In this chapter we set up a classical Lagrangian and Hamiltonian dynamics for matrix models. The fundamental idea is to set up an analog of classical dynamics in which the phase space variables are non-commutative, and the basic tool that allows one to accomplish this is cyclic invariance under a trace. Since no assumptions about commutativity of the phase space variables (such as canonical commutators/anticommutators) are made at this stage, the dynamics that we set up is not the same as standard quantum mechanics. Quantum mechanical behavior will be seen to emerge only when, in Chapters 4 and 5, we study the statistical mechanics of the classical matrix dynamics formulated here.

In Section 1.1, we introduce our basic notation for bosonic and fermionic matrices, and give the cyclic identities that will be used repeatedly throughout the book. In Section 1.2, we define the derivative of a trace quantity with respect to an operator, and give the basic properties of this definition. In Section 1.3, we use the operator derivative to formulate a Lagrangian and Hamiltonian dynamics for matrix models. In Section 1.4, we introduce a generalized Poisson bracket appropriate to trace dynamics, constructed from the operator derivative defined in Section 1.2, and give its properties and some applications. Finally, in Section 1.5 we discuss the relation between the trace dynamics time evolution equations, and the usual unitary Heisenberg picture equations of motion obtained when one assumes standard canonical commutators/anticommutators.

Bosonic and fermionic matrices and the cyclic trace identities

We shall assume finite-dimensional matrices, although ultimately an extension to the infinite-dimensional case may be needed. The matrix elements of these matrices will be constructed from ordinary complex numbers, and from complex anticommuting Grassmann numbers.

In Section 2.4, we illustrated the trace dynamics formalism by constructing the trace dynamics analog of a simple field theory model, in which a Dirac fermion interacts with a scalar Klein–Gordon field. Much of the recent literature in quantum field theory has concerned itself with supersymmetric theories, in which invariance under the Poincaré group has been extended to invariance under the graded Poincaré group, and theories of this type are considered likely to play a central role in the ultimate unification of the forces. Our aim in this chapter (which can be omitted on a first reading) is to show that the trace dynamics formalism naturally extends to globally supersymmetric theories. Specifically, we shall see that, when there is a global supersymmetry, there is a conserved trace supersymmetry current with a time-independent trace supercharge Qα, that together with the trace four momentum obeys the Poincaré supersymmetry algebra under the generalized Poisson bracket of Eq. (1.11a). We shall illustrate this statement with three concrete examples, the trace dynamics versions (Adler 1997a,b) of the Wess–Zumino model (Section 3.1), the supersymmetric Yang–Mills model (Section 3.2), and the so-called “matrix model for M theory” (Section 3.3). These three examples are worked out using component field methods; we close in Section 3.4 with a short discussion of a superspace approach, and of the obstruction that prevents the construction of a trace dynamics theory with local supersymmetry.

The Wess–Zumino model

We begin with the trace dynamics transcription of the Wess–Zumino model.

To keep the discussion of this book self-contained, a number of topics that are briefly mentioned in the text are treated in more detail in the appendices that follow.

The notation of the appendices follows that generally used in the text. Throughout this book, we indicate sums explicitly, except that the usual Einstein summation convention is used for sums over Greek letter four-vector and tensor indices, and a summation convention is used for the indices i, j in Section 3.3. Our Minkowski metric convention is ημν = diag(1, 1, 1, –1), and we have taken the velocity of light to be unity, so that c does not appear in the equations. However, Planck's constant is generally retained (except in the formulas of Sections 1.5 and 6.3 through 6.5 where we set = 1), because our approach implies that it has a dynamical origin.

Fermionic quantities are represented throughout by Grassmann numbers. Real Grassmann numbers are constructed as products of a basis Xr of real Grassmann elements, that obey the anticommutative algebra {Xr, Xs} = 0 for all r, s, which implies that the square of any Grassmann element vanishes. Complex Grassmann numbers have real and imaginary parts that are real Grassmann numbers. A product of an even number of Grassmann basis elements, multiplied by a complex number coefficient, is called even grade or bosonic, and a product of an odd number of Grassmann basis elements, with a complex number coefficient, is called odd grade or fermionic. Grassmann integration is defined by and so is effectively the same as differention from the left.

In 1957 H. Everett published a paper setting out what seemed to be a radically new interpretation of quantum mechanics. His approach has recently received increasing attention. He did not refer to the ideas of de Broglie of thirty years before nor to the intervening elaboration of those ideas by Bohm. Yet it will be argued here that the elimination of arbitrary and inessential elements from Everett's theory leads back to, and throws new light on, the concepts of de Broglie.

Everett was motivated by the notion of a quantum theory of gravitation and cosmology. In a thoroughly quantum cosmology, a quantum mechanics of the whole world, the wave function of the world could not be interpreted in the usual way. For this usual interpretation refers only to the statistics of measurement results for an observer intervening from outside the quantum system. When that system is the whole world, there is nothing outside. This situation presents no particular difficulty for the traditional (or ‘Copenhagen’) philosophy, which holds that a classical conception of the macroscopic world is logically prior to the quantum conception of the microscopic. The microscopic world is described by wave functions which are determined by and have implications for macroscopic phenomena in experimental set-ups.

It has been argued that quantum mechanics is not locally causal and cannot be embedded in a locally causal theory. That conclusion depends on treating certain experimental parameters, typically the orientations of polarization filters, as free variables. Roughly speaking it is supposed that an experimenter is quite free to choose among the various possibilities offered by his equipment. But it might be that this apparent freedom is illusory. Perhaps experimental parameters and experimental results are both consequences, or partially so, of some common hidden mechanism. Then the apparent non-locality could be simulated.

This possibility is the starting point of a paper by Clauser, Home and Shimony (CHS hereafter), which is valuable in particular for a careful mathematical formulation of the assumption which excludes such a conspiracy. In this connection they severely criticize my own ‘theory of local beables’ (B hereafter). Much of their criticism is perfectly just. In B there were jumps in the argument, and the assumption in question was not stated at the appropriate place, but only later and inadequately. However, I do not agree with CHS that this assumption, when carefully formulated, is an unreasonable one.

I will organize these remarks around the three phrases in which I belatedly formulated the hypothesis in B, Section 8.

Introduction

The paradox of Einstein, Podolsky and Rosen was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables. These additional variables were to restore to the theory causality and locality. In this note that idea will be formulated mathematically and shown to be incompatible with the statistical predictions of quantum mechanics. It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty. There have been attempts to show that even without such a separability or locality requirement no ‘hidden variable’ interpretation of quantum mechanics is possible. These attempts have been examined elsewhere and found wanting. Moreover, a hidden variable interpretation of elementary quantum theory has been explicitly constructed. That particular interpretation has indeed a grossly non-local structure. This is characteristic, according to the result to be proved here, of any such theory which reproduces exactly the quantum mechanical predictions.

Formulation

With the example advocated by Bohm and Aharonov, the EPR argument is the following. Consider a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions. Measurements can be made, say by Stern–Gerlach magnets, on selected components of the spins σ1 and σ2.

Introduction

Bohm's 1952 papers on quantum mechanics were for me a revelation. The elimination of indeterminism was very striking. But more important, it seemed to me, was the elimination of any need for a vague division of the world into ‘system’ on the one hand, and ‘apparatus’ or ‘observer’ on the other. I have always felt since that people who have not grasped the ideas of those papers … and unfortunately they remain the majority … are handicapped in any discussion of the meaning of quantum mechanics.

When the cogency of Bohm's reasoning is admitted, a final protest is often this: it is all nonrelativistic. This is to ignore that Bohm himself, in an appendix to one of the 1952 papers, already applied his scheme to the electromagnetic field. And application to scalar fields is straightforward. However until recently, to my knowledge, no extension covering Fermi fields had been made. Such an extension will be sketched here. The need for Fermi fields might be questioned. Fermions might be composite structures of some kind. But also they might not be, or not all. The present exercise will not only include Fermi fields, but even give them a central role. The dependence on the ideas of de Broglie and Bohm, and also on my own simplified extension to cover spin, will be manifest to those familiar with these things. However no such familiarity will be assumed.

Introduction

I have borrowed the title of a characteristic paper by Schrödinger (Schrödinger, 1952). In it he contrasts the smooth evolution of the Schrödinger wavefunction with the erratic behaviour of the picture by which the wavefunction is usually supplemented, or ‘interpreted’, in the minds of most physicists. He objects in particular to the notion of ‘stationary states’, and above all to ‘quantum jumping’ between those states. He regards these concepts as hangovers from the old Bohr quantum theory, of 1913, and entirely unmotivated by anything in the mathematics of the new theory of 1926. He would like to regard the wavefunction itself as the complete picture, and completely determined by the Schrödinger equation, and so evolving smoothly without ‘quantum jumps’. Nor would he have ‘particles’ in the picture. At an early stage, he had tried to replace ‘particles’ by wavepackets (Schrödinger, 1926). But wavepackets diffuse. And the paper of 1952 ends, rather lamely, with the admission that Schrödinger does not see how, for the present, to account for particle tracks in track chambers … nor, more generally, for the definiteness, the particularity, of the world of experience, as compared with the indefiniteness, the waviness, of the wavefunction. It is the problem that he had had (Schrödinger, 1935a) with his cat. He thought that she could not be both dead and alive.

Surely, after 62 years, we should have an exact formulation of some serious part of quantum mechanics? By ‘exact’ I do not of course mean ‘exactly true’. I mean only that the theory should be fully formulated in mathematical terms, with nothing left to the discretion of the theoretical physicist … until workable approximations are needed in applications. By ‘serious’ I mean that some substantial fragment of physics should be covered. Non-relativistic ‘particle’ quantum mechanics, perhaps with the inclusion of the electromagnetic field and a cut-off interaction, is serious enough. For it covers ‘a large part of physics and the whole of chemistry’. I mean too, by ‘serious’, that ‘apparatus’ should not be separated off from the rest of the world into black boxes, as if it were not made of atoms and not ruled by quantum mechanics.

The question, ‘… should we not have an exact formulation …?’, is often answered by one or both of two others. I will try to reply to them: Why bother? Why not look it up in a good book?

Why bother?

Perhaps the most distinguished of ‘why bother?’ers has been Dirac. He divided the difficulties of quantum mechanics into two classes, those of the first class and those of the second. The second-class difficulties were essentially the infinities of relativistic quantum field theory. Dirac was very disturbed by these, and was not impressed by the ‘renormalisation’ procedures by which they are circumvented. Dirac tried hard to eliminate these second-class difficulties, and urged others to do likewise.

Simon Capelin, of Cambridge University Press, suggested that I send him my papers on quantum philosophy and let him make them into a book. I have done so. The papers, from the years 1964–1986, are presented here in the order, as far as I now can tell, in which they were written. But of course that is not the order, if any, in which they should be read.

Papers 18 and 20, ‘Speakable and unspeakable in quantum mechanics’ and ‘Six possible worlds of quantum mechanics’, are nontechnical introductions to the subject. They are meant to be intelligible to nonphysicists. So also is most of paper 16, ‘Bertlmann's socks and the nature of reality’, which is concerned with the problem of apparent action at a distance.

For those who know something of quantum formalism, paper 3, ‘The moral aspect of quantum mechanics’, introduces the infamous ‘measurement problem’. I thank Michael Nauenberg, who was co-author of that paper, for permission to include it here. At about the same level, paper 17, ‘On the impossible pilot wave’, begins the discussion of ‘hidden variables’, and of related ‘impossibility’ proofs.

More elaborate discussions of the ‘measurement problem’ are given in paper 6, ‘On wavepacket reduction in the Coleman–Hepp model’, and in 15, ‘Quantum mechanics for cosmologists’. These show my conviction that, despite numerous solutions of the problem ‘for all practical purposes’, a problem of principle remains. It is that of locating precisely the boundary between what must be described by wavy quantum states on the one hand, and in Bohr's ‘classical terms’ on the other.

I suppose one could imagine laws of physics which would dictate that a world be exactly so, and not otherwise, allowing no detail to be varied. But what could dictate that those laws of physics be ‘the’ laws of physics? By considering a spectrum of possible laws, one could again consider a spectrum of possible worlds.

In fact the laws of physics of our actual world, as presently understood, have no such dictatorial character. So that even with the laws given, a spectrum of different worlds is possible. There are two kinds of freedom. Although the laws say something about how a given state of the world may develop, they say nothing (or anyway very little) about in what state the world should start. So, to begin with, we have freedom as regards ‘initial conditions’. To go on with, the future that can evolve from a given present is not uniquely determined, according to contemporary orthodoxy. The laws list various possibilities, and attach to them various probabilities.

The relation between the set of possibilities and the unique actuality which emerges is quite peculiar in modern ‘quantum theory’ – the contemporary all-embracing basic physical theory. The absence of determinism, the probabilistic nature of the assertions of the theory, is already a little peculiar… at least in the light of pre-twentieth-century ‘classical’ physics. But after all everyday life, if not classical physics, prepares us very well for the idea that not everything is predictable, that chance is important.