INDIVIDUALITY, BELL AND NON-SUPERVENIENT RELATIONS

INDIVIDUALITY, BELL AND NON-SUPERVENIENT RELATIONS

The de Broglie-Bohm position is a form of 'hidden variables' interpretation. Such interpretations typically claim that quantum mechanical systems can be characterized by some set of 'hidden' factors (position in the de Broglie-Bohm case) which eliminate the ontological indeterminacy represented by the wave function. They were originally thought to be ruled out by von Neumann's anti-hidden variables proofs but these contain certain loopholes, which were subsequently exploited by Bohm for example. In the 1960s the cost of adopting such an interpretation was revealed by Bell via his eponymous theorem: if we assume some form of 'locality' condition-broadly understood as the constraint that th 18318s1810s e possessed value of some attribute belonging to one member of a pair, say, of previously interacting but now spatially separated systems, should not be affected by an alteration in the physical arrangement of an apparatus interacting with the other member-then in the context of a hidden variables interpretation one can derive an inequality which is violated by quantum mechanics. ¹³² Perhaps the most important virtue of this fundamentally important result is that it is experimentally testable and to date the experimental results have been in favour of quantum mechanics. ¹³³ The implication of this result is that in such situations quantum mechanics manifests 'non-local' interactions in some sense. These situations are precisely those in which the relevant systems are in superposed or 'entangled' states and the concern is that this non-locality might bring quantum mechanics into conflict with relativistic considerations.

As originally developed, this result applied to deterministic hidden variables interpretations, in which the set of such variables determines the values of the outcomes of certain measurements. In response, attention shifted to indeterministic or stochastic forms of such interpretations, in which the hidden variables determine only the probability of an outcome. In this context it proved more difficult to generate a Bell-type inequality and appeal has to be made to two logically independent conditions: ¹³⁴

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Since (1) is equivalent to the locality constraint underpinned by Special Relativity, which most (but by no means all) physicists would be reluctant to abandon, the violation of the inequality based on these conditions has led to attention being focused on (2). ¹³⁵ Thus it has been noted that if it is the latter that is targeted, rather than (1), then there is the possibility of 'peaceful coexistence' with the constraints of Special Relativity. ¹³⁶ However, there is still the issue of the metaphysical basis of (2) and in particular the question arises as to how we should understand its violation.

Further light has been shed on these issues by Howard's interpretation of (2) as a 'Principle of Separability' and it is this which ties these considerations to our account of quantum individuality. ¹³⁷ The Principle embodies the idea that spatially separated systems possess distinct states. Its origins lie in the field-theoretic approach, where, as we saw in Chapter 2, field quantities are defined at space-time points and the (mathematical) separability of the points is inherited (ontologically) by these field quantities which in turn represent properties of systems. According to Howard, it was this notion of separability that was the real focus of Einstein's concerns in the EPR debate. ¹³⁸ It is interesting, given our discussion in Chapter 3 of Born's view of objectivity in physics, that Einstein presents this notion in a 1948 letter commenting on Born's work. He writes that,

. if one abandons the assumption that what exists in different parts of space has its own, independent, real existence, then I simply cannot see what it is that physics is meant to describe. For what is thought to be a 'system' is, after all, just a convention, and I cannot see how one could divide the world objectively in such a way that one could make statements about parts of it. ¹³⁹

Later that same year Einstein sent Born a draft of his paper in which the incompleteness of quantum mechanics, as standardly formulated, was derived from its failure to satisfy separability. ¹⁴⁰ Again, this notion is articulated in terms of the 'independent existence' of spatially separated systems and Einstein insists that,

[u]nless one makes this kind of assumption about the independence of the existence (the 'being-thus') of objects which are far apart from one another in space-which stems in the first place from everyday thinking-physical thinking in the familiar sense

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would not be possible. It is also hard to see any way of formulating and testing the laws of physics unless one makes a clear distinction of this kind. This principle has been carried to extremes in the field theory by localizing the elementary objects on which it is based and which exist independently of each other, as well as the elementary laws which have been postulated for it, in the infinitely small (four-dimensional) elements of space. ¹⁴¹

Born's response is worth noting. ¹⁴² He argues that Einstein's 'axiom' fails for light, which exhibits the kind of coherence which violates separability. Hence if quantum mechanics is deemed incomplete on the basis of such a violation, then the theory of light must be regarded as incomplete as well. Of course, the analogy is not entirely satisfactory: in order to obtain coherent behaviour from a light beam, we typically need to split it but in the case of particles we assume we have two to begin with. It is this assumption that generates the problem, as how can two entities, each with their own 'being-thus' (individuality) as it were, exhibit such coherence? This is the root of Einstein's concern but Born insists that it must simply be accepted, ¹⁴³ without, however, spelling out the metaphysical implications.

These are taken up by Howard, who urges that separability should be understood as a principle of individuality: ¹⁴⁴

. we can now understand the larger point Einstein intended to make about field theories. It is that by modeling a physical ontology upon the ontology of the mathematical manifold, we take over as a criterion for the individuation of physical systems and states within field theories the mathematician's criterion for the individuation of mathematical points. This criterion is the existence between two points of a non-vanishing interval, which gets interpreted as a three-dimensional spatial interval in classical electrodynamics, and as a four-dimensional spatio-temporal or metrical interval in general relativity. In this way, field theories-as understood by Einstein-necessarily satisfy the separability principle. ¹⁴⁵

Thus Howard takes separability as a necessary condition for individuality and suggests that what the violation of Bell-type inequalities shows is that, first, we must give up separability for quantum particles in entangled states

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and hence, secondly, that when in such states these particles cannot be regarded as individuals. In other words, Bell's Theorem implies non-individuality. ¹⁴⁶

How should we respond to this result? One way is to simply follow the determined Bohmian, who accepts a form of non-locality via the operation of the quantum potential and thus has no fear of Bell's Theorem. Hence we could maintain the Bohmian's sense of individual particles, grounded ultimately on Space-Time Individuality, at a cost of admitting non-locality into our metaphysical scheme. There is an alternative, however, which maintains the individuality of the particles and shifts the metaphysical peculiarities of quantum mechanics onto the relations manifested in these 'entangled' states.

The heart of this alternative lies in Teller's claim that this notion of 'entanglement' can be thought of as an expression of 'non-supervenient' relations holding between the particles. ¹⁴⁷ Now, the notion of 'supervenience' that is used here is more typically found in discussions of the philosophies of mind and morality. ¹⁴⁸ According to this characterization, a dyadic relation R is supervenient upon a determinable non-relational attribute P if and only if:

where a physical attribute P is understood as 'non-relational' if and only if it is possible that there exists an x such that x has P and no individual physical thing wholly other than x exists (in other words, non-relational attributes are independent in the sense that they can be instanced in a universe containing only

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one individual). What these conditions mean can be spelled out as follows. If R is genuinely supervenient upon P then (a) implies that the relation R cannot possibly appear in the absence of each of its relata instancing the requisite reductive property P, whereas (b) states that there must exist one or more pairs of determinate monadic properties (of kind P) whose exemplification alone is sufficient to guarantee the appearance of R.

This then yields two forms of 'non-supervenience', according to whether both (a) and (b) or (b) only are violated, respectively:

Strong Non-Supervenience: a relation is 'strongly' non-supervenient upon a determinable non-relational attribute if the appearance of this relation is neither dependent upon nor determined by its relata bearing the non-relational attribute concerned.

Weak Non-Supervenience: a relation is 'weakly' non-supervenient upon a determinable non-relational attribute if the appearance of this relation is dependent upon the instantiation of the non-relational attribute in the sense that the relation could not possibly be exemplified in the absence of each of its relata separately bearing the determinable non-relational attribute in question, but there exist no determinate non-relational attributes whose manifestation is sufficient for the appearance of the relation.

The question now, of course, is whether the inter-particle relations expressed by the entangled states of quantum mechanics can be described as 'non-supervenient' in either of these senses. ¹⁴⁹ To answer this let us recall the case of two particles distributed over two one-particle states and the state-functions given as (5), (6), (7) and (8) above. Assuming the standard Eigenstate-Eigenvalue link again, these can all be taken to represent properties of the particles. (5) and (6), although relational, can be factorized into properties of each particle separately. Thus there exist, in this case, pairs of monadic properties whose exemplification alone is sufficient to guarantee the appearance of the relations expressed by (5) and (6) and these relations satisfy condition (b) and hence also (a) of Cleland's characterization. In other words, they are not 'inherent' relations, in Teller's sense of not supervening on the relevant non-relational properties.

However, it is not so immediately clear that this is also true for the kinds of relations expressed by (7) and (8). We have already seen that one cannot ascribe to each particle an individual state function on the basis of either (7) or (8), since these state functions are not the product of the separate state functions of the particles. It is this, of course, which reveals the peculiar non-classical holism of quantum mechanics. We recall that the odd nature of these states is revealed when one considers correlation measurements between observables relating to each particle, such as joint measurements in the two-particle system relating to the probability of finding both particles in the same one-particle state, as we discussed in the section on PII. These statistical correlations-to the effect that, for example, if one fermion is occupying a given state, all other fermions are excluded from that state-express a relation between, or relational property of, the particles, in Cleland's sense. The question now is, are such relations 'inherent' or not?

A negative answer could be given if one were willing to entertain non-classical forces coupling between certain properties of the particles, as exemplified by the so-called 'exchange forces' which we've already mentioned. This would then effectively reduce the relational properties represented by (7) and (8) to non-relational attributes of the relata. ¹⁵⁰ We have already indicated the objections that have been made to such approaches. First of all, such forces would have to be non-local, acting 'at a distance', and possibly would have to be acausal as well. ¹⁵¹ This may not unduly perturb the advocates of this sort of line, as they are generally happy to accept non-local, but not necessarily acausal interactions. ¹⁵² Secondly, this sort of approach is typically accompanied by a further-and sometimes quite dramatic-expansion of our ontology; thus, in addition to the Bohm potential we have these peculiar forces, not manifested in terms of the familiar four (or three) and mediated by some kind of 'sub-aether'. ¹⁵³ At this point, the price of reducing our relations of entanglement may seem unacceptably high. ¹⁵⁴

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If we are unwilling to pay this price, then a straightforward argument can be given to the effect that these quantum statistical correlations manifest relations which are 'inherent' in the sense of not supervening upon properties of the relata. The argument runs as follows: ¹⁵⁵

In other words, the relations manifested by these statistical correlations are irreducible to properties represented by, for example, (5) and (6) and do not supervene upon the latter. Let us now return to our question, are they 'strongly' non-supervenient or only 'weakly' so?

It is worth comparing the quantum case with Cleland's own example of spatio-temporal relations. She argues that these are only weakly non-supervenient upon non-relational attributes of material bodies, since such relations are dependent on non-relational attributes such as size and shape, thus satisfying (a) above, but are not determined by such attributes, thus violating (b). However, this conclusion crucially hinges on our being able to imagine a situation, a possible world, in which the size and/or shape of an object are stripped away and in such a case we would say that the object ceases to be at any distance from another. The important point to note is that it is because an object can possess both size/shape and certain inter-object relational properties simultaneously that we have to move to a possible world in which the former are removed in order to see if the latter disappear as well. In quantum mechanics the situation is quite different, as our arguments above show. In this case the properties represented by (6) and (7), for example, are mutually exclusive in the sense that a system in one cannot simultaneously be in the other. Hence there is no need to invoke possible worlds here since it is already a 'built in' feature of the theory that these properties are independent

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of one another. This, of course, is the thrust of the argument that (7) cannot be reduced to (5) or (6). As Teller remarks, referring to the Bell-type situation:

. the superposition characterises an independently identifiable property with distinctive experimental implications for the 1-2 system as a whole. ¹⁵⁶

If the properties represented by (7) and (8) are not dependent upon those represented by (5) or (6), then clearly they are not determined by them either. Therefore, in contrast with Cleland's inter-object distance relations, the quantum mechanical entangled or correlation relations are strongly non-supervenient with respect to the relevant non-relational properties. Furthermore, if the bundle view of individuality is rejected, then these relations are strongly non-supervenient with respect to the intrinsic properties of the particles as well. The conclusion, then, is that the 'entangled' relations of quantum mechanics are strongly non-supervenient with respect to non-relational or monadic properties in general.

It is hard to avoid the implication that such relations must be regarded as having a physical reality qua relations, transcending all the other properties of the relata, and should be taken as basic constituents of the world along with particulars and monadic properties. ¹⁵⁷ Teller himself calls this view 'Relational Holism' and contrasts it with what he calls 'Particularism', which states that:

. the world is composed of individuals, that the individuals have nonrelational properties and that all relations between individuals supervene on the nonrelational properties of the relata. ¹⁵⁸

What quantum mechanics tells us, he insists, is that particularism is simply wrong and that we must endorse 'Relational Holism' and accept that there are non-supervenient relations. Nevertheless, the metaphysical implications have yet to be fully spelt out. Perhaps the most obvious way of embarking

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on such a project would be to start with the traditional account of relations in general.

As is well known, this issue of how we are to understand relations ontologically is one that crops up again and again throughout the history of philosophy. The commonly held view that relations should be regarded as reducible to non-relational properties is reflected in the claim that every statement of the form R(x, y) can be reduced to one of the form P(x), where P stands for 'being R to y'; that is, relational statements can always be 'reduced' to statements of subject-predicate form. Of course, as Cleland notes, ¹⁵⁹ there is something rather unsatisfactory about this way of simply 'parsing out' relations. For one thing, it amounts to nothing more than a restructuring of a statement so that it contains a predicate which looks as if it refers to a non-relational property but contains an ineliminable reference to a particular, unlike genuine non-relational properties which are exemplified by particulars, rather than having them as constituents.

One response would be to simply deny the connection between physical and logical reduction in this case-a response which acquires a certain plausibility in the context of the quantum logic programme. ¹⁶⁰ Teller himself drew on Stairs's realist quantum logic approach, which attempts to formalize the idea that,

. there can be facts about a pair of quantum systems which are, in a clearly specifiable sense, about the whole system but are neither reducible to nor implied by facts about the parts. ¹⁶¹

Quantum correlations are then explained by appealing to the 'unusual' relations which quantum systems bear to one another and the way these then relate to probability. ¹⁶² One can interpret this programme as an attempt to bring the logical situation back in line with the physical one.

Are there any other non-supervenient relations? The fact is, spatio-temporal relations aside, the relations which are of interest to physicists all supervene on the intrinsic properties of the particles. From the phenomenological perspective, Weyl suggested that this may have its origin ". within the domain of sense data, which-it is true-can yield but quality and not relation". ¹⁶³ This might explain why relations between 'classical' objects are not regarded as irreducible in Teller's sense, but it should not be taken as binding in the quantum realm.

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James, on the other hand, grounded his 'radical empiricism' on the 'statement of fact' that ". the relations between things, conjunctive as well as disjunctive, are just as much matters of direct particular experience, neither more nor less so, than the things themselves". ¹⁶⁴ However, calling a composite such as a quantum system in an entangled state an irreducible 'concatenated union', ¹⁶⁵ sheds little further light on the nature of this union.

More recently, Ladyman has suggested that this view of entangled relations might be naturally understood in a structuralist context. ¹⁶⁶ However, this seems clearly different from the case of Saunders's approach above, in that here the individuality of the particles is not understood as constituted by the relations but as given independently of them. Nevertheless, by broadening our metaphysical pantheon, the idea of non-supervenient relations might be seen as opening up ontological space for structuralist conceptions. ¹⁶⁷ Such conceptions have also drawn support from the contrast between metaphysical 'packages' we have been exploring here. In our final section we shall look at the nature of that contrast a little more closely.