kind of nonseparability even thoughit may be formulated as a gauge theory. Weatherall (2016) argues thata distinction exists, but it is not enough to rule out spatiotemporalholism in general relativity. If there are more than one spatialdimensions, an analogue of the Aharonov-Bohm effect in generalrelativity (using the directions of parallel-propagated vectors)theoretically exhibits a similar type of nonseparability as thatobserved in gauge theory.

12. Quantum Field Theory

Certain phenomena that arise within quantum field theory have beentaken to challenge principles of separability or to involve holism.These have been analyzed most intensively by mathematical physicistsand philosophers taking an algebraic approach to quantum theory, eventhough many empirical successes of quantum field theory have beenachieved by following other approaches.

Algebraic quantum field theory (AQFT) represents the state in a regionof spacetime by means of a function from an algebra of associated“field” or “observable” operators: the valueof this function for a self-adjoint operator represents the expectedresult of a measurement of the corresponding observable on thatregion. A state is said to be decomposable (some say separable) acrossalgebras \(R_{A}, R_{B}\) associated with regions \(A, B\) if itsrestriction \(\omega\) to the algebra \(R_{AB}\) generated by \(R_{A},R_{B}\) is a product state—i.e. satisfies\(\omega(XY)=\omega(X)\omega(Y)\), for all \(X\in R_{A}, Y\in R_{B}\);or if \(\omega\) is a limit of convex combinations of product states:otherwise, it is said to be entangled across \(R_{AB}\) (see e.g.Valente 2010, pp. 1031–2). This is a natural reformulation of ageneralization to mixed states of the first condition for entanglementgiven insection 8.

Entanglement is endemic in AQFT. Summers and Werner (1985) proved thatthe vacuum state of a quantum field is not only entangled acrossalgebras associated with certain spacelike separated regions ofMinkowski spacetime, but that it also maximally violates Bellinequalities for algebras associated with these regions. They alsoproved (1988) thatevery state on a pair of space-likeseparated open regions whose closures share a single point ismaximally entangled across their algebras. For each state, the degreeof entanglement decreases rapidly with spatial separation. But if andonly if \(R_{A}, R_{B}\) possess what is called thesplitproperty is any state decomposable across these algebras.

The split property (Valente 2010, p. 1035) is a strengthening of thecondition of microcausality (observables on space-like separatedregions commute). Summers (2009) argues that it is meaningful to speakof independent subsystems in relativistic quantum theory if they canbe localized in spacetime regions \(A, B\) whose algebras \(R_{A},R_{B}\) posses the split property; and that most, if not all,physically relevant models of quantum field theories have thisproperty (for sufficiently space-like separated regions \(A, B)\).

The split property is a kind of independence condition. Rédei(2010) argues that, by conforming to this and other independenceconditions, an AQFT can meet all the requirements Einstein (1948)considered necessary for a quantum theory satisfactorily to realizethe field-theoretic ideal. These were the requirement that physicalthings be arranged in a space-time continuum (Spatiotemporality); thatthings located in space-like separated regions have their own distinctstates (Independence); and that if \(a, b\) are located in space-likeseparated regions \(A, B\) respectively, then an external influence ona has noimmediate effect onb(Local Action). (The first two names are Rédei’s: the last isEinstein’s.) Rédei takes AQFT to satisfySpatiotemporality because of its basic assumption that observables arelocalized in space-time regions; that an AQFT’s satisfaction ofthe split property and other members of a hierarchy of independenceconditions establishes Independence; and that an AQFT obeys LocalAction insofar as it meets a condition he calls operationalseparability.

In evaluating Rédei’s argument it is important to askwhat counts as a physical thing. Einstein mentioned two possiblecandidates: bodies and fields.Howard’s separability principle permits a natural transposition of Einstein’sreal state separability principle to field theory. Entangled states in AQFT violate section 5’sstate separability principle just as they do in nonrelativistic quantum mechanics eventhough the split property and related independence conditions on theiralgebras hold. So if the content of a space-time region were specifiedby its algebra in AQFT, considered as a physical system with realphysical state given by a state on that algebra, then Howard’sseparability principle would fail. (Although the failure of the splitproperty or other algebraic independence conditions for certainregions in a quantum field theory would represent a more radicalthreat to the separate existence of such physical systems in thoseregions than mere entanglement). But it is doubtful that Einsteinwould have counted observables or their algebras as physical things.If instead these are taken to represent magnitudes of physical fieldsor the spatio-temporal region on which they are defined then meetingRédei’s Independence requirement is still consistent withfailure of Howard’s (stronger) separability principle. Finallymeeting Rédei’s operational separability condition onlyfornon-selective operations would not suffice to ensureconformity to Local Action. Einstein’s reasons for rejecting thecompleteness of quantum mechanical description are naturally extendedto AQFT: if a state on its local algebra completely specifies the realstate of a space-time region then either the natural extension of hisreal state separability principle or his principle ofLocal Action fails.

Metaphysical holism presupposes a division of a whole into parts. To apply the part/wholedistinction here one must address the ontology of quantum fieldtheory. Taking spacetime regions to be the relevant physical objects,one could understand the system/subsystem and part/whole relations interms of spatiotemporal inclusion. To assessphysical property holism ornonseparability we need to determine the qualitative intrinsic properties andrelations pertaining to spacetime regions in quantum field theory.

Arageorgis (2013) gives an example of quantum field states entangledacross two regions which nevertheless, he argues, fail to exhibit thesame kind ofstate nonseparability as singlet and triplet spin states of a pair of quantum particles(see Maudlin 1998). But he suggests that his example does exhibit akind of epistemological state nonseparability, in so far as an agentconfined to a single region cannot determine its state by operationsconfined to that region. By applying the eigenvalue-eigenstate link tohis example, Arageorgis (2013) argues that the energy of a particularcompound quantum field system is not determined by the energies (orany other qualitative intrinsic properties and relations) of itscomponent subsystems. He concludes that this example manifestsphysical property holism.

Wayne (2002) has suggested that quantum field theory is bestinterpreted as postulating extensive holism or nonseparability. Onthis interpretation, the fundamental quantities in quantum fieldtheory are vacuum expectation values of products of field operatorsdefined at various spacetime points. The field can be reconstructedout of all of these. Nonseparability supposedly arises because thevacuum expectation value of a product of field operators defined at ann-tuple of distinct spacetime points does notsupervene on qualitative intrinsic physical properties defined atthosen points, together with the spatiotemporalrelations among the points. But it is not clear that vacuumexpectation values of products of field operators defined atn-tuplesof distinct spacetime points represent eitherqualitative intrinsic physical properties of thesen-tuplesor physical relations between them. Improvedassessment of the extent to which quantum field theory illustratesholism ornonseparability must await further progress in the interpretation of quantum fieldtheory. (Kuhlman, Lyre and Wayne (2002) represents a relevant firststep: but see also Fraser (2008), Baker (2009).)

13. String Theory

String theory (or its descendant,M-theory) hasemerged as a speculative candidate for unifying much of fundamentalphysics, including quantum mechanics and general relativity. Existingstring theories proceed by quantizing classical theories of basicentities that are extended in one or more dimensions of a space thathas 6 or 7 tiny compact dimensions in addition to the three spatialdimensions of ordinary geometry. If these additional dimensions areappropriately considered spatial, then it is natural to extend theconcepts ofspatial andspatiotemporal separability to encompass them. In that case, processes involvingclassical strings (orp-branes with \(p \gt 0\))would count as (spatiotemporally) nonseparable, even though allparticles and their properties conform to spatial separability.

The status ofnonseparability within a quantized string field theory is not so easy to assess,because of the general problems associated with deciding what theontology of any relativistic quantum field theory should be taken tobe.

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• Grangier, Philippe, “Contextual Objectivity and Quantum Holism”, manuscript at arXiv.org.
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