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Consideration of the classic measurement paradox of quantum mechanics raises the question: What is the relationship between acceptable theories of the physical world at different levels? It is suggested that it is similar to the relationship between maps of different types and that, while theories at different levels need not be derivable from one another, they must at least be mutually consistent in their predictions. The relationship between quantum mechanics, in its standard interpretation, and classical physics fails this test. Existing attempts to resolve the measurement paradox are briefly reviewed, and it is suggested that one avenue has been insufficiently explored; namely, the possibility that the complexity of a physical system may itself be a relevant variable which may introduce new physical principles. Possible reasons for this lacuna are discussed, and it is pointed out that some of the relevant questions are now within the reach of an experimental test.

In this essay I shall try to defend three claims. The first is that the classic quantum measurement paradox, so far from being a non-problem, is a

sufficiently glaring indication of the inadequacy of quantum mechanics as a total world-view that it should motivate us actively to explore the likely direction in which it will break down. The second is that, as a consequence of ingrained reductionist prejudices and perhaps to some extent of sociological factors, we may have been looking in precisely the wrong direction. And the third is that we are already at the threshold of some very significant experiments in what just might turn out to be the right direction. The third claim is one which I have discussed rather extensively elsewhere, so I will deal with it here rather briefly, without much technical detail.

Let us remind ourselves briefly what the quantum measurement paradox is all about. A quantum-mechanical system drawn from an ensemble in a pure state is, according to the axioms as presented in most textbooks, most completely characterized by a wave function ψ, which may or may not be an eigenfunction of any particular quantity we wish to measure on it. Suppose it is not, and that the quantity in question is described by an operator  with eigenfunctions φi and eigenvalues ai. Then in the standard way we write ψ as a linear combination of the φi (the structure of the theory guarantees that this

can always be done):

[1]

and the prediction is then that, if the measurement in question is actually performed, the probability of obtaining the result ai is Once the result ai has

been obtained, the system must be assigned (in the case of an ‘ideal’ measurement) to a new ensemble whose wave function is φi. However, it is not

correct to think of the description [1] as implying that before the measurement the system was already in some (unknown) one of the ensembles whose wave functions are φi (i.e., in technical language, that its density matrix corresponded to a ‘mixture’ of the φi). To demonstrate the incorrectness of this

conclusion (or rather its incompatibility with the usual interpretation of quantum mechanics) it is sufficient to consider the results of a measurement of some quantity which fails to commute with Â. In general the ‘mixture’ description will predict a result quite different from that which one would expect on the basis of the pure-state wave function of equation [1].

There are, of course, a number of well-known thought-experiments which illustrate this feature of quantum mechanics, of which the best-known is probably the classic Young’s slits experiment. In this case the operator  in effect has eigenvalues corresponding to passage through one or the other of the two slits in the first screen, and the quantity is the position of arrival at the detecting screen. (Or more precisely the operator which evolves into this under the time evolution of the system; see reference 1, equation [4.4].) Actually, an even more spectacular experiment has been carried out in real life recently using a neutron interferometer.2 _{In this experiment a beam of spin-polarized neutrons was split into two beams, well separated from one another}

in space; the spins of the neutrons in one beam were flipped and the beams then allowed to recombine. By blocking off one beam at a time it was possible to examine the properties of the other at the point of recombination, and in this way it was explicitly established that the spin properties of the complete ensemble (the recombined beams) were quite different from those of the mixture of the two sub-ensembles (the separated beams). Since the average flux of neutrons was low

enough that the probability of two being close enough to affect one another’s dynamics was totally negligible, this experiment provides rather spectacular confirmation of the conclusion that one should not think of the neutron as ‘being in’ one or other beam until it has actually been measured to be so.

So, crudely speaking, even in situations where a measurement will reveal a microscopic system to have one of a set of possible values of a variable, quantum mechanics forbids us to conclude that it actually had that value before the measurement was made. This picture is no doubt paradoxical (in the original meaning of the term, ‘against expectation’) and might lead us to guess that we will some day obtain a more intuitively pleasing description, but in itself it is not obviously internally inconsistent, nor does it lead in any obvious way to any discrepancy with our everyday view of the macroscopic world—provided that we are prepared to accept the notion of ‘measurement’ as given from outside the theory. Indeed, Bohr,3 _{and with greater}

sophistication Reichenbach,4 were able to develop an interpretation of the quantum-mechanical formalism which is consistent within its self-imposed limits precisely by postulating a radically different ontological status for microscopic entities such as electrons or neutrons and the macroscopic apparatus which performs the measurement. In the words of a famous quotation from Bohr: ‘Atomic systems should not even be thought of as possessing definite

properties in the absence of a specific experimental set-up designed to measure these properties.’ On the other hand, Bohr also maintains that in order to secure unambiguous communication of our results, we must describe the macroscopic apparatus and its behavior in the language of classical physics (in which, of course, the apparatus must possess definite properties). Thus, it is the act of measurement that is the bridge between the microworld, which does not by itself possess definite properties, and the macroworld, which does. Indeed, according to most textbooks it .is the act of measurement which causes the ‘collapse of the wave function’ from a linear superposition into an eigenstate of the measured quantity. Thus, the concept of measurement is,

prima facie at least, absolutely central to the interpretation of the quantum-mechanical formalism.

Now, the problem is not that quantum mechanics itself provides no criterion for when a measurement has taken place. That would not necessarily matter if we could find some criterion elsewhere (and, as we shall see, some alleged resolutions of the measurement paradox do in effect try to do just this). The problem is that quantum mechanics absolutely forbids a measurement to take place, if by a ‘measurement’ is meant a process which has the features ascribed to it in the standard textbook account. This statement needs some amplification. The point is that, if we believe (as most physicists at least implicitly seem to) that quantum mechanics is a universal theory, then it applies not only

to single atoms and molecules but to arbitrarily large and complex collections of them, and in particular to the special collections which we have chosen to use as measuring devices (photographic plates, Geiger counters, etc.). So, although it is not obviously necessary to describe these objects, and their interaction with the microsystems whose properties are to be measured, in explicitly quantum-mechanical terms, it is at any rate legitimate to do so. Let us then initially (for the sake of simplicity of exposition only) assume that the measuring device starts in an initial state which is represented by a quantum mechanical wave function Ψ0. As before, we suppose that the property of the microscopic system which is to be measured is represented by a quantum-

mechanical operator  with eigenfunctions φi and eigenvalues ai. Suppose, first, that a particular microsystem entering the apparatus is drawn from an

ensemble whose quantum state is represented by the wave function φi. Then, if we are to be able to read off the value of ai from the behavior of the

macroscopic measuring device, the interaction between the microsystem and the device must induce the latter to make a transition into a different state, with a wave function we shall label Ψi. (Itispossiblethat one (but no more than one) of the Ψi is identical to Ψ0, in which case we speak of a ‘negative-

result’ measurement. This feature in no way affects the general argument.) Moreover, the different states Ψi must be not only mutually orthogonal but also

distinguishable by purely macroscopic measurements (e.g. inspection with the naked eye). (A discussion (considerably more sophisticated than the above one, cf. below) of how some common measuring devices fulfil this condition is given in reference 5.) At the end of the measurement process the ‘universe’ (system plus apparatus) is in a pure state χ which (for an ‘ideal’ measurement) is a product of φi and Ψi, i.e.:

[2]

The properties of the ‘universe’ are, as we have noted, macroscopically different for the different values of i. Now, what happens in the case that the microsystem was drawn from an ensemble whose quantum state is the linear superposition [1]? Quite irrespective of the details of the systems involved, it is a general feature of the quantum-mechanical formalism that if, under specified conditions, an initial state evolves into a final state χi, then a given

superposition will evolve into the corresponding superposition of final states, i.e. into ∑ciχi. Thus, from [2], we have:

[3]

The right-hand side of [3] is a linear superposition of states of the universe possessing macroscopically different properties. If we interpret a linear superposition at the macroscopic level in the same way as we

have learned to do at the microlevel, then the only possible interpretation of [3] is that the macroscopic state of the universe is not well-defined until some

further, unspecified, ‘measurement’ is performed. In other words, the notion of ‘measurement’ has on closer inspection dissolved before our eyes; there

is no magic ingredient in the process of interaction of a microsystem with a measuring device which could lead to the reduction of the wave packet postulated in the standard textbook discussions of the axioms of quantum mechanics.

It is necessary to remark that the above discussion of the quantum-mechanical description of the measurement process is, of course, quite naive. In the first place, in real life we rarely if ever know enough about the initial state of the apparatus to assign to it a single pure quantum-mechanical state, and would need in practice to describe it by a density matrix; the final state of the universe would therefore also be described by an appropriate density matrix. Secondly, the case where the state of the microsystem is unchanged by its interaction with the measuring device (‘ideal’ measurement) is the exception rather than the rule, and the description of the final state may have to be modified to allow for this. Thirdly, the measuring device is in practice itself an open system which interacts with outside influences such as the vacuum electromagnetic field, and the ‘universe’ which is described by the superposition [3] (or the appropriate density-matrix generalization) contains many other such degrees of freedom. These are technical details which do not (at least in the present author’s opinion) in any way blunt the force of the paradox, and will not be discussed here. (For a conclusive refutation of the conjecture that taking account of the first feature would resolve the paradox, see reference 6. The second feature has never to my knowledge been exploited in any alleged resolution. The third, which has, is discussed by implication below.)

The quantum measurement paradox, then, consists in the fact that an extrapolation of the quantum-mechanical formalism to the scale of the macro- world leads under certain circumstances to a description, namely equation [3], which is prima facie quite incompatible with the commonsense everyday picture we have of the world around us. In a nutshell, in quantum mechanics events don’t (or don’t necessarily) happen, whereas in our everyday world- view they certainly do: the Geiger counter does or does not fire; the photographic plate is or is not blackened at a definite point; and so on. So the first question we might ask ourselves is: Why should we find this state of affairs even surprising, let alone intellectually intolerable? To answer this question it is necessary to make a digression and investigate what we expect of those conceptions of the world around us which we are prepared to dignify by the name of scientific ‘theories.’

may be conceived) and what we human beings may say or think about it is of course one of the oldest problems in philosophy, and the particular sets of thoughts which have come to constitute the subject-matter of the natural sciences as we know them today have no claim to exemption from the rigorous philosophical questioning to which their less technical counterparts are regularly subjected.7 _{While most professional scientists, including the present}

author, lack the competence (and for that matter the time) to pursue such an analysis in detail, it is clear that their reactions to questions such as the ones raised by the measurement paradox will be determined by their (often implicit) perception of the basic functions of language in general and scientific language in particular. So it is appropriate that I should try to make explicit at this point the prejudices with which I myself approach this subject.8

I would suggest that a helpful way of looking at scientific theories is not as something totally divorced from the everyday language in which we describe our experience, but as an extension of its resources; that, at least in those aspects of language which we would normally be happy to call ‘factual

description,’ we are trying in some crude sense to build for ourselves and others maps of the world; and that this ‘map-making’ function of language is made much more precise and explicit in the language developed in modern scientific theories. This no doubt sounds not only unoriginal but naive; and indeed, so long as we take the concept of a ‘map’ as equivalent to that of a ‘picture,’ it is a view which has deservedly and repeatedly been shot through with holes by successive generations of philosophers. But let us for a moment experiment with the idea of taking the notion of a ‘map,’ as such, deadly seriously. In that case it is obvious after a moment’s thought that a map is certainly not equivalent to a picture. What kinds of maps do we know? There are Ordnance Survey (or USGS) maps; there are the road maps put out by the motoring organizations; there are maps prepared for military use; there are demographic maps; maps of the city subway system; and so on. What do they have in common? In the first place, a negative feature; a map is not a picture of anything. (On looking down from a low-flying plane, one does not see contour lines twisting around the hillsides, nor does the red or brown with which Ordnance Survey maps mark the roads designated as A- or B-class by the UK Ministry of Transport bear any relation to the actual color of the road.) Indeed, while many maps (e.g. Ordnance Survey or military maps) do bear a metric or at least a topological correspondence to the objects they describe, even this feature is not essential; the maps of the London Underground (subway) system displayed in the stations are certainly not metrically accurate, and (as far as I remember) maps of Charles de Gaulle airport do not show the complicated topology of the connecting tubes in detail. Why do we, nevertheless, regard these as adequate and useful

maps? Quite simply, because they fulfil the basic function of a map, which is to convey, in the form of a visual gestalt, an amount of information adequate to permit us to plan whatever activity we had in mind (the ascent of a mountain, a car journey to Scotland, an airport security operation, or whatever it was). Since different kinds of maps have different functions, it is not surprising that they have little in common, except for being two-dimensional displays (though in future we may no doubt get used to holographic maps). Mountaineers do not usually complain because their map fails to show the names of roads, nor do London subway travellers complain that, if you believe the wall maps, all subway lines travel in one of only eight directions; both maps are perfectly adequate for the particular purpose for which they are intended. It would be totally ridiculous to complain that a map of one kind is somehow inadequate because it is not in one-to-one correspondence with a map of a quite different kind. There is not and could not be any ‘ultimate map.’

Now, I would like to suggest that in many cases, where two different types of scientific theory apparently cover the same subject-matter and are not obviously reducible to one another (as for example in the case of statistical mechanics and thermodynamics) we should view their relationship as

resembling that of two different types of map of the same area; that is, they are different because they are trying to answer different types of question, but they are not therefore mutually incompatible. I believe that this type of relationship between theories at different ‘levels’ (to use a rather question-begging word) is much more common in science generally, and even in physics, than one would think if one reads either most physics textbooks or most

philosophers of science. For example, most textbooks of solid-state physics give the impression 9 _{that the whole goal of the subject is to solve}

Schrödinger’s equation for a collection of, say, 1023 nuclei and associated electrons, and that any models which we may build at an intermediate or macroscopic level (for example, the Drude model of electrons in metals, the Debye model of an elastic continuum, the Landau-Fermi liquid theory, and so on) are rather regrettable props which are in essence attempts to cover up our inability to do this by giving approximately valid solutions to the problem. Popular as this way of thinking may be among solid-state physicists, I believe that it is totally mistaken. In the first place, even if we could solve the

appropriate Schrödinger’s equation for an arbitrary, specified set of boundary conditions, we should never be able to apply the solution since we should in practice never be able to determine these boundary conditions with sufficient accuracy. Secondly, and far more importantly, what would be the value in having a complete and exact solution anyway? We should be rather in the position of the ancient Babylonians, who had (or so it is said) an elaborate system of recipes which enabled them to predict astronomical phenomena with a high degree of accuracy, but

without any unifying principle or anything which we could call a model. Such a solution would by itself in no way help us to predict qualitatively new