The report by Born and Heisenberg on ‘quantum mechanics’ may seem surprisingly difficult to the modern reader. This is partly because Born and Heisenberg are describing various stages of development of the theory that are quite different from today's quantum mechanics. Among these, it should be noted in particular that the theory developed by Heisenberg, Born and Jordan in the years 1925–6 and known today as matrix mechanics (Heisenberg 1925b [1], Born and Jordan 1925 [2], Born, Heisenberg and Jordan 1926 [4])a differs from standard quantum mechanics in several important respects. At the same time, the interpretation of the theory (the topic of Section II of the report) also appears to have undergone important modifications, in particular regarding the notion of the state of a system. Initially, Born and Heisenberg insist on the notion that a system is always in a stationary state (performing quantum jumps between different stationary states). Then the notion of the wave function is introduced and related to probabilities for the stationary states. At a later stage, probabilistic notions (in particular, what one now calls transition probabilities) are extended to arbitrary observables, but it remains somewhat unclear whether the wave function itself should be regarded as a fundamental entity or merely as an effective one. This may reflect the different routes followed by Born and by Heisenberg in the development of their ideas. The common position presented by Born and Heisenberg emphasises the probabilistic aspect of the theory as fundamental, and the conclusion of the report expresses strong confidence in the resulting picture.

The fifth Solvay conference is usually remembered for the clash that took place between Bohr and Einstein, supposedly concerning in particular the possibility of breaking the uncertainty relations. It might be assumed that this clash took the form of an official debate that was the centrepiece of the conference. However, no record of any such debate appears in the published proceedings, where both Bohr and Einstein are in fact relatively silent.

The available evidence shows that in 1927 the famous exchanges between Bohr and Einstein actually consisted of informal discussions, which took place semiprivately (mainly over breakfast and dinner), and which were overheard by just a few of the participants, in particular Heisenberg and Ehrenfest. The historical sources for this consist, in fact, entirely of accounts given by Bohr, Heisenberg and Ehrenfest. These accounts essentially ignore the extensive formal discussions appearing in the published proceedings.

As a result of relying on these sources, the perception of the conference by posterity has been skewed on two counts. First, at the fifth Solvay conference there occurred much more that was memorable and important besides the Bohr–Einstein clash. Second, as shown in detail by Howard (1990), the real nature of Einstein's objections was in fact misunderstood by Bohr, Heisenberg and Ehrenfest – for Einstein's main target was not the uncertainty relations, but what he saw as the non-separability of quantum theory.

Introduction

Under this name at present two theories are being carried on, which are indeed closely related but not identical. The first, which follows on directly from the famous doctoral thesis by L. de Broglie, concerns waves in three-dimensional space. Because of the strictly relativistic treatment that is adopted in this version from the outset, we shall refer to it as the four-dimensional wave mechanics. The other theory is more remote from Mr de Broglie's original ideas, insofar as it is based on a wave-like process in the space of position coordinates (q-space) of an arbitrary mechanical system. We shall therefore call it the multi-dimensional wave mechanics. Of course this use of the q-space is to be seen only as a mathematical tool, as it is often applied also in the old mechanics; ultimately, in this version also, the process to be described is one in space and time. In truth, however, a complete unification of the two conceptions has not yet been achieved. Anything over and above the motion of a single electron could be treated so far only in the multi-dimensional version; also, this is the one that provides the mathematical solution to the problems posed by the Heisenberg–Born matrix mechanics. For these reasons I shall place it first, hoping in this way also to illustrate better the characteristic difficulties of the as such more beautiful four-dimensional version.

Background

In September 1923, Prince Louis de Brogliea made one of the most astonishing predictions in the history of theoretical physics: that material bodies would exhibit the wave-like phenomena of diffraction and interference upon passing through sufficiently narrow slits. Like Einstein's prediction of the deflection of light by the sun, which was based on a reinterpretation of gravitational force in terms of geometry, de Broglie's prediction of the deflection of electron paths by narrow slits was made on the basis of a fundamental reappraisal of the nature of forces and of dynamics. De Broglie had proposed that Newton's first law of motion be abandoned, and replaced by a new postulate, according to which a freely moving body follows a trajectory that is orthogonal to the surfaces of equal phase of an associated guiding wave. The resulting ‘de Broglian dynamics’ – or pilot-wave theory as de Broglie later called it – was a new approach to the theory of motion, as radical as Einstein's interpretation of the trajectories of falling bodies as geodesics of a curved spacetime, and as far-reaching in its implications. In 1929 de Broglie received the Nobel Prize ‘for his discovery of the wave nature of electrons’.

Introduction

Professor W. L. Bragg has just discussed a whole series of radiation phenomena in which the electromagnetic theory is confirmed. He has even dwelt on some of the limiting cases, such as the reflection of X-rays by crystals, in which the electromagnetic theory of radiation gives us, at least approximately, a correct interpretation of the facts, although there are reasons to doubt that its predictions are truly exact. I have been left the task of pleading the opposing cause to that of the electromagnetic theory of radiation, seen from the experimental viewpoint.

I have to declare from the outset that in playing this role of the accuser I have no intention of diminishing the importance of the electromagnetic theory as applied to a great variety of problems. It is, however, only by acquainting ourselves with the real or apparent failures of this powerful theory that we can hope to develop a more complete theory of radiation which will describe the facts as we know them.

In this chapter, we address proposals (by Einstein and by Bohr, Kramers and Slater) according to which quantum events are influenced by ‘guiding fields’ in 3-space. These ideas led to a predicted violation of energy-momentum conservation for single events, in contradiction with experiment. The contradiction was resolved only by the introduction of guiding fields in configuration space. All this took place before the fifth Solvay conference, but nevertheless forms an important background to some of the discussions that took place there.

Einstein's early attempts to formulate a dynamical theory of light quanta

Since the publication of his light-quantum hypothesis in 1905, Einstein had been engaged in a solitary struggle to construct a detailed theory of light quanta, and to understand the relationship between the quanta on the one hand and the electromagnetic field on the other. Einstein's efforts in this direction were never published. We know of them indirectly: they are mentioned in letters, and they are alluded to in Einstein's 1909 lecture in Salzburg. Einstein's published papers on light quanta continued for the most part in the same vein as his 1905 paper: using the theory of fluctuations to make deductions about the nature of radiation, without giving details of a substantial theory. Einstein was essentially alone in his dualistic view of light, in which localised energy fragments coexisted with extended waves, until the work of de Broglie in 1923 – which extended the dualism to all particles, and made considerable progress towards a real theory (see Chapter 2).

The classical treatment of x-ray diffraction phenomena

The earliest experiments on the diffraction of X-rays by crystals showed that the directions in which the rays were diffracted were governed by the classical laws of optics. Laue's original paper on the diffraction of white radiation by a crystal, and the work which my father and I initiated on the reflection of lines in the X-ray spectrum, were alike based on the laws of optics which hold for the diffraction grating. The high accuracy which has been developed by Siegbahn and others in the realm of X-ray spectroscopy is the best evidence of the truth of these laws. Advance in accuracy has shown the necessity of taking into account the very small refraction of X-rays by the crystal, but this refraction is also determined by the classical laws and provides no exception to the above statement.

The first attempts at crystal analysis showed further that the strength of the diffracted beam was related to the structure of the crystal in a way to be expected by the optical analogy. This has been the basis of most work on the analysis of crystal structure. When monochromatic X-rays are reflected from a set of crystal planes, the orders of reflection are strong, weak, or absent in a way which can be accounted for qualitatively by the arrangement of atoms parallel to these planes.

As we discussed in Section 2.4, in his Solvay lecture of 1927 de Broglie presented the pilot-wave dynamics of a non-relativistic many-body system, and outlined some simple applications of his ‘new dynamics of quanta’ (to interference, diffraction and atomic transitions). Further, as we saw in Section 10.2, contrary to a widespread misunderstanding, in the general discussion de Broglie's reply to Pauli's objection contained the essential points needed to treat inelastic scattering (even if Fermi's misleading optical analogy confused matters): in particular, de Broglie correctly indicated how definite quantum outcomes in scattering processes arise from a separation of wave packets in configuration space. We also saw in Section 10.4 that de Broglie was unable to reply to a query from Kramers concerning the recoil of a single photon on a mirror: to do so, he would have had to introduce a joint wave function for the photon and the mirror.

De Broglie's theory was revived by Bohm 25 years later (Bohm 1952a,b) (though with the dynamics written in terms of a law for acceleration instead of a law for velocity). Bohm's truly new and very important contribution was a pilot-wave account of the general quantum theory of measurement, with macroscopic equipment (pointers, etc.) treated as part of the quantum system. In effect, in 1952, Bohm provided a detailed derivation of quantum phenomenology from de Broglie's dynamics of 1927 (albeit with the dynamical equations written differently).

Probability and interference

According to Feynman, single-particle interference is ‘the only mystery’ of quantum theory (Feynman, Leighton and Sands 1965, ch. 1, p. 1). Feynman considered an experiment in which particles are fired, one at a time, towards a screen with two holes labelled 1 and 2. With both holes open, the distribution P12 of particles at the backstop displays an oscillatory pattern of bright and dark fringes. If P1 is the distribution with only hole 1 open, and P2 is the distribution with only hole 2 open, then experimentally it is found that P12 ≠ P1 + P2. According to the argument given by Feynman (as well as by many other authors), this result is inexplicable by ‘classical’ reasoning.

By his presentation of the two-slit experiment (as well as by his development of the path-integral formulation of quantum theory), Feynman popularised the idea that the usual probability calculus breaks down in the presence of quantum interference, where it is probability amplitudes (and not probabilities themselves) that are to be added. As pointed out by Koopman (1955), and by Ballentine (1986), this argument is mistaken: the probability distributions at the backstop – P12, P1 and P2 – are conditional probabilities with three distinct conditions (both slits open, one or other slit closed), and probability calculus does not imply any relationship between these. Feynman's argument notwithstanding, standard probability calculus is perfectly consistent with the two-slit experiment.

The early Solvay conferences were remarkable occasions, made possible by the generosity of Belgian industrialist Ernest Solvay and, with the exception of the first conference in 1911, planned and organised by the indefatigable Hendrik Antoon Lorentz. In this chapter, we shall first sketch the beginnings of the Solvay conferences, Lorentz's involvement and the situation in the years leading up to 1927 (Sections 1.1 and 1.2). Then we shall describe specifically the planning of the fifth Solvay conference, both in its scientific aspects (Section 1.3) and in its more practical aspects (Section 1.4). Section 1.5 presents the day-by-day progress of the conference as far as it can be reconstructed from the sources, while Section 1.6 follows the making of the volume of proceedings, which is the main source of original material from the fifth Solvay conference and forms Part III of this book.

What is quantum theory?

For much of the twentieth century, it was widely believed that the interpretation of quantum theory had been essentially settled by Bohr and Heisenberg in 1927. But not only were the ‘dissenters’ of 1927 – in particular de Broglie, Einstein and Schrödinger – unconvinced at the time: similar dissenting points of view are not uncommon even today. What Popper called ‘the schism in physics’ (Popper 1982) never really healed. Soon after 1927 it became standard to assert that matters of interpretation had been dealt with, but the sense of puzzlement and paradox surrounding quantum theory never disappeared.

As the century wore on, many of the concerns and alternative viewpoints expressed in 1927 slowly but surely revived. In 1952, Bohm revived and extended de Broglie's theory (Bohm 1952a,b), and in 1993 the de Broglie–Bohm theory finally received textbook treatment as an alternative formulation of quantum theory (Bohm and Hiley 1993; Holland 1993). In 1957, Everett (1957) revived Schrödinger's view that the wave function, and the wave function alone, is real (albeit in a very novel sense), and the resulting ‘Everett’ or ‘many-worlds’ interpretation (DeWitt and Graham 1973) gradually won widespread support, especially among physicists interested in quantum gravity and quantum cosmology.