Introduction: the theory of local beables

This is a pretentious name for a theory which hardly exists otherwise, but which ought to exist. The name is deliberately modelled on ‘the algebra of local observables’. The terminology, be-able as against observ-able, is not designed to frighten with metaphysic those dedicated to realphysic. It is chosen rather to help in making explicit some notions already implicit in, and basic to, ordinary quantum theory. For, in the words of Bohr, ‘it is decisive to recognize that, however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms’. It is the ambition of the theory of local beables to bring these ‘classical terms’ into the equations, and not relegate them entirely to the surrounding talk.

The concept of ‘observable’ lends itself to very precise mathematics when identified with ‘self-adjoint operator’. But physically, it is a rather woolly concept. It is not easy to identify precisely which physical processes are to be given the status of ‘observations’ and which are to be relegated to the limbo between one observation and another. So it could be hoped that some increase in precision might be possible by concentration on the beables, which can be described in ‘classical terms’, because they are there. The beables must include the settings of switches and knobs on experimental equipment, the currents in coils, and the readings of instruments. ‘Observables’ must be made, somehow, out of beables.

The editor has asked me to reply to a paper, by G. Lochak, refuting a theorem of mine on hidden variables. If I understand correctly, Lochak finds that I failed somehow to allow for the effect on these variables of the measuring equipment. I will try to explain why I do not agree. The opportunity will also be taken here to comment on another refutation, by L. de la Peña, A. M. Cetto and T. A. Brody, and on another by L. de Broglie. Yet another refutation of the same theorem, by J. Bub, has already been refuted by S. Freedman and E. P. Wigner.

Let us recall a typical context to which the theorem is relevant. A ‘pair of spin ½ particles’ is produced in a space-time region 3 and activates counting systems, preceded by Stern–Gerlach magnets, in space–time regions 1 and 2. The system at 1 is such that one of two counters (‘up’ or ‘down’) registers each time the experiment is done; correspondingly we label the result there by A (= + 1 or – 1). Likewise the system at 2 is such that one of two counters registers each time the experiment is done, giving B (= + 1 or – 1).

I will try to interest you in the de Broglie–Bohm version of non-relativistic quantum mechanics. It is, in my opinion, very instructive. It is experimentally equivalent to the usual version insofar as the latter is unambiguous. But it does not require, in its very formulation, a vague division of the world into ‘system’ and ‘apparatus,’ nor of history into ‘measurement’ and ‘nonmeasurement.’ So it applies to the world at large, and not just to idealized laboratory procedures. Indeed the de Broglie–Bohm theory is sharp where the usual one is fuzzy, and general where the usual one is special. This is not a systematic exposition, but only an illustration of the ideas with a particularly nice example, and then some remarks on the role of the density matrix – in tribute to the title of this conference.

No one more eloquently than John A. Wheeler has presented the delayed-choice double slit experiment. A pulsed particle source S (see Fig. 1) is so feeble that not more than one particle is emitted per pulse. The associated wave pulse B falls on a screen with slits 1 and 2. The transmitted pulses B′ are focussed by off-centred lenses into intersecting plane wave trains which fall finally on particle counters C1 and C2 – unless a photographic plate P is pushed into the region where the two wave trains interpenetrate. The decision, to interpose the plate or not, is made only after the pulse has passed the slits.

Introduction

In a very elegant and rigorous paper, K. Hepp has discussed quantum measurement theory. He uses the C* algebra description of infinite quantum systems. Here an attempt is made to give a more popular account of some of his reasoning. Such an attempt seems worthwhile because many people not familiar with the C* algebra approach, and even somewhat intimidated by it, have been intrigued by the following statement in Hepp's abstract:

This could look like a clean solution at last to the infamous measurement problem. But it is not so, nor thought by Hepp to be so. Here we will take one of his models and analyse it in elementary text-book terms. It will be insisted that the ‘rigorous reduction’ does not occur in physical time but only in an unattainable mathematical limit. It will be argued that the distinction is an important one.

We will work at first in the Schrödinger picture, but later, with the extension to relativistic systems in mind, it will be argued that such considerations become particularly clear in the Heisenberg picture.

Introduction

When I was a student I had much difficulty with quantum mechanics. It was comforting to find that even Einstein had had such difficulties for a long time. Indeed they had led him to the heretical conclusion that something was missing in the theory: ‘I am, in fact, rather firmly convinced that the essentially statistical character of contemporary quantum theory is solely to be ascribed to the fact that this (theory) operates with an incomplete description of physical systems.’

More explicitly, in ‘a complete physical description, the statistical quantum theory would … take an approximately analogous position to the statistical mechanics within the framework of classical mechanics …’.

Einstein did not seem to know that this possibility, of peaceful coexistence between quantum statistical predictions and a more complete theoretical description, had been disposed of with great rigour by J. von Neumann. I myself did not know von Neumann's demonstration at first hand, for at that time it was available only in German, which I could not read. However I knew of it from the beautiful book by Born, Natural Philosophy of Cause and Chance, which was in fact one of the highlights of my physics education. Discussing how physics might develop Born wrote: ‘I expect … that we shall have to sacrifice some current ideas and to use still more abstract methods. However these are only opinions.

I have for long thought that if I had the opportunity to teach this subject, I would emphasize the continuity with earlier ideas. Usually it is the discontinuity which is stressed, the radical break with more primitive notions of space and time. Often the result is to destroy completely the confidence of the student in perfectly sound and useful concepts already acquired.

If you doubt this, then you might try the experiment of confronting your students with the following situation. Three small spaceships, A, B, and C, drift freely in a region of space remote from other matter, without rotation and without relative motion, with B and C equidistant from A (Fig. 1).

On reception of a signal from A the motors of B and C are ignited and they accelerate gently (Fig. 2).

Let ships B and C be identical, and have identical acceleration programmes. Then (as reckoned by an observer in A) they will have at every moment the same velocity, and so remain displaced one from the other by a fixed distance. Suppose that a fragile thread is tied initially between projections from B and C (Fig. 3). If it is just long enough to span the required distance initially, then as the rockets speed up, it will become too short, because of its need to Fitzgerald contract, and must finally break. It must break when, at a sufficiently high velocity, the artificial prevention of the natural contraction imposes intolerable stress.

Introduction

There is an ongoing series of symposia, at Tokyo, on ‘Foundations of Quantum Mechanics in the Light of New Technology’. Indeed new technology (electronics, computers, lasers, …) has made possible new demonstrations of quantum queerness. And it has made possible practical approximations to old gedankenexperiments. Over the last decade or so have appeared beautiful experiments on ‘particle’ interference and diffraction, with neutrons and electrons, on ‘delayed choice’, on the Ehrenburg–Siday–Aharonov–Bohm effect, and on the Einstein–Podolsky–Rosen–Bohm correlations. These last are of particular relevance for the particular themes of this paper. But those themes arise already in the context of technology which is neither new or advanced, as is illustrated by the following passage:

These notions, of cause and effect on the one hand, and of correlation on the other, and the problem of formulating them sharply in contemporary physical theory, will be the themes of my talk.

Introduction

The philosopher in the street, who has not suffered a course in quantum mechanics, is quite unimpressed by Einstein–Podolsky–Rosen correlations. He can point to many examples of similar correlations in everyday life. The case of Bertlmann's socks is often cited. Dr. Bertlmann likes to wear two socks of different colours. Which colour he will have on a given foot on a given day is quite unpredictable. But when you see (Fig. 1) that the first sock is pink you can be already sure that the second sock will not be pink. Observation of the first, and experience of Bertlmann, gives immediate information about the second. There is no accounting for tastes, but apart from that there is no mystery here. And is not the EPR business just the same?

Consider for example the particular EPR gedanken experiment of Bohm (Fig. 2). Two suitable particles, suitably prepared (in the ‘singlet spin state’), are directed from a common source towards two widely separated magnets followed by detecting screens. Each time the experiment is performed each of the two particles is deflected either up or down at the corresponding magnet. Whether either particle separately goes up or down on a given occasion is quite unpredictable. But when one particle goes up the other always goes down and vice-versa. After a little experience it is enough to look at one side to know also about the other.

The quantum revolutions: from concepts to technology

The development of quantum mechanics in the beginning of the twentieth century was a unique intellectual adventure, which obliged scientists and philosophers to change radically the concepts they used to describe the world. After these heroic efforts, it became possible to understand the stability of matter, the mechanical and thermal properties of materials, the interaction between radiation and matter, and many other properties of the microscopic world that had been impossible to understand with classical physics. A few decades later, that conceptual revolution enabled a technological revolution, at the root of our information-based society. It is indeed with the quantum mechanical understanding of the structure and properties of matter that physicists and engineers were able to invent and develop the transistor and the laser – two key technologies that now permit the high-bandwidth circulation of information, as well as many other scientific and commercial applications.

After such an accumulation of conceptual – and eventually technological – successes, one might think that by 1960 all the interesting questions about quantum mechanics had been raised and answered. However, in his now-famous paper of 1964 – one of the most remarkable papers in the history of physics – John Bell drew the attention of physicists to the extraordinary features of entanglement: quantum mechanics describes a pair of entangled objects as a single global quantum system, impossible to be thought of as two individual objects, even if the two components are far apart.

Introduction

To know the quantum mechanical state of a system implies, in general, only statistical restrictions on the results of measurements. It seems interesting to ask if this statistical element be thought of as arising, as in classical statistical mechanics, because the states in question are averages over better defined states for which individually the results would be quite determined. These hypothetical ‘dispersion free’ states would be specified not only by the quantum mechanical state vector but also by additional ‘hidden variables’ – ‘hidden’ because if states with prescribed values of these variables could actually be prepared, quantum mechanics would be observably inadequate.

Whether this question is indeed interesting has been the subject of debate. The present paper does not contribute to that debate. It is addressed to those who do find the question interesting, and more particularly to those among them who believe that ‘the question concerning the existence of such hidden variables received an early and rather decisive answer in the form of von Neumann's proof on the mathematical impossibility of such variables in quantum theory.’ An attempt will be made to clarify what von Neumann and his successors actually demonstrated. This will cover, as well as von Neumann's treatment, the recent version of the argument by Jauch and Piron, and the stronger result consequent on the work of Gleason. It will be urged that these analyses leave the real question untouched.

The subject–object distinction is indeed at the very root of the unease that many people still feel in connection with quantum mechanics. Some such distinction is dictated by the postulates of the theory, but exactly where or when to make it is not prescribed. Thus in the classic treatise of Dirac we learn the fundamental propositions:

So the theory is fundamentally about the results of ‘measurements’, and therefore presupposes in addition to the ‘system’ (or object) a ‘measurer’ (or subject). Now must this subject include a person? Or was there already some such subject–object distinction before the appearance of life in the universe? Were some of the natural processes then occurring, or occurring now in distant places, to be identified as ‘measurements’ and subjected to jumps rather than to the Schrödinger equation? Is ‘measurement’ something that occurs all at once? Are the jumps instantaneous? And so on.

I have been invited to speak on ‘foundations of quantum mechanics’ – and to a captive audience of high energy physicists! How can I hope to hold the attention of such serious people with philosophy? I will try to do so by concentrating on an area where some courageous experimenters have recently been putting philosophy to experimental test.

The area in question is that of Einstein, Podolsky, and Rosen. Suppose for example, that protons of a few MeV energy are incident on a hydrogen target. Occasionally one will scatter, causing a target proton to recoil. Suppose (Fig. 1) that we have counter telescopes T1 and T2 which register when suitable protons are going towards distant counters C1 and C2. With ideal arrangements registering of both T1 and T2 will then imply registering of both C1 and C2 after appropriate time decays. Suppose next that C1 and C2 are preceded by filters that pass only particles of given polarization, say those with spin projection + ½ along the z axis. Then one or both of C1 and C2 may fail to register. Indeed for protons of suitable energy one and only one of these counters will register on almost every suitable occasion – i.e., those occasions certified as suitable by telescopes T1 and T2. This is because proton–proton scattering at large angle and low energy, say a few MeV, goes mainly in S wave. But the antisymmetry of the final wave function then requires the antisymmetric singlet spin state.

The notion of morality appears to have been introduced into quantum theory by Wigner, as reported by Goldberger and Watson. The question at issue is the famous ‘reduction of the wave packet’. There are, ultimately, no mechanical arguments for this process, and the arguments that are actually used may well be called moral. This is a popular account of the subject. Very practical people not interested in logical questions should not read it. It is a pleasure for us to dedicate the paper to Professor Weisskopf, for whom intense interest in the latest developments of detail has not dulled concern with fundamentals.

Suppose that some quantity F is measured on a quantum mechanical system, and a result f obtained. Assume that immediate repetition of the measurement must give the same result. Then, after the first measurement, the system must be in an eigenstate of F with eigenvalue f. In general, the measurement will be ‘incomplete’, i.e., there will be more than one eigenstate with the observed eigenvalue, so that the latter does not suffice to specify completely the state resulting from the measurement. Let the relevant set of eigenstates be donoted by ϕfg. The extra index g may be regarded as the eigenvalue of a second observable G that commutes with F and so can be measured at the same time.

‘… the history of cosmic theories may without exaggeration be called a history of collective obsessions and controlled schizophrenias; and the manner in which some of the most important individual discoveries were arrived at reminds one of a sleepwalker's performance …’

This is a quotation from A. Koestler's book The Sleepwalkers. It is an account of the Copernican revolution, with Copernicus, Kepler, and Galilei as heroes. Koestler was of course impressed by the magnitude of the step made by these men. He was also fascinated by the manner in which they made it. He saw them as motivated by irrational prejudice, obstinately adhered to, making mistakes which they did not discover, which somehow cancelled at the important points, and unable to recognize what was important in their results, among the mass of details. He concluded that they were not really aware of what they were doing … sleepwalkers. I thought it would be interesting to keep Koestler's thesis in mind as we hear at this meeting about contemporary theories from contemporary theorists.

For many decades now our fundamental theories have rested on the two great pillars to which this meeting is dedicated: quantum theory and relativity. We will see that the lines of research opened up by these theories remain splendidly vital. We will see that order is brought into a vast and expanding array of experimental data. We will see even a continuing ability to get ahead of the experimental data … as with the existence and masses of the W and Z mesons.

Introduction

Cosmologists, even more than laboratory physicists, must find the usual interpretive rules of quantum mechanics a bit frustrating:

It would seem that the theory is exclusively concerned with ‘results of measurement’ and has nothing to say about anything else. When the ‘system’ in question is the whole world where is the ‘measurer’ to be found? Inside, rather than outside, presumably. What exactly qualifies some subsystems to play this role? Was the world wave function waiting to jump for thousands of millions of years until a single-celled living creature appeared? Or did it have to wait a little longer for some more highly qualified measurer – with a Ph.D.? If the theory is to apply to anything but idealized laboratory operations, are we not obliged to admit that more or less ‘measurement-like’ processes are going on more or less all the time more or less everywhere? Is there ever then a moment when there is no jumping and the Schrödinger equation applies?