What was Bohr's response to the problem of wave-particle duality? Before considering this question in detail it will be useful to survey briefly Bohr's work in physics between 1911 and 1923.

Bohr and the ‘old’ quantum theory

In 1911 Bohr completed his doctoral dissertation on the classical electron theory of metals. Developed by Sir J.J. Thomson and Lorentz, the theory explained electrical, magnetic and thermal properties of metals in terms of the motions of free electrons. Although Bohr made no use of the quantum hypothesis in his dissertation, he noted that classical electrodynamics was incapable of explaining Planck's radiation law and entailed the ultraviolet catastrophe.

After completing his dissertation Bohr pursued his post-graduate studies under Thomson at Cambridge University. Finding that Thomson was little interested in his work, Bohr went on to Manchester University in March 1912, to work under Ernest Rutherford. Bohr's move to Manchester was very opportune: the quantum hypothesis was just beginning to receive a wider recognition, thanks largely to the first Solvay Conference in 1911, which Rutherford attended; Rutherford's laboratory, moreover, was one of the most active centres in the investigation of radio-activity.

In order to explain the results of experiments on the passage of alpha and beta particles through matter Rutherford had revived Hantaro Nagaoka's (1904) planetary model of the atom. On this model an atom consists of a positively charged nucleus which is orbited by negatively charged electrons, the number of which corresponds to the number of charges on the nucleus.

Bohr's response in 1925 to the failure of the Bohr–Kramers–Slater theory was, as we have seen, to regard wave–particle duality as having a formal, rather than realistic, significance; he saw it as a ‘limitation of our usual means of visualization’. That same year, however, wave–particle duality was generalised by Louis-Victor de Broglie to cover the theory of matter. Moreover, the advent of quantum mechanics (i.e. matrix mechanics and wave mechanics), rather than dispelling the duality problem, heightened it. How did Bohr respond to these new developments in the quantum theory?

A matter of waves

In the autumn of 1923 de Broglie put forward the extremely bold hypothesis that any particle whatever is associated with a wave whose wavelength λ and frequency ν are proportional to the particle's momentum and energy respectively: λ = h/p, v = E/h. De Broglie regarded his hypothesis as the basis of a new mechanics, a wave mechanics related to classical mechanics in the same way as wave optics is related to ray optics. On this view the trajectory of a particle corresponds to the ray of the wave associated with it (i.e. the normal to the equiphase surface of the wave), and the velocity of the particle corresponds to the group velocity of the wave. De Broglie suggested, moreover, that in certain circumstances particles should exhibit diffraction effects: a stream of electrons, for example, which passes through a slit in a screen should undergo diffraction.

Bohr's general thesis of complementarity conveys the idea that we can, and perhaps must, make use of the classical physical concepts in quantum physics, notwithstanding the inadequacies of these concepts; but we can use them only within the limits circumscribed by the quantum of action – beyond these limits the classical concepts cease to be well defined. This idea originated in his thesis that the conceptual structure of the quantum theory is in a sense a generalisation of the conceptual scheme of classical mechanics; and this thesis was a corollary of the correspondence principle.

During the first months of 1927 Bohr regarded quantum mechanics as the quantum-theoretical generalisation of classical mechanics to which he had been looking forward. He came also to think that quantum mechanics required in a sense the generalisation of the two classical models of particle and wave. In classical physics these two models generally pertain to different theories, the particle model to the mechanical theory of matter, the wave model to the electromagnetic theory of radiation. In quantum physics, however, the two models pertain to one theory, quantum mechanics, and each is applicable to matter and radiation. Where discontinuity obtains, the standard classical models of matter and radiation cease to be well defined, and the alternative, non-standard models may be used. There is no logical inconsistency in this dual use of the models since each is appropriately applicable only in different empirical circumstances.

In Bohr's view, what ontological significance does measurement have in quantum physics? How, exactly, are we to conceive of the physical properties which we attribute to microphysical objects on the basis of measurement? As we have seen, Bohr regards such properties as objective in the sense that in general they characterise the object immediately before the measurement. But what exactly is their ontological status? In order to answer this question, we shall have to consider in detail Bohr's view of the ontological significance of kinematic-dynamic complementarity. It is generally believed that Bohr held that in quantum physics an object cannot meaningfully be said to possess simultaneously an exact position and an exact momentum. Did he, and if so, why did he?

The interactive-properties theory

Some hold that the process of measurement does not so much create the value of the measured observable as disturb a pre-existing value. On this view the measured value characterises the observable only after the measurement, and not before: the measured value may be said to be a ‘created’ or ‘disturbed’ value.

Bohr, as I have already said, seems to have toyed with this view in his earlier years: even as late as 1933 he says:

Niels Bohr's theory of complementarity was an attempt to solve the enormous problems of interpretation – especially the problem of wave particle duality – that beset the quantum theory in the mid-twenties. By 1920 electromagnetic radiation could be conceived of either in terms of the wave model or in terms of the particle model, though neither conception alone was wholly adequate to the empirical data. By the mid-twenties the duality problem was quite general, applying to matter as well as to radiation. Quantum mechanics, the new quantum theory, could be interpreted partially in terms of one or other of the two models but not comprehensively in terms of either.

Bohr's theory can be fully understood only against the background of the interpretative problems which it was intended to solve; and the root of these problems goes back to the very origin of the quantum theory, viz. the quantum hypothesis to the effect that in certain physical processes energy is transferred discontinuously in discrete amounts. Wave-particle duality was a development of the continuity-discontinuity duality which the quantum hypothesis introduced into physics. How did wave-particle duality arise?

Having discussed the meaning of complementarity in general and wave particle complementarity in particular, I wish now to explicate Bohr's subtle argument for the thesis that certain pairs of classical concepts which are mutually compatible in classical physics are mutually exclusive in quantum physics, i.e. the thesis of kinematic–dynamic complementarity. It is generally held that Bohr regarded the mutual exclusiveness in question as being not only epistemic but also ontic or semantic, in the sense that the uncertainty principle expresses a restriction not only on the joint measurability of canonically conjugate observables but also on the joint instantiation of the properties corresponding to these observables. I shall consider the question of the ontic or semantic construal of kinematic–dynamic complementarity in Chapter 7. Bohr's argument for complementarity in the epistemic sense is, however, more fundamental: it goes to the heart of his theory of observation and measurement in quantum physics. What, then, is Bohr's argument?

The mutual exclusiveness of kinematic and dynamic properties

Kinematic and dynamic attributes in quantum mechanics are mutually exclusive in the sense that they cannot be simultaneously measured; they are, in this sense, epistemically incompatible. Why did Bohr think this? His reasons may seem intricate, but they are basically quite simple. Two main factors, he maintains, account for the epistemic incompatibility: (a) measurements of these two different sorts of properties require mutually exclusive experimental arrangements, incompatible measurement procedures; (b) the indeterminability of the interaction between the object and the instrument of measurement precludes extrapolation of the different measurement results to one and the same time.

Now that my account of Bohr's philosophy of physics has been completed, it remains for me to give a general assessment of it. What value should we assign to Bohr's main ideas – the notions of correspondence and complementarity? How does his interpretation of quantum physics fare in comparison with some of its rivals? What should our assessment of the Bohr–Einstein debate be? I shall take up these questions in the reverse order.

Einstein or Bohr? The final verdict

The Bohr–Einstein debate is primarily a dispute about the philosophical foundations in terms of which the interpretation of quantum mechanics should be based: specifically it is a dispute about the intrinsic-properties theory, i.e. about the scope of a realist interpretation of quantum mechanics. Einstein wished to maintain the two realist theses, the intrinsic-properties theory and the objective-values theory, whereas Bohr wished to hold only the latter. My own philosophical leanings are in the direction of realism. To my mind the notion of a physical reality the existence and constitution of which are independent of human thought and perception is coherent and plausible: what physical objects there are, and what properties they have, is not determined by human thought or perception, or by human practical interests. I have sympathy, therefore, with Einstein's position concerning the indefinability thesis and the question of the completeness of quantum mechanics.

What exactly is Niels Bohr's interpretation of quantum physics? And what general philosophical position underlies it? In the following pages I try to present clear and thorough answers to these questions. Bohr's interpretation is undoubtedly of major importance, but there is no universal agreement about what that interpretation is, owing largely to the notorious obscurity of his writings and the fact that his own views are often confused with those of others within the Copenhagen tradition which he founded. My reading is based solely on Bohr's own writings, and it is supported with liberal citations; I defend it against rival readings, but I try to keep polemical discussion to a minimum. When it illuminates Bohr's views, I have made use of material in the Niels Bohr Archive, much of which has now been published in Niels Bohr: Collected Works (six vols., eds. Léon Rosenfeld & Erik Rüdinger, North-Holland, Amsterdam, 1972–85). All translations are my own, unless the contrary is stated.

Bohr's interpretation of quantum physics, I believe, contrary to a widely held view, was not the outcome of a ready-made philosophy; rather, it developed gradually from his day-to-day grappling with the problems of the physics. It is best understood, therefore, when seen against the historical background of these problems. My account of this background in the first few chapters is selective, highlighting only those ideas and problems which played a prominent part in the formation of Bohr's views.

The EPR paper and Bohr's reply to it brought to an end the public debate between Einstein and Bohr. In the aftermath, Bohr's construal of quantum mechanics established itself as the basis of the orthodox interpretation – the ‘Copenhagen interpretation’ as it came to be called. The vast majority of physicists regarded Einstein's criticism as reactionary, and dismissed it with disdain. One of the main causes of this disdainful attitude was the general opinion that Einstein's criticism committed him to the existence of hidden states, and the existence of these, it was thought, had been conclusively ruled out by the work of von Neumann and others. What light does the subsequent work on the question of hidden states, and in particular the work of Bell, throw on the Bohr–Einstein dispute?

Completeness and hidden states

There are three distinct questions concerning hidden states. First, might there be such states, unbeknown to quantum mechanics? Second, can quantum mechanics, as it stands, take account of such states? Third, could some theory containing descriptions of such states reduplicate the predictions of quantum mechanics? Concerning the first of these questions, Einstein's view was that it is very likely that such states exist, and since quantum mechanics takes no account of them, it is incomplete and best regarded as a statistical theory analogous in some respects to classical statistical mechanics.

I propose now to examine Bohr's informal theory of measurement from a somewhat more formal point of view, and to consider how his theory relates to the orthodox theory of measurement in quantum mechanics. This is a difficult task, since Bohr nowhere discusses the theory of measurement in formal terms. My discussion, therefore, will be less closely tied to Bohr's own statements. I shall also consider some important objections to the theory.

The objective-values theory of measurement

Before we look into this theory, it will be useful to have a brief sketch of some of the relevant formal elements of quantum mechanics.

In the Hilbert space formulation of quantum mechanics the measurable physical properties (‘observables’) of an object are represented by linear Hermitian operators A, B, … which have complete orthonormal sets of eigenvalues an, bn, … corresponding to the subspaces (closed linear manifolds) of a Hilbert space H. These operators are in one-to-one correspondence with the projection measures PA, PB, … on the Hilbert space such that PA = ΣnanPn, where Pn are the projections of the eigenvectors ψn belonging to the eigenvalues an of A, and an are discrete, non-degenerate values (for simplicity I shall deal only with operators which have such values, and shall denote observables and the corresponding operators by the same symbols).

Niels Bohr solved the problem of interpreting quantum mechanics once and for all. That, until recently, was the orthodoxy in physics, especially among writers of textbooks on quantum mechanics, who mostly wanted to get off the philosophy as quickly as possible and on with the physics. Niels Bohr, incidentally, would not have agreed with them.

Now the fashion is for physicists to say that quantum mechanics is so peculiar that no one understands it, least of all themselves. Perhaps the change was wrought by Bell's theorem in the 1960s; perhaps it really was just a matter of fashion.

Of course, in saying that quantum mechanics is incomprehensible, one is not saying that it is false, only that the human mind is not tuned in to the way the world is. The philosophy of quantum mechanics deals with the question: What is the way the world is, if quantum mechanics is true? It would be nice if, in answering that question, one were also to make quantum mechanics comprehensible, something philosophers tend to feel they can pull off.

Philosophers often have another motive, one which is inspired by their intellectual cussedness. Quantum mechanics is most easily interpreted antirealistically, that is, as a theory which, though it works, does not describe the way the world is. Therefore, philosophers go out of their way to interpret it realistically. Realism in the philosophy of quantum mechanics means the idea that quantum systems are really like classical particles. Everything points against it.

The debate between Bohr and Einstein on the consistency, completeness, and finality of quantum theory ranks as one of the most significant philosophical debates of the twentieth century. It was a philosophical debate not in the sense that it had a considerable impact on the relatively technical field of the philosophy of physics. It was philosophical because at its root there lay two rival conceptions of physical reality and of our capacity to comprehend physical reality.

With the possible exception of the Leibniz-Clarke correspondence in which Leibniz and Clarke - with the latter acting as Newton's frontman - pressed their respective conceptions of space and time in the early part of the eighteenth century, the Einstein-Bohr dialogue has no parallel in the history of physics. Unlike the Leibniz-Clarke correspondence, the Bohr-Einstein debate contained no bitterness and was not based on any personal rivalry.

The debate was continuous from about 1927 and lasted right up to Einstein's death in 1955. For the contemporary philosopher of physics the most important episodes took place in the years between 1927 and 1935. However, in one respect at least the differences between Bohr and Einstein predated quantum mechanics itself.

One is tempted to view the disagreement between Bohr and Einstein as the opposition of Bohr the revolutionary and Einstein the conservative, and in his best and clearest account of it Bohr certainly writes as if that were so.

Quantum mechanics comes in a variety of different forms and formalisms. There is wave mechanics, matrix mechanics, Dirac's version of quantum mechanics, von Neumann's version of quantum mechanics. The philosopher of physics has to ask not only what quantum mechanics means, but also what it is.

Orthodoxy in the philosophy of physics treats quantum mechanics as the general theory of microphysics which takes Hilbert space as its state space and which associates observables with certain special operators on that space. This amounts to a decision as to what quantum mechanics is. However it is a far from arbitrary decision. For the Hilbert space, or von Neumann, formalism summarizes and captures everything that is in the other versions of quantum mechanics, without any of the dubious mathematics of, for example, Dirac's version.

Contemporary philosophers of physics take a very formal view of what quantum mechanics is, a view which has many advantages and some disadvantages. Whatever the disadvantages of focussing on the formalism of quantum mechanics - and it may lead to losing sight of the ‘physics' and possibly to a bias against Bohrian Copenhagenism - it is certainly true that the literate philosopher of physics must have some acquaintance with Hilbert space.

There are in fact several topics we have to deal with, though we deal with them only very cursorily. (This chapter is no substitute for Jauch's classic text Foundations of Quantum Mechanics.)

Of all the many rival interpretations of quantum mechanics, none is more important or influential than the Copenhagen interpretation. Unfortunately, there is no one Copenhagen interpretation of quantum mechanics. If one uses the expression, and everyone does, one should not think of the Copenhagen interpretation as a single consistent interpretation of the theory.

Philosophers of physics tend to use the expression ‘the Copenhagen interpretation of quantum mechanics’ as a label covering a variety of different interpretations all of which either originate with Niels Bohr or were invented by him, his colleagues and guests at the Institute for Theoretical Physics in Copenhagen beginning in the middle 1920s. Perhaps one should say that the Copenhagen interpretation in this diffuse sense goes back farther than this, back even before the rise of the new quantum mechanics.

Heisenberg tells the story of a boat trip he, Bohr, and some friends took from Copenhagen to the island of Fyn around the time when the then new quantum mechanics was being developed. Bohr, ‘full of the new interpretation of quantum theory’, talked of ‘the difficulties of language, of the limitations of all our means of expressing ourselves’ and of how these limitations had been expressed ‘in the foundation of atomic theory in a mathematically lucid way’. ‘Finally,’ Heisenberg ends the story, ‘one of the friends remarked drily, “But, Niels, this is not really new, you said exactly the same ten years ago”

Why should anyone think that what is so deeply puzzling about the quantum world is really a matter of logic? Why should anyone think that studying the logic of quantum mechanics should be a route to a proper understanding of quantum mechanics?

There is a tradition in the philosophy of quantum mechanics, a tradition stronger among philosophical commentators than among physicists, which sees the subject as the business of understanding quantum mechanical language and the way that language relates to the world. It is an idea more strongly held by philosophers – it comes much more naturally to them – because the analytic tradition, in which most Anglo-American philosophers of science are raised, is generally much exercised about words and how they hook on to things.

The analytic tradition in the philosophy of quantum mechanics is not unreasonable. Everyone admits that we cannot picture the microphysical world, that a graphic or iconic representation of quantum systems is impossible. Therefore understanding quantum mechanics must be a matter of understanding the logic of the words and the mathematics of quantum mechanics. If this seems implausible, it is because of an ambiguity in our use of the verb ‘to understand’.

We understand the physical world, and we understand physics. But physics is not the physical world. It is something of an entirely different sort, a human product, a way of representing the world.