Chapter 11 Quantum Mechanicsusers.ox.ac.uk/~jrlucas/reasreal/quanchp3.pdf · Chapter 11 Quantum Mechanics \The quantum theory of elds is the contemporary locus of meta-physical research",

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Chapter 11Quantum Mechanics

\The quantum theory of �elds is the contemporary locus of meta-physical research", H.Stein1

x11.1 Unhistoryx11.2 The Inner Cavex11.3 Discreteness and Continuityx11.4 From von Neumann to Kochen-Speckerx11.5 From EPR via JSB to GHZx11.6 Non-localityx11.7 The \Measurement Problem"x11.8 Knowing and Beingx11.9 The Uncertainty Principlex11.10 Nullary Qualitiesx11.11 Indiscernability and xHaecceitasx11.12 Quantum Realismx11.13 Quantum Philosophy

x11.1 UnhistoryHistorically, quantum mechanics was forced on physicists by em-pirical evidence. Black-body radiation and the photo-electric e�ectcould not be explained in terms of classical physics, and quantummechanics was developed in order to make some sort of sense ofbizarre experimental observations. But with the bene�t of hind-sight, we can begin to see the rational pressures that shaped ourtwentieth-century attempts to understand the physical world.

1 \On the Notion of Field on Newton, Maxwell and Beyond", in R.H.Stuewer,ed., Historical and Philosophical Perspectives in Science, Minnesota Stud-ies in the Philosophy of Science, 5, Minneapolis, Minn., USA, 1970, p.285;quoted by M.L.G.Redhead, \A Philosopher Looks at Quantum Field The-ory", in Harvey Brown and Rom Harr�e, eds., Philosophical Foundations ofQuantum Mechanics, Oxford, 1988 (pbk 1990), p.9.

307

308 Reason and Reality x11.1

It is a dangerous exercise.2 All too easily we read into the pastprinciples that exist only in our imagination, and endow our reasonwith a sensitivity it would never have developed, without the brutegiven-ness of experimental evidence. Nevertheless, it is a useful ex-ercise, enabling us to see well-worn truths in a new light. We arenow able to see the Special Theory as the culmination of Newtoniantheories of space and time, though it took the Michelson-Morley ex-periment to force us to re-think in that direction.3 With quantummechanics, if we set it against the foil of classical corpuscularianism,we see the di�erent alternatives that we might have in a theory ofmatter; and the weaknesses of classical corpuscularianism enhancethe cogency of its development into quantum mechanics.

The Corpuscularian philosophy leaves the following loose ends:1. It yokes together discrete point-particles and continuous space;2. It assumes a certain set of primary qualities, which presuppose3. 3-dimensional, Euclidean space;4. It entirely ignores epistemology.

For quantum mechanics is not simply an alternative to classicalmechanics. Rather, it accepts the physics of classical corpuscular-ianism, but seeks to go beyond it. It seeks to remedy some of theblemishes on the corpuscularian schema of explanation, by going astep further than corpuscularianism, and o�ering more fundamen-tal explanations, both physical and metaphysical. Explanation isa rational activity. Although constrained by the need to save theappearances, explanations are not explanatory unless they conformto certain rational ideals. And in the development of quantum me-chanics, we can detect the pressure of certain canons of rationality,and dimly discern some rational ideals.

2 Also undertaken in x10.4.3 See J.R.Lucas and P.E.Hodgson, Spacetime and Electromagnetism, Oxford,1990, ch.8.

x11.2 Quantum Mechanics 309

x11.2 The Inner CaveQuantum mechanics is a metaphysics of the Cave. Behind theworld of appearances lies a more fundamental and more real world,which explains how things appear to us, and the underlying courseof events, and which is accessible to thinkers who think hard, andreally want to know the nature of things. As with classical cor-puscularianism, the explanatory power of quantum mechanics isits warrant of truth. It stands or falls by its ability to account forphenomena, and explain what hitherto had been un-understood.But though there is no independent access to it by means of someinsight into the Form of the Good, explanatory power is itself a �efof Reason. The Greek atomists and many of the early corpuscu-larians were persuaded, in advance of empirical vindication, by therational merits of its tenets. Even more so with quantum mechan-ics, where in spite of the extraordinary di�culty in making sense ofit, there seems to be some intellectual necessity about the theory,quite apart from its con�rmation by experimental observation.

Unlike classical corpuscularianism, however, quantum mechan-ics posits not just a Cave, but an Inner Cave. When the corpus-cularians thought that they had escaped from the cave of mereappearances into the light of day, they misled themselves|theyhad only come from an inner cave to an outer one, in which we stillsaw only shadowy representations of the real cosmos. The corpus-cularian philosophy assumed a certain set of primary qualities. Ifwe had microscopic eyes, we would be able to see the primary qual-ities of the ultimate constituents of matter, and it was these thatdetermined their secondary qualities, and all their other powersand properties. The great economy of explanation was a weightyargument in favour of corpuscularianism, but raised the question ofwhy the primary qualities were so important, and whether they didnot need to be explained in their turn. Quantum mechanics, beingnot so much a rival to classical corpuscularianism as a developmentof it, takes us further|if not right into the outside world of fullenlightenment, at least remedying some of the defects of currenttheories. A better Fundamental Theory of the Universe would gobehind Locke's primary qualities to ones more basic still|\nullaryqualities", so to speak; and posit a cosmos with properties thatwere inherent, not contingent.

Optative PhysicsA Desirable Fundamental Theory of the Universe would1. Go behind Locke's primary qualities to ones more basic

still|\nullary qualities"??2. Posit a space with properties that were inherent, not

contingent.


Quantum mechanics does this. The -functions it ascribes arehighly abstract, and remote from experience, but have intellectualcoherence and rational appeal. The Fourier analysis of periodicfunctions requires an abstract in�nite-dimensionsal Hilbert spacein which Parseval's theorem serves as an analogue of Pythagoras'theorem in establishing a quasi-Euclidean metric.4 We thus havea picture. A periodic function can be expressed as a superpositionof pure sinusoidal functions of di�erent frequencies, and these lat-ter can be represented by di�erent axes in Hilbert space, and theirsuperposition as the corresponding ray in Hilbert space. AlthoughHilbert space in general has a denumerable in�nity of dimensions,we can make do with three. We can picture a -function, repre-senting the state of a quantum-mechanical system, as a searchlightbeam moving across the sky, searching for enemy bombers. The di-rection of the beam will be measured by the angle it makes with theNorth, the East and the vertical axes (or any other three orthog-onal axes we choose to adopt), say �, � and �, with Pythagoras'theorem yielding:

cos2� + cos2�+ cos2� = 1

Each squared cosine gives the probability that if the relevant oper-ator is applied, the beam will align itself along the correspondingaxis.5

Fourier analysis is bad news for the learner. Although Schr�odin-ger's wave approach and Heisenberg's Hilbert-space approach areau font equivalent, there being two approaches have given rise toa bewildering variety of symbols and ideas. Nearly all involve theletter . In Schr�odinger's approach is a function, a periodic func-tion representing a wave, on which we perform various operationswith operators|energy operators, momentum operators, positionoperators, and the like|and then �nd solutions for the resultingequations, which tell us what energy levels, what values for momen-tum or position, are possible. Heisenberg represented quantum-mechanical operators as matrices, for which we could work out

4 See above, x9.7.5 According to standard expositions of quantum mechanics it should beseen as a single ray, but, as will be argued in x11.7, a beam is a betterrepresentation.


what vectors were \eigen-vectors", vectors that is which were trans-formed by the matrix into a vector aligned in the same direction,but with a magnitude multiplied by some real number (possibly,just the number 1), its eigen-number. Dirac developed this into anextremely elegant formulation in terms of vector spaces, in whichthe state of a quantum-mechanical system was represented by a\ket vector", j i, which can be expressed as a sum of orthogonalvectors in Hilbert space. The fact that these approaches are equiv-alent adds greatly to the power, the beauty and the profundity ofquantum mechanics: but the way most writers about quantum me-chanics slide around the di�erent symbolism and di�erent methodsof argument make for very di�cult reading.

Classical corpuscularianism has a problem in relating the worldoutside the cave|the world of science|to what we see in thecave|the world of experience. Quantum mechanics has a simi-lar problem in relating the world of corpuscularianism to a moreprofound one behind it. The relation is represented by operators|self-adjoint Hermitian operators which operate on the -functionsor j i vectors that represent the state of a quantum system. Theseoperators have certain eigen-values, the set of eigen-values depend-ing on which operator we are considering, and for each eigen-valuethere is a probability, depending on which -function we are con-sidering, of getting that eigen-value. These eigen-values are oftencalled values of òbservables'. It is a deeply misleading name. Mostof the magnitudes of which the eigen-values are measures are sev-eral steps remote from actual observation. I do not observe energyor momentum, and the natural philosophers of the SeventeenthCentury had di�culty in distinguishing them, and in framing ad-equate concepts. The alternative term, `dynamical variable', ispreferable, but I shall talk of `classical physical magnitude'. Itmakes the distinction, overlooked by Bohr, between the inner cave,in which there are material objects, inter-personal communicationand shared experiences, and the outer cave of classical corpuscu-larianism, in which there are particles possessing mass and movingin space. It also �ts with Einstein's phrase èlement of physicalreality', but uses the word `classical' to limit the force of the word`physical': there should be no suggestion that only elements of clas-sical physical reality are real; -functions and j i are not elementsof classical physical reality, but are, we shall argue, physically realnone the less.6

6 See below, x11.12.


x11.3 Discreteness and ContinuityClassical corpuscularianism found it di�cult to reconcile discrete-ness with continuity. Its ultimate entities, the atoms, were discrete,but they existed in a continuous space. Any pluralist metaphysic isbound to deal with a plurality of discrete ultimate entities, but facesa problem of how they can interact. If we reject Leibniz' heroic ex-pedient of denying interaction altogether, we are bound to posit anall-encompassing medium for them to interact in, a medium whichis uni�ed topologically, as space is. Classical corpuscularianism hasa pluralist metaphysic of discrete individual entities, but integratesthem into a monistic all-encompassing continuous space.

But there were problems in accommodating absolutely discretecorpuscles in an entirely continuous space. In order to be absolutelydiscrete, the corpuscles had to be point-particles, �-��o�o� (atomoi),logically unsplittable, as well as being impenetrable, so as to enableeach to maintain its separate existence. Though point-particleswere impenetrably solid, they could be packed into an in�nitesimalvolume|and would, granted the inverse-square law of attractionall collapse into an in�nitesimal volume, which would be alwaysgetting smaller, but could never actually be a single point. More-over, point-particles in a more-than-one-dimensional space wouldalmost never collide; and if they did, would have to be subject toin�nite forces in order to e�ect an instantaneous change of velocity.

Quantum mechanics marries discreteness and continuity in anentirely di�erent way. Beginners in quantum mechanics are en-couraged to think of an orbit like the petals of a ower: a stableorbit is a standing wave which has one, two, three, four, �ve, sixpetals, but not two and a half petals. There are other manifes-tations of discreteness within a matrix of continuity: a polygoncan have three, four, �ve, six sides, but not two and a half sides.And although some mathematicians have been able to work withfractional dimensions, in our ordinary understanding the numberof dimensions of a space is necessarily a whole number.

In quantum mechanics discreteness and continuity are inte-grated by means of periodic functions. A periodic function is onethat regularly returns to the same value after a certain interval, forexample

sin(x) = sin(2n� + x):Although such a function is continuous, the number of periods iscounted in whole numbers.7 This discreteness is not the logical7 See more fully below, x11.10.


discreteness of entities unsplittable because they have no size, butan as-it-happens discreteness of intervals whose magnitude is de-termined by physical laws. It is not linked with individuality, andis the discreteness of an accountant's accounts, rather than of anontological pluralist. Quantum mechanics can, in consequence, ac-commodate solidity. It tells a long story, about energy levels, andpermitted jumps from one to another, instead of the excessivelyshort story, told by the corpuscularians, of logically impenetrableand unsplittable point-particles. The quantum-mechanical accounthas more \give" in it, which enables it to explain the phenomenaof colliding molecules of gas, and the elastic rigidity of metals, buthas to take for granted the value of Planck's constant as a givenfact.

Classical physics admits only discrete truth-values, and is en-tirely Either-Or. Quantum mechanics admits a continuum ofprobabilities, and can often allow a Both-And. Where classicalphysics insists on impenetrability, which is a concomitant of exclu-sive space-occupancy, quantum mechanics allows the superpositionof alternative states of a�airs. This parallels a large part of ourthought, which is Both-And. We often consider cumulative cases,with arguments on either side. These arguments continue to co-exist: it is only when we have to decide, that we come down onone side or the other. The thinkers who in the early days of quan-tum mechanics suggested that electrons had free will, were mis-understanding what the physicists were trying to tell them. Butthe parallel between the process of making up one's mind, and asuperposition of di�erent states collapsing into one de�nite eigen-state, is a genuine parallel, which has important implications forthe nature of time.


x11.4 From von Neumann to Kochen-SpeckerThe marriage of discreteness and continuity gave issue in probabil-ity theory. Quantum mechanics uses probability theory, and needsthe probabilities it uses to be objective probabilities, not cloaks forour partial ignorance of the exact composition of ensembles. Johnvon Neumann had seen this, but his argument was not generallyaccepted, and in due course David Bohm faulted it by producing ahidden-variable theory none the less, and in 1966 John Bell showedwhich of von Neumann's assumptions could be challenged. Yetthough von Neumann's actual argument could be faulted, it wasnot refuted: it was revealing the fundamental di�culty in addinghidden variables to quantum mechanics: the whole point of hid-den variables was that they had values, and for any speci�ed valueeither de�nitely did have, or de�nitely did not have, that particu-lar value; quantum mechanics, on the other hand, was combiningideals of continuity with those of discreteness in its own idiosyn-cratic fashion, which could not accommodate the requirements ofdiscreteness inherent in a hidden variable theory.

This is brought out by Gleason's Theorem,8 which can be seenas a development and re�nement of von Neumann's putative proof,and exploits the di�culty of introducing hidden variables, yieldingdiscrete truth-values, into an essentially probabilistic theory. Inordinary quantum mechanics a system, j i, does not, unless itis an eigen-vector of some operator, have a de�nite value of thatphysical magnitude, but only a probability that IF the operatoroperates on the system, THEN the value of the physical magni-tude would turn out to be one of the eigen-values of that operator.A hidden-variable theory claims that there is a hidden variable,�, which together with j i determines what the value of the dy-namic variable is. It acts like a projection operator, picking outone eigen-value, and rejecting the others. But, provided that thereare three or more possible eigen-values, it is impossible to do thatconsistently.

The proof is a topological one, turning on the impossibility ofcontinuously mapping a discrete set onto a connected surface. Thenthere cannot be a smooth transition for occasions when a physicalmagnitude, say energy, has one de�nite value, say 12 electron voltsto occasions when it has another, say 16 electron volts; there willhave to be jumps.

8 C.Piron, Foundations of Quantum Physics, New York, 1976, p.86.


An intuitive way of seeing that this cannot be,9 is to picture theinside of a three-dimensional sphere: a projection operator is likea set of orthogonal axes, that is, three diameters at right angles toone another, one of them picking out two preferred antipodal points(which could be coloured red) and the other two marking the twoantipodal pairs of rejected points (which could be coloured blue).Then if the North and South Poles were red, the whole equatorwould have to be blue; similarly for any other pair of antipodalpoints, there would have to be a great circle of blue points, andwe should be unable to place enough red points, to get each set ofdiameters at right angles to one another having one of the diameterson red.

Gleason's theorem can be proved rigorously, though the assump-tions on which their proof is based, can be questioned. The ar-gument given by Kochen and Specker is likewise rigorous,10 butformidably di�cult to follow, involving 117 di�erent unit vectors,which cannot all be coloured so that any triad of orthogonal axeswill have one red and two blues. This is an argument where thepath of wisdom is not to try and make the argument one's own,but to accept it on the authority of the many clever and dedicatedmen who have examined it closely, and found no fault in it.11

Even the Kochen-Specker proof can be evaded if we impose suf-�ciently radical surgery on our normal understanding of science.Many physicists have tried to formulate a version of quantum me-chanics which is determinist while remaining faithful to the facts,This can be done|in a manner of speaking. We can formulate thefollowing \Hidden Variable Theory".

9 Due to F.J.Belinfante, A Survey of Hidden-Variable Theories, Oxford,1973, pp.38-39, a fuller discussion is given on pp.63-67; see also M.L.G.Red-head, Incompleteness, Non-Locality and Realism, 1st ed., Oxford, 1987,pp.124-125.

10 S.Kochen and E.Specker \The Problem of Hidden Variables in QuantumMechanics", Journal of Mathematics and Mechanics, 17, 1967, pp.293-328.; reprinted in C.A.Hooker, ed., The Logico-Algebraic Approach toQuantum Mechanics, vol.1, Dordrecht, 1975.

11 For a careful, but intelligible, exposition of these arguments, seeM.L.G.Redhead, Incompleteness, Non-locality and Realism, Oxford, 1987and 1989, ch.1, x1.5, pp.27-30, and ch.5, x5.1, pp.119-131; I am greatlyindebted to this work.


Spoof Hidden Variable TheoryThe state of a quantum system is given by

1. a ket vector j i, and2. a hidden variable �.� takes positive integral values

For any dynamical variable (\observable") A, with associ-ated operator A, and eigen--values a1; a2; a3; : : : :[ordered by magnitude]the A-value of (j i; �) is a�

This is fully determinate: for any system we can tellwhat the value of � was by making a measurement, andseeing which eigen--value emerged. Admittedly, we do nothave a law of development; the Schr�odinger equation tellsus how j i evolves, but not how � does|but one cannothave everything.

This spoof hidden variable theory fails because there is no law ofdevelopment for the hidden variable, �. The theory allows us to saywhat � was, after the event, but not what � is going to be, beforewe measure it; � is a dangler; it has no grip on the theory, it playsno part, except to register what we already know ex post facto. Ahidden variable theory can be constructed, provided we su�cientlyrelax the conditions on what constitutes a respectable theory, butat the cost of their credibility. In order to be a genuine hidden-variable theory, a theory must satisfy various criteria, which forthe most part have not been fully formulated, and often are onlyrecognised when a theory emerges which seems unsatisfactory insome respect. It is only because we have implicit standards of whatcan properly count as a theory, that we can say that serious theorieswith hidden variables are inconsistent with quantum mechanics.

If we accept that quantum mechanics is not going to be sup-planted or supplemented, the implications for philosophy are mo-mentous. In the �rst place we are led, as we have seen,12 to objec-tive probabilities for singular propositions, and hence to reject pro-jectivism in one of its most favourable cases. Probabilities cannotalways be explained as projections of our partial con�dences, norswept aside as a merely statistical property of frequencies found

12 See above, x5.6.


in ensembles, but are to be ascribed objectively either as gener-alised truth-values to singular propositions, or as propensities toparticular quantum-mechanical systems. It tells against projec-tivism generally. Although we still can explain secondary qualitiesand values as the projections of our sense-experience or attitudesonto a colourless, valueless real world, the argument for our doingso for reasons of metaphysical economy no longer applies.13

Secondly, if quantum mechanics requires objective probabil-ities, it cannot be completely determinist. For many physicalproperties|the mass of non-radioactive elements, the position ofmaterial objects, the movement of the planets|predictions can bemade with an overwhelming probability of their coming true, butfor some physical properties there will be a non-negligible probabil-ity of their becoming this, and a non-negligible probability of theirbecoming that. And sometimes it will make a signi�cant di�erencewhich way they turn out. If our fundamental physical theory isnot deterministic, many Laplacian arguments, which seem to un-dermine the autonomy of di�erent types of explanation, can nolonger be sustained.14

Quantum mechanics is philosophically importantbecause:1. It refutes projectivism,2. It refutes determinism.

13 See above, x5.7.14 See below, x13.2.


x11.5 From EPR via JSB to GHZEinstein was a determinist. He did not believe that God playeddice. In the positivist atmosphere of the 1930s, and against thebackground of the widespread (mis-)understanding of Heisenberg'sUncertainty Principle15 a realist construal of objective probabili-ties seemed implausible, and in view of the extreme di�culty inmaking sense of \the collapse of the wave function" the force ofvon Neumann's argument was parried.

Einstein tried to prove the existence of hidden variables|andpaved the way for a further argument showing them to be impos-sible. In his original example he considered two particles, initiallytogether which then separated to be at some considerable distancefrom each other.16 By measuring the momentum of one, we could,thanks to the conservation of momentum, know the momentum ofthe other. But we could also measure the position of the other,and hence know both its position and its momentum, contrary tothe Heisenberg Uncertainty Principle. This seemed to show thatalthough quantum mechanics was not able in general to measureboth the position and the momentum of a particle at any one time,it made sense to ascribe both. Each particle had a de�nite positionand a de�nite momentum, even though we could not measure themboth at once. And from this it would follow that quantum mechan-ics was in some sense incomplete: there was a fact of the matter|\an element of physical reality"|which went beyond the purviewof quantum mechanics. This argument supported the then prevail-ing view that quantum mechanics was like statistical mechanics:it gave useful, statistical information about what happened in alarge number of cases, but fuller information about each individualparticle was in principle available, and from it a completely deter-ministic account of its future trajectory could be calculated. Thefuller information was comprised by the hidden variables, which,together with those already in use in quantum mechanics, wouldbelong to a complete theory, containing quantum mechanics as apart.

15 See below x11.9.16 A.Einstein, B.Podolsky and N.Rosen, \Can Quantum-Mechanical Descrip-

tion of Reality be Considered Complete?", Physical Review, 47, 1935,pp.777-780; reprinted in J.A.Wheeler and W.H.Zurek, eds., Quantum The-ory and Measurement, Princeton, 1983, pp.138-141.


Modern versions of the EPR (Einstein-Podolsky-Rosen) argu-ment use examples based not on position and momentum, buton \spin"|a quantum-mechanical concept akin to polarization inclassical optics. Two photons, which originally were together andpolarized in some direction, are separated. When they are at a con-siderable distance from each other, and so, presumably, no longerinteracting, each comes to a polaroid screen, which will either letit through or absorb it, always letting it through if the angle of po-larization of the screen is the same as the angle of polarization ofthe photon, always absorbing it, if the angle of polarization of thescreen is at right angles to the angle of polarization of the photon,and if the angle of polarization of the screen is at �o to the angle ofpolarization of the photon, having a probability of cos2� of gettingthrough, and a probability of sin2� of being absorbed.17 Accord-ing to quantum mechanics the polarization of the two photons iscorrelated: that is to say, if the two polaroid screens are both setwith their direction of polarization the same, then either both pho-tons get through, or both photons are absorbed. This correlationis analogous to the assumption in the original EPR paper that thetwo particles had the same momentum, though in opposite direc-tions. The correlation decreases if the polaroid screens are inclinedto each other, and if they are at right angles, then a photon getsthrough one if and only if its partner does not get through theother. In general, if the polaroid screens are inclined to each otherat an angle of �o, the probability of the photons either both gettingthrough or both being absorbed is, cos2�o.

Einstein argued that this correlation could not be due to thepolaroid screen on one side in uencing the behaviour of the pho-ton on the other side, and that it must be due to each photonhaving acquired some common property while they were interact-ing, which would account for their linked behaviour subsequently.But this cannot be so. If, for instance, each photon had the samedirection of polarization, then IF the polaroid screens were bothoriented to be aligned in that direction, we should, indeed, �ndthat they both would let through photons from the same pair; andsimilarly IF they were both aligned at right angles to the directionof the photons' polarization, they both would absorb photons from

17 I am greatly indebted to T.Maudlin, Quantum Non-Locality and Relativity,Oxford, 1994 and 2002, for this exposition.


the same pair: But if the polaroid screens were set at di�erent an-gles to each other, discrepancies would inevitably occur, since theircorrelation, according to quantum mechanics, would depend onlyon the alignment of the screens with each other, while according tothe hidden variable hypothesis, it should depend on the alignmentof each screen with the supposed direction of polarization of thephoton pair. The mathematics works out easily if we take eachpolaroid screen being aligned at an angle of 30o, one clockwise andone anti-clockwise to the supposed direction of polarization of thephoton pair, so that the screens are inclined at 60o to each other.Then, since cos260o = 14 , and cos230o = 34 , the probability of apair of photons of both getting through or both being absorbed isonly 14 according to quantum mechanics, but would be 34 � 34 ;= 916 ,were the hidden variable hypothesis true.

Against Einstein-Podolsky-RosenQuantum mechanics says that the results of making mea-surements on correlated entities depends only on the set-tings of the measuring apparatus.The hidden-variable theory says that the results of makingmeasurements on correlated photons depend on the direc-tion of polarization of the photons as well as the settingsof the measuring apparatus.In some circumstances the supposed e�ect of the directionof polarization on the two results does not cancel out.So the hidden-variable theory is incompatible with quan-tum mechanics.It is for experiments to determine which is right.

This is only one example of a hidden variable account, andmakes many assumptions which can be faulted. J.S. Bell producedan entirely general argument to show that no hidden variable ac-count could be consistent with quantum mechanics.18 Instead ofconsidering correlated photons, whose passing through or absorp-tion by a polaroid screen could be measured, Bell considered anti-correlated particles whose spin in a particular direction could be18 J.S.Bell, \On the Einstein Podolsky Rosen Paradox", Physics, 1, 1964,

pp.195-200; reprinted in J.A.Wheeler and W.H.Zurek, eds., Quantum The-ory and Measurement, Princeton, 1983, pp.403-408.


measured; and instead of supposing that the hidden variable wasthe direction of polarization of the pair of photons, he merely sup-posed that there was some hidden variable �, which might be aset of variables or even a set of functions, but which, for conve-nience sake, was represented as a single continuous variable. It wasstill the case that each measurement could yield just two possi-ble results, which were symbolized as +1 and �1 (instead of \gotthrough" and \absorbed"). Thus there were two measurements tobe made, one on each particle, the results of which would dependon1 the settings of the measuring device,and, if the hidden variable hypothesis were true,2 �;and the result of each measurement would be +1 or �1.

Bell considers the case where each measuring device can be setto measure spin in two directions, which we can represent by ~aand ~a0 for the settings of measuring device A, and ~b and ~b0 forthe settings of measuring device B; Thus each particle could bemeasured in two ways, depending on the settings of the measuringdevices. If the hidden variable hypothesis were true, the resultof measuring will be a function of the hidden variable, �, and ofthe setting, which we can represent as: A(�;~a) or A(�;~a0), andB(�;~b) or B(�;~b0). For the sake of conciseness, let us representthese results as a; a0; b and b0, and the joint results as ab; a0b; ab0 anda0b0, remembering that these are, according to the hidden variabletheory, functions of �. Now consider the value of

ab+ a0b+ ab0 � a0b0:

It can be factorised as

a(b+ b0) + a0(b� b0):

Each term is �1, so that either (b+b0) = 0; or (b�b0) = 0; hence thewhole expression must have the value �2. If we take the modulusof this, we can say that the hidden variable theory assigns de�nitevalues to what would be the result of measuring the spin of the twoparticles, such that the function of those values:

jab+ a0b+ ab0 � a0b0j = 2:


But this result is inconsistent with quantum mechanics. Accord-ing to quantum mechanics the correlation between the results ofmeasuring the spin of two anti-correlated particles depends onlyon the settings of the measuring device, and if they are inclinedat an angle of �, the correlation is cos2�. If, once again, we aligntwo of the settings of the measuring devices, so that ~a = ~b, have ~a0inclined to ~a at an angle of 30o clockwise, and ~b0 inclined to ~b at anangle of 30o anti-clockwise, so that ~b0 is inclined to ~a0 at an angleof 60o, then ab = �1, a0b = � 34 , ab0 = � 34 and a0b0 = � 14 so that

jab+ a0b+ ab0 � a0b0j = 214 :

Bell's ArgumentQuantum mechanics says that the results of making mea-surements on correlated entities depends only on the set-tings of the measuring apparatus.Hidden-variable theories say that the results of makingmeasurements on correlated entities depend on the value ofthe hidden variable as well as the settings of the measuringapparatus.Bell constructed a function of the results of alternativemeasurements on an anti-correlated pair of entities. Ifthese results depended on the value of a hidden variable,they would be con�ned between certain limits. But forcertain pairs of alternative measurements, quantum me-chanics yields values outside those limits.Contrary to the EPR argument, quantum mechanics withregard to some functions is complete. There is no room toinsert a hidden variable.

Since Bell's original paper, there have been many variants putforward, with di�erent assumptions and simpler working.19 In par-ticular, Greenberger, Horne and Zetlinger have devised a proof in-volving three particles that avoid some of the doubts and di�culties19 See for example, J.F.Clauser, M.A.Horne, A.Shimony, and R.A.Holt,

Phys.Rev.Lett. 23, 1983, pp.880 �.; P.Heywood and M.L.G.Redhead,Fund. Phys., 13, 1983, pp.481 �.; A.Stairs, Philosophy of Science, 50,1983, pp.587 �.; H.R.Brown and G.Svetlichny, Found. Phys., 20, 1990,pp.1379 �.; R.Penrose, Shadows of the Mind, Oxford, 1994, pp.246-249.


raised by earlier versions.20 But the arguments are di�cult. Al-though we can be clear about what quantum mechanics predicts,it is harder to get a grip on hypothetical hidden variable theories.In particular we may well suspect that the Bell Inequality involvesmultiplying numbers which could only be elicited under incompat-ible conditions. If the measuring device A is set at ~a, it cannot beset at ~a0, and so the expression jab+ a0b+ ab0� a0b0j is a nonsense.But that is to misconstrue the nature of the argument. The ar-gument is a Reductio ad Absurdum. The hypothesis is that somehidden variable theory is true, rather than quantum mechanics.And in a hidden variable theory each particle had de�nite proper-ties, whether or not they are measured. On that basis it would beperfectly acceptable to consider what would be the result of mea-suring a property, even if the measurement is not, and cannot be,carried out. Hidden variables confer counter-factual legitimacy. So,granted the assumption of there being some hidden variable, thereare, under any conditions, de�nite values ~a and ~a0 at any giventime, even if we cannot discover what they both are. Hence also,IF some hidden variable theory is true, THEN jab+a0b+ab0�a0b0jis not a nonsense, and has a de�nite value, which is inconsistentwith some particular predictions of quantum mechanics.

Any standard hidden variable theory makes predictions whichare di�erent from those made by quantum mechanics for some ofthe possible settings of the measuring device. It could, of course,be that quantum mechanics was wrong|not merely incomplete,as Einstein supposed, but de�nitely wrong. Experiments to decidebetween quantum mechanics and a hidden variable theory were dif-�cult to conduct, but �nally Aspect and his collaborators in Pariswere able to devise one that excluded the possibility of some com-munication between the measuring devices at the speed of light,and was sensitive enough to allow for ine�ciencies in the measur-ing devices. The results were inconsistent with hidden variable the-ories, and vindicated quantum mechanics.21 The results, though

20 D.M.Greenberger, M.A.Horne, and A.Zetlinger, \Going Beyond Bell's The-orem", in Bell's Theorem, Quantum Theory, and Conceptions of the Uni-verse, ed.M.Kafatos, Dordrecht, 1989, pp.69-72; L.Hardy, Phys.Rev. Lett.,68, 1992, pp.2981 �.

21 A.Aspect, J.Dalibard, G.Roger, \Experimental Tests of Bell's Inequalities,using time-varying analyzers", Physical Review Letters, 49, 1982, pp.1804-1807.


not absolutely conclusive|further experiments might yield otherresults|seem decisive.22 In any case quantum mechanics has beencon�rmed by other experimental observations to a very high de-gree of accuracy. Any theory to supplant or supplement quantummechanics needs to have very strong credentials. These are con-spicuously lacking in those that call in question the assumptions onwhich either the Aspect experiment or the Kochen-Specker proofis based.

The EPR argument, intended to show that quantum mechanicswas incomplete, and needed to be supplemented by a hidden vari-able theory, achieved the opposite, proving that quantum mechan-ics was as complete as possible, that no reasonable hidden variabletheory could be true, that quantum mechanics was through andthrough probabilistic, and hence that physical determinism mustbe false.

In BriefQuantum mechanics cannot be supplemented by a hiddenvarible because:1. They won't �t in; we cannot paint discrete truth-values

onto a continuum of possible probabilities(von Neuman, Gleason and Kochen-Specker).

2. There is no room for them; (Bell and Aspect).

But there is an even more surprising import of Bell's argumentand Aspect's experiment: quantum mechanics is not only indeter-minist, but is also non-local.

22 See R.Penrose, Shadows of the Mind, Oxford, 1994, pp.247-248.


x11.6 Non-localityThe Einstein-Podolsky-Rosen argument assumed the Principle ofLocality: the behaviour of one particle could not be in uenced bythe setting of a distant measuring device. Aspect's experimenttook great pains to exclude in uences propagated with the speedof light. The result of his experiment showed that, even whenapparently separated, the two particles still formed one entangledquantum state. It might be possible to explain this by positingsome curious topology for quantum-mechanical systems, wherebyapparently distant entities were really quite close to each other,23but failing that, we should have to allow some instantaneous oralmost instantaneous transmission of information, quite contraryto the thrust of Einstein's Special Theory.

It is easy to misunderstand this result. Some people havethought that we could construct a \Bell telephone", by meansof which we could send \unstoppable and unjammable" messagesfaster than light.24 But we cannot choose what message to send.We can only choose what question to ask about the quantum-entangled system: we can choose which direction to set our spindetector; that is, we can choose whether to ask \Is it spin up orspin down? in the direction that points towards the Pole Star?",or \Is it spin up or spin down? in the direction that points to-wards Sirius?", or \Is it spin up or spin down? in the directionthat points towards Betelgeuse?", or in any other direction we like,but we cannot choose what answer we shall get. And it is onlythe answer, together with the question that it is the answer to,that determines the answer given a long way o� to a similar ques-tion. Aspect's experiments do not show that there is \action at adistance", but only \passion at a distance".25

Even though the non-locality revealed by Aspect's experimentsdoes not reinstate action at a distance, it has signi�cant impli-cations, some of them disturbing. If locality is not required inquantum mechanics, neither need it be required of hidden variabletheories intended to supplement quantum mechanics. The deter-ministic theories devised by Bohm are consistent with quantum

23 As in Minkowski spacetime; see above, x10.3.24 See N.D.Mermin, Physics Today, April 1985, p.38.25 I owe this phrase and much else to M.L.G.Redhead, Incompleteness, Non-

locality and Realism, Oxford, 1987; and 1989.


mechanics, operating with \pilot waves" to guide the particles intothe right paths. And even the Kochen-Specker contradiction couldbe escaped at the cost of further non-locality assumptions.26 Thesetheories are generally rejected out of hand, for failing to meet ournormal requirements of what a theory should be, much as the spoofhidden variable theory of x11.4 would be rejected as unworthy ofserious attention. But if quantum mechanics itself is non-local,why should we not accept alternatives to quantum mechanics thatare no more bizarre?

But non-locality is not bizarre|not unless Newton's theory ofgravitation was bizarre. Admittedly, Action at a Distance wasfelt to be a blemish, but one that could be accepted, given thetheory's intellectual elegance and empirical support. It was onlyin the Twentieth Century, with the advent of Einstein's Specialand General Theories, that the principle of locality was elevatedinto an absolute principle. Aspect's experiments force us to re-think again. Locality assumptions have two functions in scienti�cthought. They tell us of the relative unimportance of distant fac-tors, and require that all causal in uences be mediated throughintervening distances by some process. The former is importantwhen we are trying to isolate a cause and e�ect:27 the latter is aregulative principle we have for an explanation's being satisfactory.More generally, we can see them as the spatial surrogates of theanalytical method. Analysis yields understanding by concentrat-ing on the parts separately. If they are spatial parts, concentratingon any one means not attending, or not attending very much, toany other. And this will not lead to our ignoring relevant factorsso long as any in uence distant factors have is mediated throughthe part under our immediate observation. Locality assumptionshold out a promise of explicability, and provide a means of accom-modating extraneous in uences, but are not essential to science.Newton's law of gravitation outed it, and if Aspect's results canbe explained only as some form of passion at a distance, so be it.

Nevertheless, the violation of locality goes deep. It makes quan-tum mechanics appear to be much more holistic than other scien-ti�c theories, not just with regard to Aspect's experiment, but inall case of quantum entanglement. Quantum systems are typically

26 M.L.G.Redhead, Incompleteness, Non-Locality and Realism, Oxford, 1987,pp.137-152.

27 J.R.Lucas, Space, Time and Causality, Oxford, 1984, p.57.


not separable into their constituent parts, but have to be consid-ered as a single whole. And that is an important message forthe metaphysician. We cannot always follow Descartes' analyticalmethod.28 Valuable though it often is to analyse problems andthings, considering each part separately, it cannot always yield thewhole truth, and therefore should not be applied without regardto circumstance. We should follow Plato rather than Descartes,and recognise that the philosopher needs to be ��o��o�& (sunop-tikos), taking the broad view as well as the narrow one on accountof the inter-connectedness of things.

Bell and AspectBell's argument and Aspect's experiments show not onlythat quantum mechanics is indeterminist,but that it is non-local.

Non-locality is important also for our understanding of time.Throughout the twentieth century it was widely believed that theSpecial Theory had demoted time. Not only was time merely oneof four dimensions of spacetime, but the distinction between past,present and future depended on the choice of frame of reference.There were no hyper-planes of absolute simultaneity, and one ob-server might regard a distant event as present, another as past,and a third as something still to take place. The Special Theory, itseemed, constituted a conclusive argument against the importanceof tense. But the \conclusive argument" depended on the premisethat the speed of light was an absolute maximum. If superluminalspeeds are possible, particularly if there are instantaneous trans-missions of information, the argument against the reality of tenseis broken; but further consideration is needed before we have apositive argument in favour.

28 Descartes, Discourse on Method, Part II, tr. J.Veitch in Everyman ed.,London, 1912, pp.15-16; E.Anscombe and P.T.Geach, Descartes Philo-sophical Writings, London, 1954, pp.20-21; J.Cottingham, R.Stootho�,D.Murdoch, Descartes: Selected Philosophical Writings, Cambridge, 1988,p.120; AT VI p.18.


SummaryAccording to quantum mechanics the correlation between the re-sults of measuring the spin of two anti-correlated particles dependsonly on the settings of the measuring device, and not on any valueof a supposed hidden variable.If the measuring devices have a space-like separation, and are notin uenced by any hidden variable, the propagation of causal in u-ence between the measuring devices must be instaneous, or at leastmuch faster than the speed of light.

|o0o|Perhaps we should not mind it too much: perhaps the SpecialTheory concerns only electromagnetic phenomena, and the instan-taneous transmission of in uence is unobjectionable elsewhere inphysics.

x11.7 The \Measurement Problem"�

Thought about quantum mechanics has been much confused by theso-called \Measurement Problem". It is a problem, but it is notprimarily a problem about measurement, though we can see howin the intellectual atmosphere of the time, it came to be construedin terms of our assigning values to physical magnitudes.

One of the strongest arguments for the subjective view of prob-ability was that it obviated the problem. A quantum state, orwave function, j i, is typically a superposition of di�erent pos-sible states, or wave functions, which, on a measurement beingmade, instantaneously or almost instantaneously collapses into asingle one. It is di�cult to make physical sense of this, but easy ifthe original j i was a probabilistic characterization of an ensem-ble, with the measuring process giving fuller information about oneparticular member of the ensemble. Probability characterizationsdo not move at all, but can change instantaneously on receipt of� In this section especially, as throughout this chapter generally, I am con-

scious of my ununderstanding of quantum mechanics. I stick my neck outat my peril. But if I am proved wrong, knowledge will have been advanceda little, whereas if I remain safely silent, my reputation may be intact, butignorance will remain unabated.


new information. I may know that a colleague is either in his roomsin college or else in the Bodleian library, and so the correspond-ing probability distribution is concentrated on his being in thesetwo areas. If I then telephone him in college, and he answers thephone, the probability distribution for his being in the library im-mediately vanishes, and that for his being in his room increases tounity. But it would be absurd to say that one part of the probabil-ity distribution had instantaneously travelled from the library intothe college, up the stairs and into his room. The instantaneouscollapse of the wave function is not due, on this interpretation ofquantum mechanics, to something's moving with a speed greaterthan that of light, because a wave function is only a probability,and a probability is not a thing at all.

The subjective view of probability looked like working for thosequantum interactions which take place when we were measuringsome physical magnitude of a quantum system, since in that caseour concern is only with information, and if j i gave only the prob-ability distribution of an ensemble of similar particles, there wouldbe no need to speak of an instantaneous motion of anything. Butmost quantum interactions are not measurements. We measurequantum systems only occasionally, but they are interacting all thetime. When we think about quantum mechanics, we too readilyabstract from the messiness of real life, and consider ideal cases,arti�cially isolated from outside interference. These are illumi-nating, but untypical. Except in carefully designed experiments,quantum systems are constantly subject to interference, and beingmade to interact with other systems. When they do, they becomeenatngled, and often \collapse" into an eigen-state of the interfer-ing system. In physics laboratories, it may well be a measuringdevice, with eigen-states corresponding to the eigen-values of thephysical magnitude being measured. Outside physics laboratories,where the interfering system has not been carefully designed, itseigen-states may be entirely di�erent from those of any measuringdevice ever designed, and may have eigen-values of no interest toany physicist.

Whether or not the interfering process is designed to elicit ameasurement, there is a problem: the system evolves discontin-uously|the \collapse of the wave function"; instead of being asuperposition of many di�erent eigen-functions, it jumps into be-ing a single eigen-function, contrary to the Schr�odinger equation,which prescribes continuous development. It is di�cult to make


sense of this. Roger Penrose suggests that the e�ect of gravitationis responsible: if gravitation is expressed as in the General Theoryas a curvature of spacetime, it is intuitively plausible that a super-position of two diferently curved spacetimes is unsustainable, andmust result in one or the other being actualised. Adding second-order terms to the Schr�odinger equation, might represent the e�ectof gravitation;29 Others have suggested that the universe is �lledwith mini-projectiles, which knock the system of course and into aneigen-state;30 an intuitively more acceptable account invokes the\Gambler's Ruin" of probability theory.31 But probability theorycan be invoked in a di�erent respect.32 We talk of probabilities asthough they were precise, but recognise in our use of them thatthey are imprecise.We give odds of three to one, but do not thinkto distinguish them from a marginally di�erent �gure; if there is a�fty-�fty chance of something happening, we often will put in anàbout'. Nor is this only a matter of uninformed usage. The in-ference from frequencies to probabilities will not go through unlesswe are ready to identify truth with the neighbourhood of 1, andfalsity with the neighbourhood of 0.33 If we accept that probabili-ties are blurred concepts, we can see that in an interaction with alarge system, there will be many, many eigen-vectors, so that thestate-vector of the system is likely to come into the neighbourhoodof some eigen-vector. We can picture the state-vector as a search-light beam. The beam moves. It is not a single ray, but a narrowpencil, with the light-intensity falling o� rapidly at the edge. Ifthe penumbra, so to speak, of the beam lights up an enemy aero-plane, the beam locks onto it with full intensity. Similarly, if the

29 Roger Penrose, The Emperor's New Mind, Oxford, 1989, and Shadows ofthe Mind, Oxford, 1994.

30 G.C.Ghirardi, A.Rimini, and T.Weber, Physical Review D, 34, 1986,pp.470�.

31 Philip Pearle, \Combining stochastic dynamical state-vector reductionwith spontaneous localization", Physical Review A, 39, 1989, pp.2277-2289.

32 See, more fully, J.R.Lucas, \Prospects for Realism in Quantum Mechan-ics", International Studies in the Philosophy of Science, vol.9, no.3, 1995,pp.225-234.

33 See above, x5.4.


state-vector of a system comes into the neighbourhood of an eigen-vector, it locks on to the eigen-vector. In most cases the di�erence,both then and subsequently, in the development of the state-vectoris minuscule, but in those cases where a measurement is really be-ing made, the accompanying di�erence can be great. Althoughthe projection of the state vector of the whole interacting systemplus measuring apparatus onto the small Hilbert sub-space of thesystem alone may initially lie well away from the projection of theeigen-vector, yet the small displacement in the large Hilbert spaceof comined system (the whole interacting system plus measuringapparatus) may rapidly lead to the state-vector of the small systemaligning itself along an eigen-vector.

The \locking on" metaphor is suggestive, but still only a meta-phor. Since the Schr�odinger equation is deterministic and preservessolid angles, something more is needed to give substance to lock-ing on. Sir Michael Dummett outlines an Intuitionist logic for afuzzy realism which may be appropiate to quantum states.34 Alter-natively, Philip Pearle's \Gambler's Ruin" provides a bridge fromcontinuity to discontinuity. if stochastic processes are continuallytaking place between di�erent eigen-vectors, then generally and inthe large, the results will mirror the initial probabilities, but occa-sionally an unlucky one will be ruined, and forced out of the game.This would happen appreciably often only when probabilities inthe neighbourhood of 0 or 1 were involved.

Thus far, we have only considered a conceptual argument aboutprobabilities. We may ask if there is anything physical to say.We need to consider not Hilbert space, but Schr�odinger's wavemechanics. Instead of a searchlight beam being nearly aligned withan eigen-vector, we should have a wave function and a Fourierexpansion of it with one of the terms having a frequency very closeto that of the wave function itself. We are familiar with closelyadjoining frequencies. In the old amplitude-modulated (AM) setsone could hear, besides the intended station, the output of otherswith neighbouring frequencies. The wireless set could resonate tothem, and if the amplitude was being altered, could select them out

34 See M.A.E.Dummett, Thought and Reality, Oxford, 2006; \Is time a Con-tinuum of Instants?", Philosophy, 75, 2000, pp.510-515. Dummett's gen-eral argument against physical magnitudes' having precise real-numbervalues is unconvincing: but that is no reason why some should not beimprecise.


of the incoming radio wave. In the quantum case the interactingsystem has very many eigen-frequencies, and it is likely that thereis one su�ciently close to that of the -function for it to resonateso strongly as to become the sole resonating frequency.35

To return from these highly speculative speculations. Quantum-mechanical systems are interacting all the time, and so their wavefunctions are collapsing all the time. Schr�odinger's cat is not leftas a superposition of dead cat and alive cat until the experimenterlooks in to see how she is: the whole experimental set-up is not|cannot be|properly isolated from outside interference, and is be-ing continually bombarded by cosmic rays, gamma rays, visiblelight and physical vibrations. Throughout the experimental periodthe superposition of the two cat states is being made to collapse:a cat-alive one if no radium atom has disintegrated and set o� theapparatus; and a cat-dead one if the apparatus has been triggered,and the hydrogen cyanide released. No superposition could existin the real world for more than a few nanoseconds without beinginterfered with by some adventitious factor and shaken down intosome eigen-state or other.

If quantum-mechanical systems are continually being confront-ed by a moment of truth, when various possibilities are winnowedout, leaving one de�nite state of a�airs, there is an ontologicaldi�erence between the future and the present and past. AlthoughI may not know it|may be unable to know it|-there is a de�nitefact of the matter, whether some atom in Betelgeuse has or has notemitted a photon. The date I should ascribe to it, in order to havemy theory of electromagnetism working smoothly, may depend onthe frame of reference adopted. But as of now, the atom either hasor has not emitted a photon: If it has not, the emission is still inthe future; if it has, the emission is either present or past.

On a realist construal of quantum mechanics, then, there is areal, world-wide distinction of tenses. Although according to theSpecial Theory there is no hyper-plane of absolute simultaneity, theSpecial Theory takes account only of electromagnetic phenomena.If we are to take account of quantum mechanics also, we see a

35 Other resolutions of the measurement problem have been suggested, mostnotably the Many-Worlds interpretation: but it is deeply implausible|it outs Ockham's razor to an extravagant degree.


world-wide hyper-surface36 of simultaneity, as interacting quantummechanical systems come to have de�nite eigen-values for theirphysical magnitudes instead of a probabilistic spread of possiblevalues. Tense is real.

Quantum mechanics is philosophically importantbecause:1. It refutes projectivism,2. It refutes determinism,3. It vindicates tense.

x11.8 Knowing and Being

The corpuscularian philosophy took no account of the problem ofknowledge: Newton thought of space as the sensorium of God, withGod knowing where everything was by immediate apprehension.The universe was bathed in divine omniscience, and we were en-couraged to take a God's-eye view of the world, in which we wereconcerned about how things were, but not about how we couldcome to know them. Quantum mechanics, by contrast, is all thetime raising the question \How do we know?". It arose originallyto accommodate di�culties in explaining radiation, the interac-tion between matter (very ontological) and radiation (light-like:rather epistemological); and works with \operators", representinginteractions with potential observers, rather than the observer-independent evolution of states. Of course, there are observer-independent states, the states on which the operators operate.But the states are remote from observation. We can posit their

36 In Newtonian mechanics it is a hyper-plane: hyper because it is three-, nottwo-dimensional; plane because Newtonian dates are entirely independentof position. If Einstein's General Theory is accepted, Spacetime is noteverywhere at, so that the hyper-surface separating the future from thepast is not a hyper-plane.


Classical physics o�ered a reasonably clear ontology, but no epis-temology. Newton took a God's-eye view of universe, knowing byimmediate omniscience where each point-particle was, and what isvelocity was. The question "How do we know?" was unaddressed,and unanswerable. Quantum mechanics by contrast includes theinteraction between knower and known as part of what it has toaccount for.

existence, but we cannot say what they are, unless they are op-erated on, when they seem to jump into an eigen-state of thatoperator.

There is a great range of questions we can ask: \Where is it?",\What is its momentum along the X-axis?" \What is its angu-lar momentum around the X-axis?" \What is its energy?", andvery many more. To any such question there will be a de�niteanswer; but the answer will not tell us what value the system hadin respect of that magnitude immediately before the question wasasked. Immediately before the question was asked, the system hadonly various probabilities of giving some one of the de�nite values(eigen-values) allowed by that question. Operating on the system|asking the question|forces the system to get into an eigen-stateof the operator|to give a de�nite answer by ticking one of theboxes provided on the questionnaire. Whereas in classical corpus-cularianism coming to know the state of a system did not seriouslya�ect what the state of the system actually was,37 in quantum me-chanics the process of coming to know does alter the state of whatwas known. After the alteration the system will have some de�niteeigen-value of the operator that brought about the alteration: thatis, from the system there will have been elicited a de�nite answerto the question asked. The answer will tell us what the state ofthe system is then, as a result of the question being asked, but notwhat it was before the question was asked.37 Thus Niels Bohr, Essays 1958-1962 on Atomic Physics and Human Knowl-

edge, London, 1963, p.59: Ultimately, this viewpoint rests on the �nenessof our senses, which for perception demands an interaction with the objectsunder investigation so small that in ordinary circumstances it is withoutappreciable in uence on the course of events. In the edi�ce of classicalphysics, this situation �nds its idealized expression in the assumption thatthe interaction between the object and the tools of observation can beneglected or, at any rate, compensated for.


x11.9 The Uncertainty PrincipleThe Uncertainty Principle is the best known corollary of quantummechanics, but also widely misunderstood. Its name, and the �rstexpositions of it, have made it appear to be an epistemological the-sis, a limit on what can be known. We cannot know the preciseposition and momentum of a quantum mechanical entity, becauseour bumbling attempts at determining the position alter its mo-mentum, and vice versa. But true though that may be, it doesnot go to the heart of the matter, for however skilful and unbum-bling we were, we still could not determine the precise position andmomentum of a quantum mechanical entity, because they are notthere to be known. Not even God can know the precise position andmomentum of a quantum mechanical entity. The trouble lies notin the apparatus|which might be improved|but in the questions,which exclude each other. It is like the problem we have tuningan old-fashioned, amplitude-modulated (AM), wireless. The moreclosely we tune in on a radio station, the more blurred the acousticresponse, and vice versa. It is not just that our condensers andaerials are imperfect|though they are|but that all there is inreality is a very complicated radio wave which we can analyse indi�erent ways. If we consider it as a very precise station frequency,we cannot register relatively rapid variations in amplitude, becausethey would constitute an alteration of the station's frequency. If,on the other hand, we want to take note of such acoustic vari-ations of amplitude, we have to take account of frequencies in aband around that of the station, and thus may pick up unwantedmessages from neighbouring stations. The di�culty lies not in theapparatus, but in the choices we make. So too with a quantummechanical entity. The more we regard it as localised in one spot,the less we can ascribe a single de�nite momentum to it; and themore we see it as moving with a uniform speed, the more widelydispersed it seems, and less concentrated in any one position.

Since the Uncertainty Principle is not an epistemological oneabout the limits of our knowledge, but an ontological one, aboutthe nature of the known, we ask how deeply it is embedded inquantum mechanics. At one level we can show how it emerges sofar as position and momentum are concerned, from the positionand momentum operators.38

38 Readers who dislike formulae can safely skip the next paragraph.


The position operator, X, is given by Xj i = xj i:the momentum operator, P , is given by P j i = �i�h @

@x j i.SoXP j i = x(�i�h @

@x j i);

andP Xj i = �i�h@x@x j i � x(i�h @

@x j i):

HenceXP j i � P Xj i

= x(�i�h @@x j i)� (�i�h@x@x (j i)� x(i�h @

@x j i))

= +i�hj i;or, leaving out the j i,

XP � P X = i�h:

From this we can see that the uncertainty principle is a con-comitant of quantum mechanics having the operators it has. So faras the j i-functions are concerned, all is determinate, just as arethe wireless waves outside the house. It is the way the macro world,for example our apparatus, interacts with them that produces theambiguity. If we adopt the Hilbert-space model, we can locate thetrouble in di�erent operators having di�erent sets of orthogonalaxes associated with their eigen-vectors. Operators with the sameset of orthogonal axes commute: roughly we can say that as a re-sult of the �rst operator, the system is in an eigen-state of thatoperator, and on applying the second operator, since the system isalready in an eigen-state of that operator too, it remains so.

More carefully,39 if the two operators, F and G, have the sameeigen-functions, we can expand the j i-function as a sum of thoseeigen-functions:

j i =X

anj in;

and work out the result of operating �rst with G and then with F :

F Gj i = F GX

anj ni = FX

anGj ni = FX

angnj ni

39 Readers who dislike formulae can again safely skip this paragraph.


=X

F angn =X

angnfn:

If we then work out the result of operating �rst with F and thenwith G:

GF j i = GFX

anj ni = GX

anF j ni = GX

anfnj ni

=X

Ganfn =X

anfngn;

we see that the result is the same, since fn and gn are real numbers,and their product is the same irrespective of their order.

Thus we see that all operators would commute if they all hadthe same eigen-vectors, but only then; that is, only if there werea preferred set of axes in Hilbert space, corresponding to a pre-ferred set of frequencies for the wave functions. We are led thento wonder whether there could be a theory analogous to quan-tum mechanics, but having all the operators commuting. Thereis no compelling reason for thinking that Hilbert space should beaxes-indi�erent. And thinkers have speculated that there might besome \unique basic rhythm of the universe".40 If this speculation isrejected|and the fact is position and momentum operators do notcommute| and we suppose the underlying waves to be as amor-phous and featureless as possible, it follows not only that thereare non-commutating operators in quantum mechanics, but, whatis often not su�ciently recognised, that there are more than justtwo families. It is not just that position and momentum are \com-plementary", but that in general eliciting the value of one classicalphysical magnitude precludes eliciting the value of many other clas-sical physical magnitudes, which often are similarly \incompatible"among themselves.

We are led also to speculate on the connexion between the op-erators and the classical physical magnitudes (or dynamical vari-ables) the values of which the operators elicit. Why does �i�h @

@x j itell us what the momentum, P , of a system is? The short answeris that it is a carry-over from classical mechanics when articulatedin its Hamiltonian form. Only if we have the operators we dohave, will the physical magnitudes they elicit from the underlyingquantum states have the relations with one another they need tohave, if they are to be plausibly identi�ed with the classical physi-cal magnitudes we already know. There is force in this contention:but still it leaves us hankering after a deeper explanation yet to bediscovered.40 G.J.Whitrow, The Natural Philosophy of Time, Edinburgh, 1961, ch.1,

p.46. See below, x12.8.


x11.10 Nullary QualitiesThere are very many self-adjoint Hermitian operators that couldoperate on a -function, or j i, and elicit one out of the set ofeigen-values for that operator. Only a few of them are of interestto contemporary physicists. The classical physical magnitudes arenot quite the same as the primary qualities, and spin is altogether anewcomer, though it might claim a distant parentage in Descartes'vortices.

The nullary qualities are not operators, but the characteristicsof the -functions, or j i vectors, themselves. These are peri-odic functions (represented by rays in Hilbert space), which canbe characterized in terms of a set of basic sinusoidal functions (ororthogonal axes in Hilbert space). There is no one preferred setof basic sinusoidal functions (or orthogonal axes in Hilbert space)but a wide variety of di�erent ones, with di�erent operators hav-ing their own set of basic sinusoidal functions (or orthogonal axesin Hilbert space). It follows that any characterization of basicsinusoidal functions (or orthogonal axes in Hilbert space) is notcanonical, being no more fundamental than any other. The basicentities are the -functions, (or j i vectors) themselves, togetherwith the fact that they are periodic, (or that they are vectors in aHilbert space).

It is easy to see that these entities are more abstract and remotefrom ordinary experience than the primary qualities of classical cor-puscularianism, but more di�cult to make out that they are morerational. Yet this can be argued. For whereas with the ordinaryspace of classical corpuscularianism, it appears to be a contingentquestion whether it is Euclidean or not,41 with wave functions (ortheir Hilbert-space representation) it is built into the very conceptof such a space that such a theorem should hold. Quantum me-chanics is thus purer than classical corpuscularianism. Instead ofprimary qualities, it has periodic functions, which under Fourieranalysis turn out to be subject to Parseval's theorem (or vectorsin Hilbert space, likewise subject to a profound Pythagorean rule).And although we use the traditional trigonometric names, sine andcosine, in discussing periodic functions, they can be characterizedpurely mathematically, without any appeal to everyday geometrywith its Euclidean presuppositions.

41 See above x9.7.


The di�erential calculus gives us the general rule for the di�er-ential coe�cient of xn, namely

dxndx = nxn�1

which, conversely, gives us an easy rule for integrating all polynomi-als|except those of the form x�1, that is to say, except those of theform 1

x . Yet clearly there is such a function, log(x), which can bede�ned in terms of its di�erential properties (by considering Rx

0 dyy ),

and it has an inverse, exp(x), which can be written ex, where eis the base of natural logarithms, 2:71828 : : : Thanks to Taylor'stheorem, we can expand the exponential function as Maclaurin'sseries:

ex = x0 + x1 + x2

1� 2 + x31� 2� 3 + : : :+ xn

n! ;(where n! = n� n� 1� n� 2� : : : 3� 2� 1)

We then consider the function eix, where i = p�1, which hasthe most remarkable property, discovered by Euler,

ei� = �1;from which it follows that e2i� = (ei�)2 = 1 and hence thate(x+2ni�) = ex. Thus we see that eix is a periodic function, withperiod 2�.

If we expand eix as Maclaurin's series, remembering that i2 =�1, and separate the real and the imaginary parts, we have

eix =�1� x2

2! +x44! : : :

�+ i

�x� x3

3! +x55! : : :

�;

which mathematical readers will recognise as cos(x)+i sin(x), or, asit is sometimes written, cis(x); only, whereas the expansions of cosand sin are ordinarily derived from their trigonometrical properties,here the expansions are used to de�ne the functions.

It is easy then to see that if we consider the expansion of e�ix,since the real part consists entirely of even powers of x, it will bethe same, that is cos(x), while the imaginary part will have all itssigns reversed, but will otherwise be the same; that is it will be� sin(x). So e�ix = cos(�x) + i sin(�x) = cos(x)� i sin(x)Also since eix � e�ix = eix�ix = 1 it follows that(cos(x) + i sin(x))� (cos(x)� i sin(x)) = cos2(x) + sin2(x) = 1;

which is normally taken to be an expression of Pythagoras' theoremin trigonometric terms. To this extent, then, quantum mechanicscan claim that its entities are more rational than those of classicalcorpuscularianism or modern materialism.


Thanks to EulerEuler's theorem (writing p�1 rather than i)

ep�1� = �1

shows us:1. that e

p�1� � ep�1� = �1��1 = +1

so that e2p�1� = 1

and hence that e2np�1�x = e

p�1xthat is to say e

p�1x is a periodic function of x with periodn;

2. that if we expand ep�1x, separating its real and imaginary

parts, as cos(x) +p�1sin(x);then since e

p�1x � e�p�1x = e(1+�1)p�1x = e0 = 1,

(cos(x) +p�1sin(x))� (cos(�x) +p�1sin(�x)) =(cos(x) +p�1sin(x))� (cos(x)�p�1sin(x)) =cos2(x) + sin2(x) = 1,which is the trigonometric version of Pythagoras' theorem,and can be generalised to any number (even an in�nite num-ber) of dimensions.

The �rst gives a deep way of combining continuity with dis-creteness, the second a deep way of securing that probabilitiesadd up to unity.

Thank you, Leonhard Euler

x11.11 Indiscernability and HaecceitasLocke held impenetrability to be a primary quality. It caused dif-�culties, but played an essential role in individuating corpuscles.It was a key feature of corpuscles in classical corpuscularianismthat each corpuscle could be individuated, having its own identity,and being in principle capable of being re-identi�ed as the samecorpuscle as it was before. Newton re-wrote Genesis to start withthe sentence: In the beginning God created atoms and the void.Although all the corpuscles were, in classical corpuscularianism,qualitatively identical, they were numerically distinct, and couldbe distinguished from each other by the requirement that two cor-puscles could not occupy the same position at the same time.


The atoms and molecules of classical mechanics give rise toMaxwell-Boltzmann statistics. In calculating the thermodynamicproperties of gases, we count as two di�erent cases the case inwhich this molecule is here and that molecule is there, and thecase in which that molecule is here and this molecule is there. Inquantum mechanics, however, such calculations give results whichare not borne out by experiment. To get the right results we needto use Bose-Einstein (or Fermi-Dirac) statistics, in which we countas only one case that in which this photon is here and that photonis there, and that in which that photon is here and this photonis there. The natural interpretation is that photons, and other\Bosons", (and electrons and protons and other \Fermions"), arenot entities to which the words `this' and `that' can be meaningfullyapplied. Indeed, as Bohr points out, \whenever customary ideasof the individuality of the particles can be upheld by ascertainingtheir location in separate spatial domains, all application of Fermi-Dirac and Bose-Einstein statistics is irrelevant . .".42

We are led to a similar conclusion by re ection on the Heisen-berg Uncertainty Principle, for it precludes our being able to usethe classical criterion for re-identi�cation: if the position of a sub-atomic particle is sharply de�ned, its momentum, and hence itstrajectory, is unde�ned, so that there is no spatio-temporal contin-uous path by means of which we can tell whether an entity at onetime is the same as, or di�erent from, an entity at another time.43

Bosons and fermions, then, lack \this-ness", or haecceitas,\haecceity", as Duns Scotus and the later Schoolmen called it. Al-though grammatically nouns, they have the logic of adjectives. Ifwe are considering a photograph, we may well think that we wouldhave a di�erent photograph if that black patch were exchanged withthis white one. But if that black patch were exchanged with thisblack one, we should not have a di�erent photograph, but the sameone. Instead of regarding them as particular particles, we should

42 Niels Bohr, Essays 1958-1962 on Atomic Physics and Human Knowledge,London, 1963, p.91.

43 For a much fuller account, in which alternative conclusions are explored,see Steven French, \Identity and Individuality in Classical and QuantumPhysics", Australasian Journal of Philosophy, 67, 1989, pp.432-446. Seealso Steven French, \Identity and Individuality in Quantum Theory" Stan-ford Encyclopedia of Philosophy, February 2000, http://plato.stanford.edu


think of them as excitations of a �eld: a �eld which is electroni-cally excited here and electronically excited there does not becomea di�erent �eld if it is electronically excited there and electronicallyexcited here.

A useful parallel is modern money. Although in the old dayswhen all transactions were made by means of coins, it would havebeen possible to trace the history of an individual coin, it is impossi-ble to do so now in the age of cheques and credit transfers. If I earn$100 from writing an article, and $100 from examining a thesis,and spend $100 on my electricity bill and $100 on my water bill,there is no saying which incoming $100 went to pay which utilitybill. Lawyers use the word `fungible' of goods whose whole pointis the function they perform, and no distinction is made betweennumerically distinct items which perform the function equally well.If I borrow a pound of caster sugar from you, I have adequatelydischarged my debt by giving you a pound of caster sugar, eventhough it is not the self-same pound|which my family has eaten|but a pound of the same sort and quality. Money is a peculiarly aptcomparison, since on account of its historical origin, it is invariablydiscrete. We can pay someone one hundredth of an American dol-lar, but not one seventh or one thousandth. Quantum mechanics isdiscrete too. Moreover, suitably favoured systems can sometimesrun short-time overdrafts, and \borrow" enough energy from theirneighbours to get them out of a hole.

Aristotle held that it was characteristic of substance that itcould be referred to as `this': �� o�-�� o�� o�� (Pasa ousia dokei tode ti semainein).44 And converselywhat lacks individual identity lacks substantiality. The subatomicentities of quantum mechanics are not substances; they lack thehard reality of the corpuscles of classical corpuscularianism andthe atoms of modern materialism. The ontological argument formaterialism fails.45 Although it remains true that we are made ofmatter, that truth is no longer the end of the story. Ultimately weare constituted not of separate impenetrable atoms, each pursuingits own path, as it collides or interacts with other atoms, but ofa shimmering ux of superposed excitations, ��!� �- ��o� �� (kumaton anerithmos gelasma), the innumerable laughterof waves.46

44 Categories, 3b10.45 See below, x13.6.46 Aeschylus, Prometheus 90. Compare Keble's \many-tinkling smile of


Quantum mechanics is philosophically importantbecause:1. It refutes projectivism,2. It refutes determinism,3. It vindicates tense,4. It refutes classical materialism.

x11.12 Quantum RealismThe insubstantiality of quantum entities raises doubts about theirreality. and has been taken by some to support some form of anti-realism. The fact that nullary qualities are more recondite thanprimary qualities, seems to tell against the reality of whatever itis that constitutes the stu� of quantum mechanics. Throughoutthe twentieth century philosophers have been drawn to instrumen-talism when they try to think about quantum mechanics. ThusSir Michael Dummett writes \. . . instrumentalists . . . . .regard theoretical entities as useful �ctions enabling us to predictobservable events; for them, the content of a theoretical statementis exhausted by its predictive power. This is one case in whichthe view opposed to realism is made more plausible by empiricalresults; for a realist interpretation of quantum mechanics seemsto lead to intolerable antinomies."47 But the antimonies are in-tolerable only if we take the primary qualities|the categories ofclassical corpuscularianism|as paradigms of reality. IF it is an es-sential mark of reality to be an individual located in space, to havea de�nite velocity, and to trace out a spatiotemporally continuouspath in spacetime, then subatomic entities fail to be real. But sodo the sounds and scents and colours of our everyday world. Scien-tists have to work hard to meet Berkeley's criticisms of their silentand colourless universe: and if by argument they can vindicatethe claims of classical corpuscularianism and modern materialismto reality, they must allow the possibility of quantum mechanics'

Ocean", and Einstein's more prosaic \There is no place in the new kindof physics for both the �eld and matter, for the �eld is the only reality."quoted by R.Sorabji, Matter, Space and Motion, London, 1988, p.40: andMilli�c �Capek, The Philosophical Impact of Contemporary Physics, Prince-ton, 1961, p.319.

47 The Logical Basis of Metaphysics, London, 1991, p.6.


making good a similar claim to a comparable reality. Quantumsystems do not have sharp values for both position and momentumat the same time, and this runs against our belief that real entitiesshould have primary qualities. But once we go beyond primaryqualities, there is no call on real entities to have them, any morethan Newtonian atoms need to have determinate colours. More-over, although quantum-mechanical states do not in general possessde�nite classical physical magnitudes, the -function, j i, is per-fectly determinate, and can be seen as an entirely determinate rayin Hilbert space, with a de�nite direction, or as a superpositionof wave functions with di�erent frequencies and di�erent weights.It is just that the relevant description is not in terms of standardprimary qualities, such as position and momentum. And even ifthe single ray is replaced by a slightly fuzzy beam, it is still mod-erately determinate. To this extent, then, quantum-mechanicalentities have a faceable claim to reality.

Still, quantum mechanics does lack the inexorable grittiness ofclassical corpuscularianism and modern materialism, being inde-terminist and lacking haecceity. But these, though they argueagainst determinist materialism, do not support a general anti-realism: rather, they shift the focus of realist concern upwards,from the sub-atomic entities themselves to the framework in whichthey operate. Instead of the time-dependent Schr�odinger equation,

i�h@@t = H = �( �h2

2m )� + U(x; y; z);

we should consider the time-independent Schr�odinger equation,

( �h2

2m )� + [E � U(x; y; z)] = 0;

which tells not how the probabilities of a quantum-mechanical sys-tem develop, but what con�gurations of energy-levels are possible.It is these that give us an understanding of physics and chemistry.Spectroscopy and valency are not explained by the history of par-ticular protons or electrons, but an understanding of the structureswithin which any photon or electron, that is, any photonic or elec-tronic excitation of the �eld, must operate. Quantum mechanicsnot only leaves room for a di�erent form of explanation, but posi-tively requires it.


Quantum mechanics is philosophically importantbecause:1. It refutes projectivism,2. It refutes determinism,3. It vindicates tense,4. It refutes classical materialism,5. It requires di�erent levels of explanation.

x11.13 Quantum PhilosophyQuantum mechanics has been a two-edged sword in philosophy. At�rst it supported a subjective or frequency theory of probability,with no great break with the received view of modern material-ism, but in the end it has entirely subverted these views, requiringan objectivist view of singular probability, and instantaneous, oralmost instantaneous passion at a distance.

The syntax of the calculus of probabilities was determined bythe need to marry Boolean and arithmetical algebra.48 If the con-junction of any proposition p with the truth-value TRUE is to bep, the probability assigned to TRUE needs to be 1; and if the con-junction of any proposition p with the truth-value FALSE is to beFALSE, the probability assigned to FALSE needs to be 0. Furtherargument shows that we can, without loss of generality, subtractthe probability from 1 to obtain the probability of the negation,have the multiplication rule generally for the conjunction of inde-pendent propositions or propositional functions, and the additionrule of the disjunction of mutually exclusive propositions or propo-sitional functions. It follows that the probability of all the alter-natives must add up to 1. Bernouilli's theorem and various Lawsof Large Numbers also follow, leading to the conclusion that in alarge ensemble the frequency of the various alternatives is likely tobe approximately proportional to their probabilities. Proportions,expressed as fractions, add up to 1, and this made Einstein's inter-pretation of quantum mechanics appear eminently reasonable. Itsfailure forces us to rethink: instead of a number of di�erent sortsof impenetrable, entirely separate, entities, we have a single entity,itself di�erently sorted; instead of proportions adding up to unity,we have the generalised Pythagoras' theorem:

cos2�1 + cos2�2 + : : : cos2�n + : : : = 1;48 See above, x5.3.


The general feel is inclusive rather than exclusive, elastic ratherthan rigid, holistic rather than exclusively analytical. Furthermore,although it may seem rather hard to accuse classical corpuscular-ianism and modern materialism of being static, there is a sense inwhich quantum mechanics is dynamic in comparison with them.Time is much less assimilated to space in quantum mechanics thanin its predecessors: it characteristically turns up as an independentvariable, most notably in the Schr�odinger time-dependent equa-tion, and it is tensed by the tide of collapse sweeping through theuniverse.

Quantum mechanics thus supports a metaphysics very di�erentfrom that of classical corpuscularianism and modern materialism.Instead of the principle of impenetrability, it has that of superposi-tion: instead of the hard Either Or, it accommodates Both And,allowing many di�erent possibilities to co-exist until the moment oftruth, when interaction forces a quantum mechanical system intoan eigen-state, with a de�nite numerical value for the appropriatemagnitude. It suggests a world of potentialities, the future beingopen to many di�erent possibilities, some of which become actualin the present, remaining �xed and unalterable thereafter.

Quantum mechanics is metaphysically importantbecause:1. It represents a further stage in pushing back the expla-

nation of things: no longer in terms of primary quali-ties, but of \nullary qualities".

2.(i) Instead of impenetrability, it has superposition,(ii) Instead Either Or, it has Both And,(iii) Instead of proportions, it has Pythagoras' theorem.

3. Not just analytic, but holistic too.4. Dynamic; takes time seriously.

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