Interpretation and identity in quantum theory

REVIEWARTICLE

Interpretation and Identity in Quantum Theory

Jeremy ButterJield* Bas C. van Fraassen, Quantum Mechanics: An Empiricist View (Oxford: Clarendon Press, 1991) xvi+541 pp., Hardback ISBN 0-19-824861-X, Paperback ISBN o-19-823980-7 E17.50. B. J. Hiley and F. David Peat (eds), Quantum Implications: Essays in Honour of David Bohm (London: Routledge, 1991), vii+455 pp., Paperback ISBN o-415-06960-2 E9.99 (Hardback published 1987). THESETWO books are both valuable contributions to the philosophy of quantum theory. They are very different. One is a monograph by a philosopher, from a largely quantum logical point of view. It advocates one interpretation of quantum theory (the ‘modal interpretation’), while also making an effort to expound the basic mathematics of quantum theory and quantum logic, so as to function as a textbook. The other is a Festschrift, largely by physicists, for David Bohm, famous for the ontological interpretation of quantum theory. Being a Festschrift, the book of course makes no claim to be a textbook. The two books are also complementary in that neither book refers to the main topic of the other! Agreed, there is no reason why authors in a Festschrift for Bohm should refer to quantum logic. But it is strange that van Fraassen does not once mention Bohm’s ontological interpretation, nor other authors in Bohm’s tradition, such as de Broglie or Vigier. Indeed, I shall argue below that van Fraassen’s discussion of quantum nonlocality is marred by this omission. I propose to concentrate on van Fraassen’s book, for several reasons. It is more philosophical; and being by a single author, it is more of a unity. Much of the Festschrift is not about the philosophy of quantum theory; indeed, about a third of the articles are not about quantum theory at all-they cover Bohm’s other interests, for example in biology. However, I firmly recommend some of the papers in the Festschrift, especially those on the ontological interpretation, by Vigier et al., and by Bell. (Other interesting authors contri-. *JesusCollege,University of Cambridge, Cambridge, CBS 8BL, U.K.

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Printed in Great Britain

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0039-3681/93

$6.00 + 0.00

Pergamon Press Ltd

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buting

on

quantum

theory

include:

Aharonov,

d’Espagnat,

Feynman,

Kilmister, Leggett and Penrose.) I shall proceed as follows. In Section 1, I shall discuss what it is to interpret a physical theory, in relation to both books. Thereafter I discuss only van Fraassen’s book. In Section 2, I give a general assessment; and from Section 3 onwards, I discuss his treatment of identical particles in quantum theory.

1. Interpretation seems to be a philosophical difference, between most of the Festschrift’s authors and van Fraassen, about the aim of an interpretation of a physical theory. As we shall see, the difference might be just an illusion; yet even the appearance will serve well as a springboard for discussion. For it will lead into the many-sided controversy between scientific realism and van Fraassen’s brand of empiricism. It seems that for Bohm himself, and for most of his commentators in the Festschrift, the interpretation of quantum theory is a stepping stone towards a better physical theory. Accordingly, the distinction between interpretation and theory is not sharp. But for van Fraassen, interpretation is an end in itself-and a distinctively philosophical, not scientific, end. In more detail: Bohm has long been dissatisfied with the cluster of orthodox interpretative ideas about quantum theory, usually called the ‘Copenhagen interpretation’. This dissatisfaction has motivated both his ontological interpretation (starting in his 1952 articles), and a continuing search for new theories. By and large, the authors in the Festschrift share this dissatisfaction; There

and accordingly, several articles report developments, either in the ontological interpretation, or in the search for new theories. Of course, this commitment to seek a better interpretation or a better theory need not involve any naive optimism that there is a unique best interpretation or best theory-let alone that any such is just over the next hill. And it need not involve any naive objectivism about the judgments of what is ‘better’: one surely must accept that there is almost always ample room for disagreement-the judgments are tentative. For example, advocates of the ontological interpretation typically claim as one of its merits that it is able to model in complete detail an individual quantum process. I agree that this is a great merit; but they can and surely must accept that it is not an a priori or logically necessary condition for an acceptable theory-other merits of another theory might outweigh it. Nevertheless, Bohm and the other authors act on these tentative judgments: they pursue single-mindedly the interpretative and theoretical ideas thereby favoured. There is every appearance of belief, albeit tentative, in those ideas, and of disbelief in the rejected alternatives.

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On the other hand, van Fraassen is famous for his constructive empiricism: the only belief involved in accepting a scientific theory is belief in its observational adequacy. (Accordingly, he might argue that the appearance of belief amongst the Festschrift’s authors is an illusion. Perhaps scientists present themselves as believing their theory, not because they really do, but for the sake of more persuasive advocacy-rather like defence lawyers presenting an alibi which they do not believe.) But for van Fraassen, the unobservable part of a theory is not to be dismissed or reconstrued as a faGon de parler, as in oldstyle instrumentalism: it is to be taken literally. And as he explains, this literal construal engenders a distinctive, and philosophical, activity of interpretation. In describing this, I shall point out that most of what van Fraassen says about interpretation can and should be accepted by scientific realists. van Fraassen first points out that scientific theories, as they come to us from science, never give an utterly precise picture of their own ontology, observable and unobservable (p. viii, p. 10). This is even so for physical theories which we typically consider as specific in topic and well-individuated, such as Newtonian mechanics-or even, say, Newtonian mechanics of point-particles under gravity. There is always an open-ended series of questions about the theory’s ontology, which the theory as it comes to us does not answer. For the Newtonian case, examples would be ‘In what sense is the theory deterministic, or causal?‘, ‘What happens at a collision?‘, and most famously ‘What is the status of space and time?‘. Answers to such questions may or may not add empirical content: van Fraassen calls these kinds of answers, respectively, ‘extensions’ and ‘interpretations’, and he sees it as the business of science to provide extensions, and of philosophy to provide interpretations. Of course, he admits that much work on such questions mingles empirical and non-empirical considerations-and rightly so, since a good answer should cohere with both kinds of consideration. But the distinction between science and philosophy, extension and interpretation, remains (p. 9, p. 209, p. 481). This may seem to consign philosophers to the role of tidying up after the heroic army of scientists has marched on. That would be even more humble a role than that famously envisaged by Locke, when he said he was content to be ‘an under-labourer . . . clearing ground a little, and removing some of the rubbish that lies in the way of knowledge’,’ since for Locke, philosophical work can at least be a useful preliminary in constructing scientific theories. But, as van Fraassen stresses, there is no reason for obsequiousness. For it is a myth that interpretation is easy work (albeit perhaps a perennial one among hard-nosed scientists!). The interpretative answers to questions like those

‘J. Locke, An Essay Concerning Human Understanding (London: 1972), p. xxxv.

Dent,

Everyman’s

Library,

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above cannot be read off the theory’s formalism, with the only prerequisites being a scientific training and clarity of mind (p. viii, p. 435). If they could be, there would surely be less controversy over rival interpretations than there in fact is. And van Fraassen emphasizes, in a liberal vein, how the development of these different rival interpretations, and the assessment of their relative merits, increases our understanding of the theory (p. 336, p. 450, p. 481). With all this, I and surely any scientific realist, can agree: and I admire the characteristic eloquence with which van Fraassen expresses it. But van Fraassen also believes something more contentious, which realists might well reject: that there is no uniquely correct or best interpretation, even when one allows for interpretations we have not formulated and perhaps cannot formulate (ibid.). As he puts it, an interpretation answers the question ‘How could the world possibly be how this theory says it is? That question does not by its form demand a unique answer; and scientific theories being complex, van Fraassen sees no reason to think that it ever has a unique answer. Nor does the goal of interpretation, namely better understanding of theories, require such uniqueness. To develop and assess interpretations-or more likely, given the complexities, sketches of interpretations: that is not only all we can expect to do, it is also all we should rationally aspire to do. Again, van Fraassen is eloquent in his statement of this pluralist and cognitively modest position. But I suspect many realists will resist it. It seems incompatible with the central realist idea of an intrinsic structure of the world which our theories aim to describe. And to many realists it will also seem too voluntarist about commitment to scientific theories.

2. Assessment

I turn to a general assessment

of van Fraassen’s

book. First I need to briefly

describe what is in it. It opens with a Chapter about interpretation of scientific theories in general, and then there follow four Parts. Part I is about determinism and probability in classical physics. Part II, ‘How the Phenomena demand Quantum Theory’, is about the violation of the Bell inequalities and (more briefly) about the basic ideas of quantum logic and quantum probability. Part III, ‘Mathematical Foundations’, expounds quantum theory in much more detail, again from a broadly quantum logical perspective. There is considerable emphasis on how the formalism describes composite systems, and thereby on interaction and the nature of measurement. But this exposition deliberately postpones controversial questions to Part IV, ‘Questions of

Interpretation

Interpretation’. standard

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and Identity in Quantum Theory

Here van Fraassen

interpretation,

first gives a critique

of what he calls the

i.e. in effect the use of the Projection

Postulate

to

explain the definiteness of pointer positions in the macroworld. The next Chapter develops his alternative, the modal interpretation. There is then another treatment of the violation of the Bell inequalities, which is independent of the modal interpretation. questions about identical

Finally, particles;

there are two Chapters on interpretative again, the discussion is almost entirely

independent of the modal interpretation. This list shows that the book has as wide a scope as one could want. van Fraassen also treats all these topics in formidable detail, while also aiming to provide a textbook in philosophy of quantum theory: Parts I and II are each about 60 pages, Part III is 100 pages, and Part IV is 250 pages. So there is a vast amount of material in the book, and no review of tolerable length can properly address even half of it. I propose a radical solution: I will properly address only the last two Chapters, on identical particles (from Section 3 onwards). I choose these Chapters partly because the literature on interpretative questions about identical particles is much smaller than that about nonlocality or measurement. That makes it worth advertising both the questions and van Fraassen’s worthy attempt to bring them into the textbook tradition. So in this Section, I will do just three small tasks. I will give examples of van Fraassen’s expository material, some of them chosen with a view to Section 3 onwards. Then I will briefly discuss some of his interpretative claims (before the last two Chapters). Finally, I will assess the book as a textbook. van Fraassen expounds a lot of technical material, often citing recent research articles. Much of it is important to later discussion, technical or interpretative. Three examples are: the idea of joint probabilities and its relation to commutativity of operators; the idea of superselection, and how it imposes restrictions on both physical quantities and states; the fact that a pure state of a composite quantum system in general determines as the states of its components (the so-called reduced states), only mixtures; (pp. 153-l 57, pp. 185-192, p. 201, respectively). These examples are doubly welcome since they are important, technically and interpretatively; and they are often mentioned only briefly in other textbooks. But there is also a good deal of more or less incidental technical material. Examples are: the elements of classical ergodic theory, and the Accardi inequalities (analogues for quantum theory of Bell’s inequalities for hidden variable theories: pp. 65-68, pp. 119-121, respectively). Turning to van Fraassen’s interpretative claims, pride of place must go to his modal interpretation (Chapter 9). It addresses what is generally accepted to be the main interpretative problem about quantum theory: measurement. To describe this problem, recall first that quantum theory is usually taken as saying:

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(i) a system’s state prescribes probabilities for the results of measuring a quantity (by the Born rule, and its generalization to mixtures); (ii) a system has a value for a quantity if and only if its state prescribes probability

1 for the corresponding

result.

(Many treatments consider only vector states, for which such a probability-l state is an eigenstate of the operator representing the quantity, and the value is the eigenvalue; so (ii) is often called the ‘eigenvalue-eigenstate link’.) These points, (i) and (ii), have of course given rise to a welter of interpretative ideas. Three examples, of increasing radicalism, are: that quantum theory is indeterministic and involves irreducible ontic probabilities; that measurement must be taken as a primitive notion in the theory; that measurement implies cognition by a mind, so that quantum theory involves a kind of subjectivity. Although van Fraassen is a liberal about interpretations (cf. Section l), his own interpretation rejects the second and third ideas. And in this rejection, I and most other workers in the field, happily join him. Indeed, I think that most workers would happily accept (i) and (ii), if there were a quantum-theoretical analysis of measurement that implied (i), so that measurement need not be a primitive notion; and that reconciled (ii)‘s widespread denial of values with the apparent existence of values for the macroscopic objects successfully treated by classical physics. But quantum theory seems to forbid such an analysis. For a quantum theoretical analysis of the interaction of, say, an electron that is indefinite (lacks a value) for momentum, with an apparatus for measuring momentum, suggests that the electron’s indefiniteness will be transmitted to the apparatus so that its pointer is in no definite position! This is the notorious ‘measurement problem’. The natural response to this problem is to deny the ‘only if’ of (ii), i.e. to try and supplement the ascriptions of values made by quantum theory. This approach faces a series of obstacles, the so-called ‘no hidden variable’ theorems. The flavour of these is that certain kinds of supplementation, though simple and natural from a formal point of view, cannot work. Some kinds lead to outright contradiction (Kochen and Specker et al.). For some others, the problem is that in order to be compatible with quantum theory’s predictions for coupled systems such as occur in the famous EPR thought-experiment, they need to be nonlocal in some precise sense (Bell). However, these theorems do not block all kinds of supplementation. So the question is: is there some natural supplementation that avoids both the measurement problem and the no hidden variable theorems? van Fraassen says Yes: the modal interpretation provides it. His basic idea is to attribute values to any system as if the so-called ignorance interpretation of mixtures were true, without in fact endorsing that interpretation. More exactly:

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when a system is in a mixture, there is some component of the mixture (vector in the density operator’s image space) such that the system has exactly those values that the eigenvalue-eigenstate link would attribute for that vector as pure state (p. 281). van Fraassen then applies this idea to an account of measurement given earlier in the book. In this account, measurement is not a primitive notion, nor is it defined anthropomorphically or in terms of macroscopic size. Rather it is a matter of a certain kind of interaction between two systems, which itself selects the quantity measured on one system and the ‘pointer’ quantity on the other (pp. 211-216). This account covers the usual eigenstate-correlating models of measurement (pp. 218-224). So van Fraassen applies his idea about valueattributions to these usual models. And by considering value-attributions, both to the composite system of object plus apparatus and its two components (with reduced states that are in genera1 mixtures), he is able to render the Born rule for quantum probabilities, not with the traditional suspicious invocation of a primitive idea of measurement, but in terms of values on systems (pp. 287-288). This is an impressive achievement. But I should report two lacunae: one of exposition, and one of defence. First, van Fraassen does not expound the relations between his interpretation and some close cousins. I have in mind particularly the interpretations of Krips, Dieks, Healey and Kochen.* An exposition of these relations would have helped anyone familiar with those interpretations to understand and assess van Fraassen’s own proposal. Agreed, van Fraassen briefly mentions the leading idea of the latter three (pp. 326-327 and fn. 19). But it would have been helpful to have more; and to consider also what seems the closest cousin, namely Krips’s interpretation. (The main difference seems to be that Krips strengthens the idea for value-attributions above. He says that for any expression of the mixture as a weighted sum of projectors onto eigenstates of a quantity Q, the system has one of the corresponding eigenvalues as its value for Q, and with the corresponding weight as its probability. He recognizes that this strengthening threatens a contradiction, based on ‘no Specker’s: and he proposes to move for which van Fraassen Second, van Fraassen does criticism of his interpretation

hidden variable’ theorems like Kochen and avoid the contradiction by ‘de-Ockhamizing’-a has no need (p. 325).) not discuss what at least seems to be a good (together with those of Dieks, Healey and

‘H. Krips, The Metaphysics of Quanfum Theory (Oxford: Clarendon Press, 1987); D. Dieks, ‘Resolution of the Measurement Problem through Decoherence of the Quantum State’, Physics Lerters A 142 (1989), pp. 43946; R. Healey, The Philosophy of Quantum Mechanics (Cambridge: Cambridge University Press, 1989); S. Kochen, ‘A New Interpretation of Quantum Mechanics’, in P. Lahti and P. Mittelstaedt (eds), Symposium on the Foundations of Modern Physics (Singapore: World Scientific, 1985), pp. 151-170. For my comment on Krips, see his pp. 34, 101-103.

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Kochen) by Albert and Loewer.3 The idea of the criticism is that real-life measurements are not perfectly accurate (in a precise sense); and this prevents these

interpretations’

proposals

for value-attributions

avowed aim, namely definite ‘pointer’ readings. how van Fraassen avoids this criticism.

from

achieving

their

I for one do not see clearly

However, these lacunae are small defects: van Fraassen gives a very detailed development of his modal interpretation. And he has a wealth of other interesting interpretative claims, spread through the rest of the book. Here are three examples: scepticism about the principle of indifference in the philosophy of probability (pp. 57-61); quantum logic as not relinquishing classical logic (pp. 134145); scepticism about solving the measurement problem by superselection rules due to the macroscopic character of the apparatus (pp. 264-271). Clearly, I must again limit myself: I shall briefly report two main claims he makes about nonlocality (in his Chapters 4, 10 and the first half of Chapter 5). I agree with the first, but not the second. van Fraassen expounds in detail the assumptions of local hidden variable models of the usual two-particle Bell experiments. He considers both deterministic and factorizable stochastic models; and he shows in detail how they are committed to Bell inequalities that are violated by experiments, thereby showing some kind of nonlocality. As to how to interpret this nonlocality, he claims (and I agree) that the nonlocality in no way threatens classical logic, or classical probability theory. This is important since this threat has been alleged, on the grounds that Bell inequalities can be derived purely as theorems of classical probability theory, so that their violation threatens classical logic or probability, rather than locality (or some related physical concept such as causality or separability). van Fraassen shows how the threat rests on an equivocation: the Bell inequalities violated by experiment are not the formally similar theorems of pure classical probability theory (pp. 102-105, pp. 107-l 10). van Fraassen also calls the local hidden variable models (both factorizable stochastic, and deterministic ones) ‘causal models’: the idea is of course that the hidden variable is to be the common cause of the results in the two wings of the experiment. Accordingly, he glosses the experiments as showing ‘phenomena that fit into no common cause pattern’ (p. 106; cf. also p. 95, p. 98, p. 112). Given his use of ‘cause’ and cognate words, that is quite right. But using familiar words as technical terms can be misleading. And the danger is all the greater, when the word is widely regarded as philosophically important. Indeed, my impression is that the question which analytic philosophers, not

3D. Albert and B. Loewer, ‘Wanted Dead or Alive: Two Attempts to Solve Schrodinger’s Paradox’, in A. Fine, M. Forbes and L. Wessels (eds), Proceedings of the 1990 Meeting of the Philosophy of Science Association, volume 1, pp. 277-285, see pp. 281-282.

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specializing in philosophy of physics, press most urgently about nonlocality, is the question ‘What does it show us about the philosophical analysis of causation?‘. So let me stress that there are at least two analyses of the Bell experiments that satisfactorily recover the actual (quantum, Bell inequality violating) statistics, and surely deserve to be called ‘causal’.4 First, one can give up a ‘screening-off’ requirement which van Fraassen takes as part of the definition of causal model (p. 83, p. 89). The requirement is roughly that a common cause must render its effects stochastically independent of one another. More precisely, the requirement is: the results in the two wings are stochastically independent, once we conditionalize on a specification of the hidden variable and the settings of the two apparatuses. This requirement, often called ‘completeness’ or ‘outcome independence’ (following Jarrett and Shimony), is crucial to the derivation of the Bell inequalities. But some argue that there is no reason why common causes should obey it. And if we give it up, we can write down common cause models (in the new liberal sense) of the Bell experiments. And although these models make the nearby result depend on the distant result in the above sense, they are local in another good sense: results do not depend on the setting of the distant apparatus. This requirement, often called ‘locality’ or ‘parameter independence’ (again following Jarrett and Shimony), is also crucial for Bell inequalities, and is again part of van Fraassen’s definition of causal model (p. 89). Indeed, in some such models the quantum-state of the particle-pair is itself the common cause of the two results.5 Second, one can give up locality! That is, one can give up the requirement that results do not depend on the setting of the distant apparatus. van Fraassen does not consider this option (nor do many philosophical discussions of nonlocality). Presumably, he would reject it out of hand; for he discusses, with evident approval, the so-called no-signalling theorem of quantum theory, that the nearby statistics do not depend on the setting of the distant apparatus (pp. 364-368). But this option is viable: it is taken by Bohm’s ontological interpretation of quantum theory. Indeed, I think van Fraassen makes a serious omission here, especially in a book that covers so much and purports to function as a textbook. Not only in discussing nonlocality, but also throughout the book, van Fraassen never mentions this interpretation, nor Bohm and associated 4As is well-known, there is a larger issue here. van Fraassen’s other writings show him to be sceptical of the notion of cause, as he is of several other notions, like law of nature, which are stock-in-trade for ‘twentieth-century medieval metaphysics’ (pp. 452, 481). But I cannot address the larger issue: I just want to prevent his scepticism getting spurious support from his choice of terms for the Bell experiments. 5For the models, cf. N. Cartwright, Nature’s Capacities and their Measurement (Oxford: Clarendon Press, 1989), see pp. 236242. For simplicity, my discussion of these two requirements sets aside apparatus hidden variables.

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authors such as Vigier. Since van Fraassen is tolerant of different interpretations (cf. Section 1), and this interpretation is often relevant to his claims (e.g. p. 43, p. 112, p. 128, pp. 357-374), I am mystified by this omission. To make amends, I shall briefly advertise the interpretation (and so the Bohm Festschrift), in application to the Bell experiments. As Bell himself emphasized in 1966, this interpretation provides a mechanism whereby the distant setting can influence the nearby result: a mechanism that has since been worked out in explicit detail for the Bell experiments involving spin-measurements on a particle pair in a singlet state. The basic ideas of the interpretation are that particles have precise positions that evolve continuously and deterministically, under the influence of a physically real wave-function that itself evolves deterministically by the Schrijdinger equation. This wave-function also provides a probability distribution over particles’ positions; so that quantum theory’s probabilities, like those of classical statistical mechanics, arise from an underlying deterministic process. Given these ideas, the mechanism for the influence in the Bell experiments is not at all mysterious. The setting of the distant apparatus is in effect the orientation of a Stern-Gerlach magnet. This can influence the nearby result by first influencing the pair’s wave-function (contiguously); the wave-function then influences the position of the nearby particle, and so the nearby result. Agreed, the details of this mechanism make it very different from, say, Newtonian gravitational interaction between the particles. But it is no more mysterious than Newtonian gravity. Besides, this mechanism does not conflict with the no-signalling theorem. The ontological interpretation can accept that for any quantum state, the distant setting does not influence the nearby statistics, although it influences each individual result, because quantum states represent averages over precise particle positions, in such a way as to wash out the individual correlations. To sum up, here is a nonlocal deterministic (and surely causal) model of the Bell experiment.6 So much for van Fraassen’s interpretative claims. Finally, does his book succeed as a textbook in philosophy of quantum theory? Yes, for students who already know some quantum theory (especially the vector space formalism, and elements of mixtures and tensor products). Such students will appreciate the book’s considerable merits: large scope, a wealth of detail, and van

%Zf. J. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge: Cambridge University Press, 1987), see p. 11; for an introduction to the interpretation cf. the articles by Bohm himself and Vigier et al. in the Festschrift; for application to the Bell experiments, cf. C. Dewdney et al., ‘Spin and Non-Locality in Quantum Mechanics, Nature 336 (1988), 536544; for reconciliation with the no-signalling theorem, cf. P. Holland and J. Vigier, ‘The Quantum Potential and Signalling in the EPR Experiment’, Foundations of Physics 18 (1988), 741-750.

Interpretation

Fraassen’s

and Identity

remarkable

sophical argument, But the answer

in Quantum

ability

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to weave precise

technical

material

argument which is typically witty and eloquent. is No, for students without such knowledge.

into philoFor

these

students, these same merits become defects. The wealth of detail (especially the technical material that is not used later) prevents them seeing the wood for the trees. And the weaving of technicalities with philosophy makes it hard to learn the technicalities: that of course needs a leisurely, rigorous textbook (or more likely, several!). van Fraassen aims to serve such students. He has purely expository sections; and he frequently puts a more rigorous presentation, or additional technical material, in ‘Proofs and Illustrations’ subsections. But still, I think it does not work. There are more technical slips and infelicities than a textbook should have. And although tastes may differ, I know that several newcomers, even very able ones, have found the device of ‘Proofs and Illustrations’ frustrating. Here is one example of slips and infelicities: pp. 141142 talk about representing a quantity as a map from vectors in a basis to possible values of the quantity. We are told that the basis is orthogonal only after a calculation using the fact. Then the next Section (pp. 147-151) tells us that the quantity is represented as a Hermitian operator, so that the map first introduced is in effect derivative and less central to the formalism. This Section describes eigenvalues, projections, the Born rule, and spectral decomposition; but does not tell us that a Hermitian operator has orthogonal eigenspaces, and real eigenvalues!

3. Three Issues about Identical Particles I turn to van Fraassen’s discussion, in his last two Chapters, of so-called puzzles to which they give rise. ‘identical particles’, and the philosophical These Chapters show the same merits and defects mentioned above: a wealth of material, and lightness of touch in presenting it. But it is hard to learn the technicalities from van Fraassen. And as will emerge below, the readable presentation can be deceptive: I often found it hard to know exactly what interpretative claim he was making. As van Fraassen points out, identical particles give us first of all a problem of terminology. Physicists often call two particles that match in their permanent intrinsic properties like mass and charge (such as two electrons, or two photons) ‘identical’. That is likely to strike philosophers as a misnomer. Surely, each electron etc. is identical only with itself. Fair comment: there are words that better connote the main idea of matching intrinsic properties, e.g. ‘dupli(The latter is also likely to seem a misnomer: one cate’, ‘indistinguishable’. tends to think that two electrons can be distinguished, as classical particles might be, by their different temporary properties. But as we shall see in Section

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5, this is not so: two electrons cannot able’ is not a misnomer.) However, ‘identical’.

be thus distinguished. So ‘indistinguishI shall follow van Fraassen in saying

Words apart, substantive issues remain. van Fraassen discusses three main ones. First, there is the puzzle of quantum statistics. Intuitively, it seems that particles should obey Maxwell-Boltzmann statistics. These are like the elementary probabilities we use for tosses of a fair coin; e.g. for two tosses, there are four equally probable cases, (heads, heads) through to (tails, tails). For us, the main point is not equiprobability, but the distinction between the outcomes (heads, tails) and (tails, heads)-a distinction that seems forced on us by the idea of two distinct tosses. But notoriously, quantum particles do not obey Maxwell-Boltzmann statistics. Instead they obey either Bose-Einstein or Fermi-Dirac statistics, both of which seem to conflate this distinction. That is, in terms of the coin example: both these statistics seem to identify the two outcomes (heads, tails) and (tails, heads). (They differ in that Bose-Einstein assigns 3 to this ‘one head’ outcome and 4 to each of (heads, heads) and (tails, tails); while Fermi-Dirac assigns 1 to the ‘one head’ outcome, 0 to the other two.) And so the puzzle: how can quantum particles obey these new statistics? Are they in some strange way not individuals? Second, there is a threat to Leibniz’s Principle of the Identity of Indiscernibles. This arises from the fact that a pair of identical quantum particles can be in the same quantum state throughout their existence, in a way that classical particles apparently cannot be. (Assume for instance that a classical particle’s position in space is part of its state, and that classical particles are impenetrable, so that no two can be in the same place.) That suggests that quantum particles refute the Identity of Indiscernibles. The third issue concerns quantum field theory. It is sometimes suggested that puzzles about the identity of quantum particles, such as the two issues above, are resolved by the use of quantum field theory, in particular by the idea that particles are not really individuals, but quanta of a field. That suggestion needs to be evaluated. I shall take up these three issues in order in the subsequent Sections. But beware: this order does not correspond to van Fraassen’s. His discussion of the first and second issues is spread over Chapter 11, and the second half of Chapter 12 (pp. 451f.); the third issue is in the first half of Chapter 12 (pp. 434451). He also intermingles exposition of the physics with his interpretative claims. In broad summary, the main claims are as follows (again: with Section l’s liberal allowance that other interpretations are viable). Quantum particles are individuals, despite the puzzling statistics; they violate the Identity of Indiscernibles; and field theory does not really abandon the idea that particles are individuals. Again very broadly, I shall agree with the first two claims, but not the third.

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To state these claims more precisely, and to assess them, it will be convenient to spend the rest of this Section summarizing the expositions he uses for the first two issues. Chapter 11 begins with the fundamental idea of permutations of particle labels represented as unitary operators P on the tensor-product Hilbert space for the system of particles. van Fraassen then states what he calls Permutation Invariance (PI: also often called the Indistinguishability Postulate): (PI) If 4 is the state of a composite system whose components are identical particles, then the expectation value of any observable A must be the same for 4 and for any permutation PI#Jof 4: (4, Ab) = (P& AP4). (PI) has a clear motivation: the physics of the composite system, as summarized in expectation values, cannot be sensitive to differences between 4 and P+i.e. to component particles’ underlying identity (in philosophers’ sense of ‘identity’!). There are two well-known ways to satisfy (PI). One way is for each state 4 to be symmetric: Pt$ = 4, for all P (a particle whose composites take only such states is a boson). Another way is for each 4 to be antisymmetric: P&J=~ if P is an even permutation, P&J= -4 if P is odd (fermions). Any such state 4, symmetric or antisymmetric, of course defines reduced states for its components (by partial tracing). And for any such 4, the reduced states are the same for all components. Given 4’s (anti)symmetry, that is mathematically unsurprising: but it of course threatens the Identity of Indiscernibles (cf. Section 5). van Fraassen then expounds three pieces of material relating to the puzzling statistics. (1) He takes pains to argue that (PI) does not entail what he calls Dichotomy: that 4 is either symmetric or antisymmetric. He shows how ‘proofs’ of this entailment (by Blokhintsev and others) always use some extra assumption (pp. 389-402, p. 447). (2) He expounds the elements of quantum statistical mechanics. He shows how the characteristic deviation of Bose-Einstein or Fermi-Dirac statistics from Maxwell-Boltzmann statistics arise by applying the quantum statistical algorithm to ignorance mixtures of symmetric or antisymmetric states; (pp. 403-409). (3) He expounds the connections between some of Carnap’s and de Finetti’s ideas, which are well-known to philosophers of probability, and quantum statistics. Thus recall Carnap’s notions: a Q-predicate is a logically strongest consistent complex predicate, formed by conjoining negated and unnegated atomic predicates; a state-description is a logically consistent conjunction of Q-predications, one for each individual constant; a structure-description is a disjunction of state-descriptions, the disjuncts being all the state-descriptions obtainable from a given one by a permutation of individual constants. Then

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Carnap’s favoured c* confirmation function is defined as giving equal probability to all structure descriptions; it corresponds to Bose-Einstein statistics. Carnap’s c+ is defined as giving equal probability to all state descriptions; it corresponds to Maxwell-Boltzmann statistics. Conditionalizing c* or c+-it does not matter which-on the proposition that no two objects fall in the same cell (are indiscernible) gives a function, cF say, that corresponds to Fermi-Dirac statistics (pp. 410-419, pp. 477480). Turning to evaluation, I applaud most of van Fraassen’s expositions of these items. In particular, I found some of his occasional cautionary remarks helpful and astute. For example, in (3) van Fraassen emphasizes that c* only corresponds with Bose-Einstein statistics when the latter uses a uniform statistical distribution over pure states (p. 417). But I found van Fraassen’s (1) unclear. I of course agree with the main point, that (PI) does not entail Dichotomy. But the real reason for this nonentailment is that the group of permutation operators has irreducible representations of dimension greater than 1. That is, the Hilbert space has subspaces of dimension greater than 1 that are invariant as a whole under all permutations; and states could be represented by such subspaces (so-called generalized rays). Admittedly, van Fraassen describes the idea of such representations. But it comes too late-namely, after the details of the critique of Blokhintsev et al., apparently en passant during a presentation of character operators (p. 399). This delay is strange and frustrating, because at the start of the Chapter van Fraassen already has the wherewithal to make this idea crystal-clear-namely from his previous presentation of superselection (pp. 185-192).’

4. The Puzzling Statistics I turn to van Fraassen’s use of these expositions interpretative issues: the puzzle about individuals

for the first of Section obeying Bose-Einstein

3’s or

Fermi-Dirac statistics. For this issue (and also in Section 5), I will confine my discussion to the quantum theory of a fixed, though arbitrary, number of particles: I postpone variable particle number (Fock space) to Section 6. As I see it, van Fraassen makes three main claims about this issue. The first two are clear enough, and do not require details about the puzzle. They are introduced at the start of Chapter 11, in terms of two tempting but false conjectures. Thus he says he will argue that the following are both false, though each has a core of truth:

‘It is a humbling thought that of this non-entailment; cf. P. University Press, 1930), Section Oxford University Press, 1938),

sixty years ago the founding fathers seem to have been well aware Dirac, The Principles of Quantum Mechanics (Oxford: Oxford 54; R. Tolman, The Principles of Sfnrisfical Mechanics (Oxford: p. 318.

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(i) (PI) accounts for the non-classical statistics of composites of quantum particles; (ii) when ‘identical’ is properly understood, it entails (PI) tautologically. van Fraassen does not tell us explicitly later on wherein lies the falsity of these conjectures, nor what are their cores of truth. But presumably, the falsity of (i) lies in the non-entailment expounded in (1) above (Section 3). That is, (i) is false in so far as ‘accounts for the non-classical’ means ‘implies Bose-Einstein or Fermi-Dirac’. And presumably, the core of truth in (i) is the claim that Dichotomy accounts for the non-classical statistics: this claim being justified by quantum statistical mechanics, as in (2) above-even if, by (3) above, there are also (partial) classical accounts of these statistics. But I do not see why (ii) is false. To me, it seems merely to summarize the motivation for (PI) which I endorsed in Section 3: that expectation values for the composite system cannot be sensitive to the differences between 4 and P~#J. van Fraassen’s third claim is mistier. In effect, he claims there is no good reason to think that individuals should obey Maxwell-Boltzmann statistics. Stated like that, the claim is perfectly clear. But his reasons for it are not. For as we shall see in the next Section, it emerges in Chapter 12 that van Fraassen rejects some traditional connotations of ‘individual’: though exactly which connotations is again not crystal-clear. In this Section, I will just consider the point van Fraassen makes specifically against the intuition that individuals should obey Maxwell-Boltzmann statistics. I will argue that the point is correct, but insufficient to solve the main puzzle-to explain how individuals can obey Bose-Einstein or Fermi-Dirac statistics. Yet I think van Fraassen himself supplies the material needed to fill the gap. van Fraassen’s point is based on his rejection of the Principle of Indifference, discussed much earlier in the book (pp. 57-61). Thus in the example of two tosses of a coin, he says that he sees no good reason to take the four outcomes, (heads, heads) to (tails, tails), as equiprobable (p. 378). I endorse this point: I have no brief to defend the Principle of Indifference. But this point does not solve the main puzzle. For as I said in Section 3, the aspect of Maxwell-Boltzmann statistics that matters for us is not equiprobability, but the distinction between the outcomes (heads, tails) and (tails, heads). Of course the distinction is not confined to tosses, but is quite general: if two individuals, a and b, each have one of the two properties, H and T, we must surely distinguish a having H and b having T, from vice versa. So the main puzzle remains: how can quantum particles obey statistics that conflate this distinction, as Bose-Einstein and Fermi-Dirac statistics seem to do? I think the solution is that they only seem to: these statistics do not really conflate the distinction. So one can maintain the ‘conservative’ interpretation that quantum particles are individuals. To explain this, I need first to clarify

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the distinction; and then to consider how one derives these statistics, and so the appearance of conflation, from applying the quantum statistical algorithm to symmetric or antisymmetric states. (Agreed, this conservative interpretation will have questionable features: I only claim, as I think van Fraassen would, that it is tenable.) So first, a philosophical point about the distinction. 1 agree that the distinction is dubious if either or both of the properties, H and T, is logically strong; for example a conjunction of all the individual’s properties, or an essential property. Thus if a# 6, and H is the conjunction of all a’s properties (perhaps including being identical to a), then we may well doubt that Hb is possible; so that the distinction fails. And if His, or includes as a conjunct, an essential property of a, and T is not such a property nor includes one, then surely Tu is impossible, and again the distinction fails. But in general, Hand T are not thus strong, and the distinction is surely mandatory. (These logically strong exceptions will come up again, in Section 5.) And when we derive the quantum statistics, by applying the quantum statistical algorithm to symmetric or antisymmetric states, it is just such typical, not logically strong, properties H and T which we are concerned with. There are two aspects of such a derivation to be considered: the observables for which one is deriving the statistics, and the states used. Again we can consider the simple ‘two toss’ case discussed above, where Bose-Einstein statistics assign 3 to each of three outcomes, and Fermi-Dirac statistics assign 1 to ‘one heads’. So first: it follows from (PI) that any observable A must be symmetric (invariant under permutation). Since in the two toss case, the operator representing ‘heads first, then tails’ is not symmetric, it is not an observable; similarly for ‘tails first, then heads’. But the sum of these two operators is symmetric, and of course represents ‘one heads’. And it is this sum observable, representing a property of the composite system, which gets the notorious probabilities: + from Bose-Einstein statistics and 1 from Fermi-Dirac statistics. The use of such a sum observable certainly does not conflate the distinction between Ha& Tb and Ta & Hb-no more than elementary probability theory does when it defines the event of ‘one head’ as the union of (heads, tails) with (tails, heads)! Furthermore, both the sum observable on the composite system, and the component observables, H and T, from which it is built, represent typical, not logically strong, properties of their respective systems. They are not conjunctions of all their system’s properties; nor is there any reason for them to be essential-they can be as transient as you please. So they are just the kind of properties for which our distinction seems mandatory. Second: nor does the use of symmetric or antisymmetric states conflate the distinction. Agreed, such states (for our two toss case) look formally like

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‘Ha& Tb+ Ta& Hb’, where the ‘ + ’ is vector addition, i.e. superposition. And so these states lead into the usual interpretative issues about value attributions, about the ‘ + ’ being neither ‘and’ nor ‘or’ etc. (Beware: since the operators ‘heads first, then tails’ and ‘tails first, then heads’ are not observables, being not symmetric, we cannot think of such a state as involving a 50% probability to yield measured value 1 on measurement of each of these two operators.) But these usual interpretative issues do not show that these superposed states conflate the distinction, any more than a conjunction conflates its conjuncts, or a disjunction its disjuncts. The same point holds, if we consider the properties assigned to components by the reduced states that are determined by symmetric or antisymmetric states. As mentioned in Section 3, those reduced states are always the same for all the components. So however we think of the properties assigned by these reduced states, all the components have the same such properties; so the distinction is certainly not conflated as regards these properties. So much by way of justifying my claim that the quantum statistics respect the distinction8 and so allow us to maintain the interpretation that quantum particles are individuals. Agreed, this interpretation is questionable, especially by people sympathetic with verificationism. They might: (i) infer from the impossibility of observing a non-symmetric operator that the corresponding quantity does not exist, so that the distinction fails; (ii) infer from the prohibition of non-symmetric states, IHa, Tb) and ITa, Hb), that the distinction fails. I deny these inferences; although, as mentioned above, I claim only that this interpretation is tenable, not that it is mandatory. Curiously, van Fraassen himself supplies the raw materials of this justification, for he presents this ‘two toss’ calculation (in his expository (2), p. 407). But he does not gloss it in the above way: his only gloss on it is a brief echo of his earlier rejection of the Principle of Indifference (p. 408). This lacuna left me wondering just what is van Fraassen’s solution to the puzzle: this prompts us to look at our second main issue.

5. The Identity of Indiscernibles van Fraassen’s discussion of the Identity of Indiscernibles has two aspects. The first aspect, treated in Chapter 11, concerns quantum theory: it also relates *The detailed derivation of these statistics adds only one ingredient to the above: the Bose-Einstein equiprobabilities, j for each, follow directly from the use of a uniform mixture of three symmetric states, and the Fermi-Dirac assignment of 1 follows directly from the use of the one antisymmetric state. French and Redhead argue that this extra ingredient can be explained dynamically: the fact that time-evolution preserves an initial symmetrization of the state means that certain states are and remain inaccessible. Cf. S. French and M. Redhead, ‘Quantum Physics and the Identity of Indiscemibles’, British Journalfor the Philosophy of Science 39 (1988), 233-246, see p. 237; S. French, ‘Identity and Individuality in Classical and Quantum Physics’, Australasian Journal of Philosophy 67 (1989), 43246, see pp. 443445.

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to the exclusion

principle

and

van

Fraassen’s

modal

interpretation.

(My

restriction from Section 4, to a fixed number of particles, remains in force till Section 6). The second aspect, treated in Chapter 12, is independent of quantum theory: van Fraassen clarifies his notion of an individual by expounding

how permutations

apply to possible

world semantics.

For the first aspect, material from Sections 3 and 4 show how the Identity of Indiscernibles is threatened. Thus we saw in Section 3 that both a symmetric and an antisymmetric state of a composite system determine the same reduced states for each of the components. And Section 4 supported the view that the components are distinct individuals. But the Principle of the Identity of Indiscernibles is a universally quantified conditional (with philosophers’ usage for ‘Identity’!), namely: for any individuals x and y, if they share all their properties, then x=y. So we conclude: if a component’s reduced state determines all its properties, then the Principle is false. How big is this ‘if’? Here, we meet a large issue, namely physicalism, which I propose to set aside. Thus there is obviously a gap between what a reduced state immediately specifies, i.e. probability distributions for physical quantities such as energy or momentum, and the general idea of a property such as being massive or red or human. The issue whether this gap, once it is made precise, can be bridged is in effect the issue whether some precise version of physicalism, i.e. reduction or supervenience of all empirical facts on microphysical facts, holds true. I myself believe that some such version holds true; but this is not the place to address the issue. Fortunately, there remain two more tractable issues which I can address. Indeed, what I say about these issues will be independent of whether physicalism is true. (In effect, the following discussion is simply confined to whichever properties do in fact supervene on microphysical states.) The first issue is about the contrast

between

intrinsic

and

extrinsic

properties;

the second

issue is

about the exclusion principle. van Fraassen discusses only the second, but I will say much more about the first. The philosophical analysis of the extrinsic-intrinsic distinction amongst properties is disputed. But the basic idea is that what makes the attribution of an extrinsic property true or false is not confined to the spatiotemporal region of the individual. Accordingly, two individuals could match each other in all their intrinsic properties (in the terminologies suggested in Section 3: can be indistinguishable, can be duplicates), and yet differ in an extrinsic property. This would happen when one individual’s environment, but not the other’s, is ‘right’ for the property. For example, being owned is an extrinsic property of, say, coins. I own this penny; but a duplicate of this penny, if there is one, need not be owned by me nor anyone else. Other examples are legion: being married is an extrinsic property of people, being part of a car is an extrinsic property of wheels etc.

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How can the philosophical distinction between intrinsic and extrinsic properties be applied to quantum systems, especially to our question about how big is the ‘if’ in ‘if a component’s reduced state determines all its properties’? The example of the duplicate pennies makes one expect that two (so-called) identical particles, say two electrons, that are components of a composite system and have the same reduced state, could differ in their extrinsic properties. If this expectation were right, then components having the same reduced state would only falsify the Principle of the Identity of Indiscernibles, in one of its stronger versions: namely, with the range of the antecedent’s quantifier ‘all properties’ excluding such extrinsic properties that can differ between components. In fact, the expectation is wrong. That is: on the most natural understanding of ‘intrinsic’ and ‘extrinsic’ within quantum theory, two identical particles that are components of a composite system (which may include other types of particle) match on all their extrinsic as well as intrinsic properties. This is true for any two such particles (fermions or bosons), any total number of particles comprising the composite, and any state of the composite. So the Principle of the Identity of Indiscernibles is falsified even in very weak forms. Since this point seems neglected in the literature, I shall expound it fully. I shall first discuss how to understand ‘intrinsic’ and ‘extrinsic’ within quantum theory, and then give an elementary result that makes the point.’ The obvious suggestion about how to understand ‘intrinsic’ is: (i) if a system is in a pure state 4, then 4 encodes only intrinsic properties of the system. Agreed, (i) can be questioned, because of the measurement problem. For if the Born-rule probabilities prescribed by C#Jare ultimately to be interpreted in terms of measurement results, then this reference to an apparatus surely makes C$encode extrinsic properties. However, I shall accept (i), for two reasons. First, I hope that the antecedent above is false, i.e. that we can ultimately avoid such reference to an apparatus. Second, even if we cannot do so, it will be clear in what follows that this reference to an apparatus is a pervasive and uniform kind of extrinsicness that leaves intact my own claims about extrinsicnesswhich will concern reference to other components of a composite system. Once we accept (i), the next question is what we should say about components of a composite system. Here again, we first need to set aside an issue:

%o far as I know, the only article making essentially this point is French and Redhead, op. cit., note 8; see pp. 240-242. My result slightly generalizes theirs, in content and method of proof: namely, they do not consider my case (3) below, and their proof does not use my (6) and (7).

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462

namely,

improper

mixtures,

which concern

in History

and Philosophy

the logically

of Science

weak extrinsic

pro-

perty of being a part of a composite. I shall be concerned instead with logically stronger extrinsic properties, namely correlations between components. Thus general

recall that if the composite is in a pure state, a component will in have as its reduced state a mixture. And notoriously, this mixture

cannot be given the ignorance interpretation (on pain of ascribing a mixture to the composite). Following d’Espagnat’s terminology, I call such a mixture ‘improper’.‘O If we ask whether such an improper mixture encodes intrinsic properties, we are pulled in two directions. On the one hand, we want to say Yes, since the mixture as a mathematical object (density matrix) prescribes probability distributions for quantities on the component, just as 4 did in (i). So once we accept (i), we want to say Yes. On the other hand, we want to say No, since the very fact that the mixture is improper implies something about the environment, namely that the component is a part of a composite. Obviously, we must divide and rule: we need to distinguish a weaker and a stronger concept of ‘reduced state’. The weaker concept is in effect the reduced density matrix: for it, we can say, in the spirit of (i), that it encodes only intrinsic properties. The stronger concept is, in effect, the reduced density matrix, together with the fact of improperness. This stronger state also encodes the extrinsic property of being a part of a composite. But the state of a composite quantum system of course encodes correlations between any two of its components. The composite’s state prescribes conditional probabilities for a given quantity on one component to take a given value, conditional on a given quantity on the other component taking a given value. One naturally thinks of such a conditional probability as a relation between the components. But one can also think of it as an intrinsic property of the composite, as (i) suggests. And however one thinks of it, it induces corresponding extrinsic properties of its relata: e.g. the property of having 50% probability for energy to take value 17, conditional on the angular momentum of the other component taking value 0. And these are the extrinsic properties

that match

between

two components,

in the promised

result.

One last philosophical comment, before giving the result. This situation-extrinsic properties of components induced by an intrinsic property of a composite-is familiar in metaphysics. In general, a composite of individuals has intrinsic properties that are not intrinsic to any component individual. One obvious example is the composite’s mass. Besides, a composite has intrinsic properties that are not determined by (supervenient upon) all its components’ various intrinsic properties. Following Lewis, I call these

‘OB. d’Espagnat, The Conceptual Foundations of Quantum Mechanics, 2nd edition Mass: Benjamin, 1976), pp. 58-61.

(Reading,

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‘external’.” Mass is not external, in Newtonian physics, for it does supervene on the components’ various intrinsic properties-in fact it is just the sum of their masses. But the spatial relations, of distance, angle etc. (or their spatiotemporal analogues) between the components are external. These clearly do not supervene on the components’ intrinsic properties. But they are surely intrinsic to the composite as a whole: a duplicate of the composite would surely have duplicate components in a duplicate spatial or spatiotemporal arrangement. (So the classical relativistic mass of a composite is also external, since it is fixed by the components’ rest masses and their mutual motions.) And similarly for the quantum conditional probabilities that we are concerned with: if we accept (i), we will no doubt say that if the composite state is pure, any such conditional probability is an external property, i.e. intrinsic to the composite but not fixed by its components’ intrinsic properties. And just as the spatiotemporal properties of any composite induce corresponding extrinsic properties of its components (e.g. being five metres from component x, or being between two more massive components), so also these conditional probabilities induce corresponding extrinsic properties of the components. And so the result: first we need notation. Let a and b be two identical (i.e. indistinguishable!) quantum systems, which are components of some composite system S. S may also contain other systems c, d. . ., perhaps nonidentical with a and b (perhaps bosons, while a and b are fermions), perhaps non-identical with each other. So the Hilbert space H for S is a tensor product with factor H, for component system x. Let QX represent the quantity Q on component x: QX=df. I@ I. . . 0 Q 0 . . . 0 Z, with Q in x’s place. It will be notationally convenient to let Q’ be a (possibly indistinct) quantity, represented on x by QY. Let q be a value of Q; and similarly, let q’ be a possibly indistinct value of Q’. (Here, twice choosing the same quantity is to be independent of twice choosing the same value: if Q’= Q, it may still be that q’ # q; and if q’ =q, it may still be that Q’# Q). We define permutations as unitary operators on H in the usual way. Let PXYbe the transposition between H, and H,,. We impose Permutation Invariance (Indistinguishability Postulate) as: (PI): for any pure state C$of S: if x and y are identical bosons, PX,,+= c$;and if x and y are identical fermions, P& = -4. Then the result is: for any such suitably symmetrized state 4:

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Studies in History and Philosophy of Science

(1) pr, (Q, has value q) = pr, (Q has value q); (2) pr, (Q, has value q/Q’b has value q’) = pr, (Qb has value q/Q’, has value q’) (3) pr, (Q, has value q/Q’, has value q’) = pr, (Q, has value q/Q’b has value q’). Proof-sketch for (2) and (3): The equations follow directly from readily proved properties of the transposition P,, namely (6) and (7) below. Thus let Q,‘s projector for value q be ‘?Qo. Then the lhs of (2) is by definition: pr, (Q, has value q/Q’b has value q’) = df. (4, 4Q0 “‘Q’&)/(c#J, 4’Q’b4)

(4)

while the rhs of (2) is by definition: pr, (Qh has value q/Q’, has value q’) = df. (4, ‘7Qb~‘Q’&)/(c$, The denominators on the rhs of (4) and (5) are equal numerators equal, we use the readily proved result,

‘J’Q’J$)

(5)

by (1). To prove

the

PabQ, Pc,b = Qb

(6)

to get:

using the fact 4 obeys (PI). The proof of (3) is very similar. Again, the denominators of its two sides are equal by (1). And for the numerators, we use as well as (6), the readily proved (7) Q.E.D. To sum up the significance of this result: we have shown state of any composite, two ‘identical’ components share:

that in any pure

(1) all their intrinsic properties, in the sense of the usual Born-rule probabilities for quantities to have values; (2) all their extrinsic properties that involve each other, in the sense of conditional probabilities for a quantity on the one to have a given value, conditional on a quantity on the other having a given value; (3) all their extrinsic properties that involve a third component, in the sense of conditional probabilities for a quantity on this third component to have a given value, conditional on a quantity on the first/second component having a value. In short,

the Identity

of Indiscernibles

is violated

even in a very weak form.

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Theory

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So much for intrinsic and extrinsic-the first of my two tractable issues raised by our initial ‘if’. To connect with van Fraassen’s text, I should briefly summarize his discussion of the second issue; it relates to the exclusion principle, and his modal interpretation of quantum theory. van Fraassen rightly stresses that the basic point against the Identity of Indiscernibles (the point with which I started this Section) holds good for fermions, just as for bosons: an antisymmetric composite state determines reduced states which are all the same, just as a symmetric state does. Thus he criticizes the common slogan form of the Pauli exclusion principle, that ‘two fermions cannot be in the same state’ (pp. 385, 424). He instead takes the exclusion principle as the theorem that antisymmetric states are superpositions of product states x, @ . . . @ x,, in which there are no repeats; i.e. xi = xj only if i = j. This has the corollary that for any antisymmetric state, the reduced state (the same for all components) has an image space of dimension at least n (pp. 385-389). Then van Fraassen uses this corollary to argue in the context of his modal interpretation, that one can make non-orthodox value attributions to quantities on the components (i.e. deny the ‘if’), in such a way as to save the Identity of Indiscernibles (pp. 4233429). The basic idea is that the large image space of the common reduced state allows us, for any two components, to find some quantity for which we can, consistently with the modal interpretation, assign distinct values to the two components. (However, this strategy for saving the Identity of Indiscernibles does not apply to bosons; pp. 429433). I now turn to the second aspect of van Fraassen’s discussion of the Identity of Indiscernibles, in which he clarifies his notion of an individual by expounding how permutations apply to possible world semantics. As I mentioned at the start of Section 4, my verdict will be that he leaves some important questions hanging. Although this material is independent of quantum theory, van Fraassen motivates it (at the start of Chapter 12), by returning to our first puzzle, about how particles, indeed any individuals, can disobey Maxwell-Boltzmann statistics. He says there have been two main responses to the puzzle. One is that to understand quantum statistics we need a new notion of individual; the other is that we need to abandon altogether the notion of individual-quantum statistics can only be understood by using quantum fields. He announces that he will argue that both responses have a false presupposition: namely, a metaphysical notion of individuality ‘that has already been moribund for centuries’ (p. 434). My Section 6 will discuss the second response. As to the first, van Fraassen says more specifically that ‘the “loss of identity” is not a new feature of the quantum world . . . The only way we will be able to see that is by looking at identity in a purely classical context . . . I shall argue specifically that . . . the

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Identity of Indiscernibles is not a matter of logic, nor inherent in the conception of an individual . . . [and] the completeness of physics does not require that its language be less than purely general’ (pp. 459460). (Here, ‘to be purely general’ turns out to mean, roughly, ‘to have as non-logical vocabulary only predicates expressing purely qualitative properties’.) To argue for the three claims I have just quoted, van Fraassen first briefly expounds how Leibniz’s own reasons for the Identity of Indiscernibles arose

from his struggles with Aquinas’s

doctrines about identity. van Fraassen thereby emphasizes that the Identity of Indiscernibles is controvertible. Fair comment, say I. Then van Fraassen gives a much longer technical exposition, apparently aimed at defending all three claims. First, he calls the claim that all factual description can be given in purely general propositions ‘semantic universalism’. (He admits, but sets aside as pragmatic, the indispensability of indexicals; p. 466.) Thus it becomes clear that his third claim does not itself assert that physics is complete, i.e. encompasses all factual description. The claim is only that if it is complete, its language can still be purely general. In other words, the third claim to be defended is just semantic universalism itself. Then van Fraassen applies permutations to possible world semantics (the exposition draws on a previous monograph”). The broad strategy is as follows: (i) to start with a ‘haecceitist’ conception of a possible worlds model, in which the individuals that inhabit the worlds are each assumed to have an underlying identity across the worlds, quite independently of which predicates they fall under in the various worlds; (ii) then to require a kind of permutation invariance in these models so as to eliminate the haecceitism and thereby defend the three claims. This elimination of haecceitism is summarized in two technical results. I shall argue that although these results are technically interesting, there are philosophical difficulties in using van Fraassen’s permutation invariance to eliminate haecceitism, or to defend the claims. Thus van Fraassen starts with a possible worlds model as a set W of worlds, where each world w is a mapping of the domain D of individuals into the set of ‘cells’, i.e. logically strongest consistent predicates, Carnap’s Q-predicates. (Incidentally: (1) in order to allow for individuals which do not exist at the world in question, one cell is to be interpreted as ‘does not exist’; (2) cells are monadic, but the discussion to follow can be extended to relations.) Taking each world as a mapping in this way, with no constraints on what cell an individual is sent to, is clearly ‘haecceitist’: each individual in D has an

12B. van Monographs,

Fraassen, Essence No. 12, 1978).

and Exisrence

(Pittsburgh:

American

Philosophical

Quarterly

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underlying identity irrespective of which cells it is assigned by the various worlds. Given a permutation, p say, of D, van Fraassen now defines a corresponding permutation of a given world w as the map p(w) defined by: for all x in D, p(w)(x) = w@(x)). That is, the individual x goes into the cell, for the world p(w), that the individual p(x) goes into for world w. The idea is that p(w) is a qualitative replica of w but differs in which individuals are in which cells-the differences specified by p. van Fraassen then defines: a proposition is purely general (relative to a model) iff its truth-value is invariant under permutation, i.e. the set of worlds (within the model) where it is true is closed under taking of permutations of worlds. Since this definition is relative to a model, it cannot on its own capture the intuitive idea of ‘purely general’. If the set W of worlds is small or contrived, it will be too easy for a proposition to be purely general. To take the extreme case: if W contains no permutations of any of its members, every proposition will be (vacuously) purely general. So van Fraassen defines a fill model as one in which W is closed under permutations. More exactly, the definition also requires that the binary accessibility relation R on worlds that explicates necessity and possibility (L and M) should be closed under permutations in a certain sense. This secures something that van Fraassen considers desirable: that the purely general propositions will be closed under L and M. (It is easy to check that (in any model) there is no problem with the Boolean operations, such as negation and conjunction: the set of purely general propositions is always closed under them.) van Fraassen now asserts that semantic universalism will require that all models be full (and of course that within full models, only purely general propositions be considered, at least for ‘factual description’). And he shows that these requirements are not as restrictive as they might seem. In particular they allow the truth of some de re modal propositions; so that enthusiasts of de re modality need not baulk at them. Thus it is easy to check that a non-modal proposition with no singular terms, such as ‘cell 1 has exactly 17 occupants’ is purely general even in full models; and similarly for Boolean compounds and modalizations with no quantifying into a modal context. What about propositions that quantify in, and so are commonly regarded as de re? van Fraassen shows that some of these are purely general, and satisfiable (i.e. true in some world), in full models (pp. 471474). van Fraassen then presents his two results about how full models eliminate haecceitism. For both results, the intuitive idea is that because a full model includes all permutations of any of its worlds, it is logically weak-it loses the information about individuals’ underlying identity across the worlds. The first result is that a haecceitist-sounding proposition is not satisfiable in a full model. Namely, van Fraassen’s proposition (Pl), on p. 471:

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there are individuals, x and y say, and cells, F and G say, such that x and y are both actually in cell F, but M(x is G) and not-MO, is G).

This proposition is haecceitist in that it makes facts about individuals’ modal properties (and so about transworld identity) not supervenient upon their qualitative properties. The second result is that there is an equivalence between full models and another notion which van Fraassen calls an ‘abstracted model’. Thus given any model of the usual sort (set W of worlds etc., not necessarily full), van Fraassen defines an abstracted model whose worlds are just maps from cells to the natural numbers: so each world is just a specification of ‘occupation numbers’, of how many objects are in each cell. So any two worlds in the usual model that are permutations of each other correspond to the same world in the corresponding abstracted model. And similarly for two different usual models that use different, but equinumerous and perhaps intersecting, basic sets of individuals D and D’: if two worlds in the two models are isomorphic (i.e. permutations of each other, module the underlying identity of individuals in D and D’), then they correspond to the same world in an abstracted model. To sum up: an abstracted model corresponds to a set, M say, of models of the usual sort. And the models in M differ from one another in: (i) their underlying set of individuals D (though the size of D is fixed by the abstracted model); and/or (ii) how many permutation-copies (for fixed D) of a given world, and which such copies, they contain. Now I can state van Fraassen’s second result: setting aside the first kind of difference (the choice of D), an abstracted model corresponds to a unique full model (pp. 475-476). As I see it, this second result promises a stronger anti-haecceitism than the first result did. For if full models are tantamount to abstracted models, they apparently make no real use of the notion of transworld identity, never mind whether such identity is primitive, or somehow supervenient on qualitative properties. Perhaps van Fraassen would agree; it is hard to say since his discussion of this result is very brief. But he also suggests something even more radical: that the result does away with the notion of individual (or perhaps just with a notion of individual ‘already moribund for centuries’). For it shows an equivalence between a picture of reality as containing individuals (though tempered by semantic universalism) and a picture without individuals, with only the pattern of instantiation of the cells as given by the occupation numbers. That is heady stuff, But I think we can see how questionable it is if we ask how van Fraassen’s two results support the three more specific claims with which he began. As regards the Identity of Indiscernibles, and the first result, van Fraassen has a good point. Possible world semantics clearly allows cells to have more than one member, violating a version of the Identity of

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Indiscernibles. And the first result strengthens this: according to semantic universalism (i.e. a restriction to full models), you cannot save the Identity of Indiscernibles, when faced with a cell with two or more members, by appealing to differences in the members’ modal properties. Since (Pl) is not satisfiable in a full model, there will be no such differences. As regards semantic universalism, van Fraassen has in effect two assertions to defend. One is semantic universalism in the original intuitive sense, that all factual description can be given in purely general propositions. The other is that his technical apparatus captures this sense; specifically his assertion that semantic universalism requires a restriction to purely general propositions within full models. As I see it, the definitions and results summarized above leave the defence of these assertions very incomplete. van Fraassen clearly does not defend semantic universalism in general. Rather he claims to capture it in his technical apparatus, and show its merits there; for instance in allowing the truth of some de re modal propositions. But even within this apparatus, the defence is incomplete, simply because van Fraassen does not relate his notions of ‘individual’ and ‘purely general’ to recent work in the metaphysics of possible worlds and properties. Since he is well-known to be no friend of metaphysics, this omission is perhaps unsurprising. But it makes for incompleteness. And worse than incompleteness. In some cases there is a danger that the gap cannot be filled in, consistently with van Fraassen’s views. I shall end this Section by describing two such gaps. (I suspect there are more. Clearly, one can question: (i) whether permutation invariance of truth-value captures the idea of ‘purely general’; and (ii) whether in interpreting his second result, van Fraassen can ignore the first kind of difference between the models in M, i.e. having different basic sets of individuals, D and D’. And does van Fraassen’s anti-haecceitism, and his apparent denial of transworld identity, based on his second result, naturally lead him to a view like that of Lewis, the arch antihaecceitist, for whom counterparthood, not identity, provides representation de re?13)

My first gap concerns the idea of a qualitative property. van Fraassen needs the cells to be (logically strongest) qualitative predicates. Of course, this is not necessary for the sheer mathematics of his results, but it is necessary for the results to somehow support semantic universalism. For if the cells are not qualitative, some of the propositions that van Fraassen defines to be ‘purely general’ will not deserve that name, even in a full model. van Fraassen’s device for erasing reference to particular individuals (viz. requiring truth-value to be preserved under permutation) will be inadequate: such reference can still be lurking in the cells themselves. And so restricting oneself to purely general ‘Tf.

D. Lewis,

op. cit., note 11; see pp. 194-197, pp. 220-227.

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propositions, as defined, within full models, will not capture semantic universalism. So this raises the question: how should we analyse ‘qualitative property’? That is a hard question (related no doubt to the other hard question at the start of this Section, ‘how should we analyse ‘intrinsic property’?‘). Since van Fraassen does not address this question, it is tempting to infer that he intends to ‘parameterize’. That is, he intends the reader to simply adjoin to his discussion their own favourite analysis of ‘qualitative property’ (though as an anti-metaphysician, he will probably not believe their analysis!). This inference may be right. But if so, van Fraassen’s position is risky. For doctrines about the analysis of qualitative property are liable to affect questions van Fraassen does address, like transworld identity. Here is one such connection: the weaker the notion of qualitative property, the more plausible is the non-supervenience of modal properties on qualitative properties. That is, the more plausible the proposition (Pl) that van Fraassen rules out. The second gap concerns the idea of an essential property. van Fraassen does not discuss these. But suppose that some objects have such properties and some such are qualitative (say, Boolean compounds of cells). Then since van Fraassen’s full models contain any permutation of any world they contain, some of these worlds will be faithless to these essential properties. That is, they violate necessary propositions about objects’ essential properties, and so do not represent a genuine possibility. That threatens the significance of results about full models. More generally, the use of essential properties obviously promises a via media, perhaps an attractive one, between the haecceitism involved in full models’ use of arbitrary permutations of worlds, and the heady idea of worlds without individuals, as in abstracted models. Furthermore, semantic universalism is surely consistent with the existence of essential properties, even if one takes ‘purely general’ as van Fraassen does, in terms of permutation invariance of truth-value. One simply weakens the requirement of invariance of truth-value, starting from a given world: one requires invariance only for those permutations that yield a genuinely possible world (that violate no essential property ascriptions). So, pace van Fraassen (p. 470), semantic universalism cannot demand that we use only full models. To sum up these last two Sections, I have largely agreed with van Fraassen’s main contentions (while questioning his reasons!): that quantum statistics, and the failure of the Identity of Indiscernibles, do not refute our notion of an individual. But I will largely disagree with him about our third and final issue.

6. Particles and Fields Our third issue is the concepts of particle and field in quantum field theory. As mentioned in Section 5, van Fraassen introduces the issue at the start of his Chapter 12: he denies that quantum field theory resolves puzzles about

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identical quantum particles, by somehow abandoning a ‘classical’ notion of individual. More specifically, he says he will expound the quantum field formalism, in order to argue that it ‘is not a different theory [from fixedparticle-number quantum theory, but] a somewhat enriched and elegantly stated theory of particles. That we can take it as a description of a world that is particle-less only masquerades as an incompatible alternative’ (p. 436). By and large, I applaud van Fraassen’s exposition of some elements of quantum field theory (pp. 436448). First, he describes occupation numbers for a fixed number of bosons. Second, he describes how given a basis in the oneparticle Hilbert space, bosonic Fock space can be built up with the creation and annihilation operators for that basis. And third, he describes how a basis change in the one-particle space induces a transformation between such operators. (For reasons that will appear, this third part is curiously entitled ‘the label-free theory’.) As before, he gives some helpful cautionary remarks. For example, he emphasizes that a number eigenstate In(l), n(2) . . . ) in bosonic Fock space is not merely an assignment of n(i) bosons to pure state i-even if we allow that the bosons are somehow indistinguishable. There are two reasons: we can always re-express the state by changing the basis in the single-boson summand space (p. 441); and-as discussed in Section 5-symmetrization means that each boson is really in (the same) reduced mixed state (p. 443). But I cannot endorse van Fraassen’s interpretative claims: neither the summary claim quoted above from the start of Chapter 12, nor his discussion after the exposition of Fock space (Section 2.4; pp. 448-451). That discussion is brief-in my opinion too brief for clarity, even when supplemented with the references he cites. But as I interpret it, its main claim is false. To state that claim, van Fraassen needs the idea of a representation theorem. The idea is that any example of a certain ‘abstract’ concept is isomorphic to one of a limited collection of ‘concrete’ examples; e.g. any group is isomorphic to a group of one-to-one mappings on some set. Such a theorem shows a kind of mutual representability. For it shows that the ‘concrete’ procedure of mathematical construction, e.g. from a set to its group of one-toone mappings, is in a sense adequate: in this example, adequate to characterize the abstract idea of group. van Fraassen says that there can be such mutual representability between two empirical theories. As examples, he suggests Schriidinger’s wave mechanics and Heisenberg’s matrix mechanics, or physical geometry founded on points, and founded (as by some mid-century mereologists) on extended regions such as spheres. He takes such representability to entail that two such theories are necessarily empirically equivalent. And this entails, on van Fraassen’s conception of interpretation, that any tenable interpretation of one theory must also be tenable of the other. And that is so,

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even if the different drawing

interpretations

incompatible

pictures

are typically

conveyed

in ways that involve

of the world (pp. 45WUl).

So far, so good, module general doubts over pluralism about interpretations (cf. end of Section 1). But now comes van Fraassen’s main interpretative claim about

quantum

reconciles

the

field

theory:

namely,

that

particle

and

particle-less

such

a representation

interpretations

theorem

of quantum

field

theory-vindicating the Chapter’s opening claim about ‘masquerading as an incompatible alternative’. And so my objection: what this theorem is, remains unclear. mutual

And on my best guess about what it is, the theorem does not show representability in a sense strong enough to entail empirical

equivalence. van Fraassen space

takes

his building

to be a ‘concrete’

up of Fock

construction.

Then

space

from

the one-particle

he cites the third

part

of his

exposition (i.e. the basis-change material) and a paper by de Muynck as proving a representation theorem: ‘all models of (elementary, non-relativistic) quantum field theory can be represented by (i.e. are isomorphic to) the sort of Fock space model constructions I described above’. He goes on: ‘Since the latter are clearly carried out within a “labelled particle” theory, we have a certain kind of demonstrated equivalence of the particle-and the particle-less-picture’ (p. 448). The ‘all models’ claim is unclear.

van Fraassen

cannot

mean merely ‘isomor-

phic as Hilbert spaces’. That would make the claim a trivial corollary of the elementary fact that all complex infinite-dimensional separable Hilbert spaces are isomorphic (indeed, isomorphic to P(co)). (The models must be infinitedimensional models

to represent

as separable,

infinite-particle-number between

particle

theorem, preserved

the canonical

as van Fraassen non-Fock

and particle-less,

more structure than by the isomorphism.

But what that extra structure

commutation implicitly

representations.) and then reconcile merely

being

is, remains

relations;

I take

the

does, so as to set aside the To capture

them by a representation

a Hilbert

unclear.

the contrast

space

needs

to be

For we are not given an

exact definition of ‘abstract’ Fock space, or of ‘abstract’ quantum field theory, that would make the isomorphism and so the theorem precise. And the material van Fraassen cites does not seem to help. His own basis-change material clearly cannot do so: it just describes a basis-change in the concrete construction. Nor does the de Muynck paper. van Fraassen summarizes de Muynck as reformulating quantum field theory ‘with the individual particle labels reinserted’ (ibid.). That is fair comment: for de Muynck in fact presents nonrelativistic quantum field theory, not with bosonic or fermionic Fock space, but with each particle-number-n summand taken as the full n-fold tensor

product

of n ‘labelled’

one-particle

spaces

(one

for each of n non-

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and Identity in Quantum Theory

473

identical

inequivalence between the usual non-relativistic

bosonic quantum field theory and his

own ‘labelled bosons’ analogue. There is another clue: van Fraassen also cites discussions by Robertson and Ginsberg of a mathematical equivalence between non-relativistic quantum field theory and many-particle elementary (‘first quantized’) quantum theory. That equivalence, first proved by Fock in 1932 and reproduced in textbooks and by Robertson,‘5 amounts to this: there is a natural one-to-one correspondence between bosonic (fermionic) Fock space states of fixed particle number, n say, and symmetric (antisymmetric) n-particle elementary states. This correspondence covers dynamics: starting from an elementary Hamiltonian H for n particles (perhaps with particle-particle interactions), one can define a Hamiltonian H’ on Fock space such that for any symmetric (antisymmetric) elementary state, the corresponding Fock state obeys the Schrijdinger equation with fl. But does this correspondence give an empirical equivalence of the kind van Fraassen announces? Surely not. As Robertson notes (p. 686), there are two widely-recognized differences between the two theories. In one respect, manyparticle elementary quantum mechanics is more general: it includes the case of n non-identical particles-just use an unsymmetrized wave function. (Admittedly, de Muynck’s more recent ‘labelled’ quantum field theory apparently captures this generality, at least for bosons.) In another respect, quantum field theory is more general: it includes states that are superpositions of particle number. There are three points to make about this second difference (in increasing order of importance). First, it is such superpositions that de Muynck uses in his closing argument for an inequivalence. So there is no prospect of de Muynck’s ‘labelled’ theory somehow overcoming this difference so as to justify van Fraassen’s claimed empirical equivalence. Second, I do not think van Fraassen can overcome this difference by assimilating it to the measurement problem. That is: he cannot argue for empirical equivalence by appealing to the fact that measurement results always

14W. Muynck, ‘Distinguishableand Indistinguishable-Particle Descriptions of Systems of Identical Particles’, International Journal of Theoretical Physics 14 (1975), 327-346, see p. 335f. ‘Tf. P. Dirac, op. cit., note 7 (fourth edition, 1958); see Section 60, pp. 23G231; for a more leisurely presentation, cf. S. Schweber, Relativistic Quantum Field Theory (New York: Harper & Row), 1961, pp. 140-146; or B. Robertson, ‘Introduction to Field Operators in Quantum Mechanics’, American Journal of Physics 41 (1973), 678-690, see pp. 684686.

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a definite

number

of particles-that

the number

of spots on a photo-

graphic plate is always an integer etc.! For there are of course processes in which particle number changes by the unitary evolution of the system, because (real, not virtual) particles are created and annihilated. (Agreed, such processes have long been treated relativistically, e.g. Fermi’s 1934 theory of b-decay where a neutron becomes a proton, electron and neutrino, while van Fraassen’s discussion is about the non-relativistic case. But one can easily write down non-relativistic interactions that do not commute with the particle number operator, so that particle number changes.) To describe such a process requires intermediate states that are superpositions of particle number to be as physically real as the initial and final eigenstates of particle number (unless one adopts some strong instrumentalism about intermediate states, which van Fraassen certainly would not). The third point is more general. Let us set aside van Fraassen’s concern with empirical equivalence, and return to his opening claim that quantum field theory’s particles are individuals. I think that superpositions of particle number make this claim false. For it seems to me that according to any reasonable notion of an individual (even a very liberal notion, not ‘moribund for centuries’), the number of individuals must be definite. In any case, van Fraassen gives no argument against this implication, or connotation, of the notion of an individual. (Agreed, to fully justify my denial of van Fraassen’s claim, I would need the more precise idea of the number of individuals in a given region, or perhaps more generally in a given ‘context’; making due allowance for individuals that are partly in, partly out. With such details filled in, the only tactic for saving van Fraassen’s claim seems to be to say that there are no individuals except when the state is a particle number eigenstate-surely a desperate tactic.) So far in this Section, I have taken van Fraassen to task for lack of clarity about his reconciliation of particle and field. But this last point leads to another that allows me to end on an irenic note, albeit without reference to van Fraassen’s text. In the space remaining, I shall argue that there is indeed a false dichotomy between particle and field, in quantum field theory. But pace van Fraassen, this is not because quantum field theory is a theory of particles that are individuals. Rather the individual is the quantum system itself: which behaves in some states like a particle or a collection of them, and in other states like a field. What, after all, do we mean by ‘particle’ and ‘field’? Clearly, the concepts get changed as we pass from classical physics, to elementary (‘first quantized’) quantum theory, to quantum field theory. So interpreting these theories, especially the last, is in part a matter of plotting those changes. And even the most cursory attempt to do that shows there are many different particle-like, and many different field-like, attributes that one can consider; as follows.

Interpretation

and Identity

in Quantum

We might list, as attributes in elementary bility.

We

quantum can

of classical

theory:

similarly

Theory

list

475

particles

a continuous new

that quantum

spacetime

attributes

of

particles

trajectory,

quantum

lack

impenetra-

particles:

the

Fourier-transformation between position and momentum, quantum statistics. In the transition to quantum field theory, definiteness and conservation of particle number go; creation and annihilation come in. Turning to fields, we might list, as attributes of classical fields that the wavefunctions of elementary quantum theory lack: energy-momentum, being mathematically real. Yet a wave-function is like a classical field in that it represents the state (‘configuration’) of the system concerned, albeit in a particular representation, namely the position representation; instantaneously or throughout time, depending on one’s definitions of wave-function and field. (Here, and in what follows, I use ‘representation’ to mean an orthonormal basis of state-vectors, not in van Fraassen’s sense of a representation theorem.) In quantum field theory, this attribute goes: the state of the system is not represented by the quantum field, i.e. by the assignment of operators to each (spatial or spacetime) point. The state is represented, as always in quantum theories, by a state-vector. With this great variety of attributes (and no doubt more) to be considered, and related to one another, there is certainly plenty to do in plotting the changes in the two notions of particle and field. But this variety should also make one wary of loose talk about a conflict between particle and field interpretations of quantum field theory. There is probably no essence in each of these two notions, one essence contradicting the other, allowing one to then try and judge which has the upper hand in interpreting quantum field theory. I think this point is supported by an apparently ‘essentialist’, and longstanding, proposal for how to understand ‘particle’ within quantum field theory: namely that a quantity is ‘particle-like’ iff it commutes with all particle number operators. I6 This proposal is certainly attractive: for instance, it makes position, momentum and spin particle-like. It also suggests that we call a representation that diagonalizes these number operators ‘a particle representation’. Such a representation will then be invaluable for describing phenomena in which particle-like quantities phenomenon will involve simultaneous particle-like quantity, and we then choose diagonalizing number and that quantity. But of course, there are other (mutually more generally sets of states, that are

are important. Typically, the eigenstates of number and some the representation by simultaneously non-commuting) representations, or invaluable for describing different

Wf. S. Schweber, ibid., p, 183, p. 193; M. Redhead, ‘Quantum Field Theory for Philosophers’, in P. Asquith and T. Nickles (eds), Proceedings of the Philosophy of Science Association I982 (East Lansing: Philosophy of Science Association 1983), pp. 57-99, see p. 65, p. 79.

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phenomena, where other quantities, not number, are important (in particular: definite in value). Two well-known examples are eigenstates of phase, and the coherent states; which are invaluable for the quantum description of the electromagnetic field. I7 Besides, since as we noted there are various connotations of ‘field’, there need be no unique best choice amongst these noncommuting representations, for a corresponding definition of ‘field-like quantity’ and ‘field representation’. Whether or not there is such a choice, the important point is this: it clearly makes no sense, on this proposal, to ask which of particle and field has the upper hand in interpreting quantum field theory. For the individual described by the theory is the underlying quantum system, with its Hilbert space of states. Particle and field are now both matters of a representation, of a selected set of states. The glory of quantum field theory is that it allows and uses all these representations, variously appropriate for describing particle-like or field-like phenomena-where the ‘like’ signals due allowance for ambiguities and changes in the concepts, as sketchily plotted above. I like to think that this irenic conclusion is what Dirac, taciturn as always, really meant by his notoriously brief remark that his quantization of the electromagnetic field gave ‘a complete harmony between the wave and light-quantum descriptions’!‘*

Acknowledgements-I am grateful to the British Academy, Leverhulme Trust and Mrs L. D. Rope Third Charitable Settlement for supporting sabbatical leave. For conversations and comments, I am very grateful to: Guido Bacciagaluppi, Thomas Breuer, Marcus Cavalier, Rob Clifton, Dennis Dieks, Gordon Fleming, Steven French, Peter Lipton, Joseph Melia, Michael Redhead, Abner Shimony, Paul Teller and Bas van Fraassen.

“Cf. e.g. R. Loudon, The Quantum Theory of Light (Oxford: 14ck153. “P. Dirac ‘The Quantum Theory of the Emission and Absorption the Royal Skiety of London A 114 (1927), 243-265, see p. 245.

Clarendon

Press,

of Radiation’,

1973),

pp.

Proceedings of