Modern cosmology envisages a universe that, at its early moments, was very hot and very small. In these extreme conditions, experience on Earth gives little guide to the behaviour of matter. We must turn to elementary particle physics in an attempt to divine what matter was like. Conversely, as mentioned at the end of the Chapter 12, particle physics has raised questions that are probably beyond laboratory experiments to answer. Maybe evidence from the universe at large can help.

There are two general features of modern particle physics that make application to the early universe particularly interesting.

• Both the gluon theory of strong interactions (Chapter 11) and the electroweak theory (Chapter 12) employ some analogies with superconductivity (Chapter 10). As the temperature of a superconducting substance is raised, there is a phase transition above which superconductivity ceases. There are good reasons to believe that something similar should happen to the vacuum of particle physics. At the high temperatures of the early universe, the Bose condensation of Higgs particles should go away, revealing the gauge symmetry in an unbroken form (as in Section 12.4). Likewise, confinement of colour should go, and the gluons behave similarly to photons. Thus the physics of the early universe may have been, in some sense, simpler than physics now.

• In non-Abelian gauge theories, coupling strengths decrease as the typical energies explored increase (Section 11.3). In the early universe, the temperature is the relevant energy, so very early, when the temperature is very high, the particles may behave nearly like free ones. Then, in spite of the extreme conditions, physics may not be as difficult as might have been supposed.

The problem of quantum state reduction

The fundamental problem of quantum mechanics, as that theory is presently understood, is to make sense of the reduction of the state vector (i.e. collapse of the wave function), denoted here by R. This issue is usually addressed in terms of the ‘quantum measurement problem’, which is to comprehend how, upon measurement of a quantum system, this (seemingly) discontinuous R-process can come about. A measurement, after all, merely consists of the quantum state under consideration becoming entangled with a more extended part of the physical universe, e.g. with a measuring apparatus. This measuring apparatus – together with the observing physicist and their common environment – should, according to conventional understanding, all also have some quantum description. Accordingly, there should be a quantum description of this entire quantum state, involving not only the original system under consideration but also the apparatus, physicist, and remaining environment – and this entire state would be expected to evolve continuously, solely according to the Schrödinger equation (unitary evolution), here denoted by the symbol U.

Numerous different attitudes to R have been expressed over many years, ever since quantum mechanics was first clearly formulated. The most influential viewpoint has been the ‘Copenhagen interpretation’ of Niels Bohr, according to which the state vector │ψ〉 is not to be taken seriously as describing a quantum-level physical reality, but is to be regarded as merely referring to our (maximal) ‘knowledge’ of a physical system, and whose ultimate role is simply to provide us with a means to calculate probabilities when a measurement is performed on the system.

Introduction

Physicists who work on canonical quantum gravity will sometimes remark that the general covariance of general relativity is responsible for many of the thorniest technical and conceptual problems in their field. In particular, it is sometimes alleged that one can trace to this single source a variety of deep puzzles about the nature of time in quantum gravity, deep disagreements surrounding the notion of ‘observable’ in classical and quantum gravity, and deep questions about the nature of the existence of spacetime in general relativity.

Philosophers who think about these things are sometimes sceptical about such claims. We have all learned that Kretschmann was quite correct to urge against Einstein that the ‘General Theory of Relativity’ was no such thing, since any theory could be cast in a generally covariant form, and hence that the general covariance of general relativity could not have any physical content, let alone bear the kind of weight that Einstein expected it to. Friedman's assessment is widely accepted: ‘As Kretschmann first pointed out in 1917, the principle of general covariance has no physical content whatever: it specifies no particular physical theory; rather, it merely expresses our commitment to a certain style of formulating physical theories’ (Friedman 1983, p. 44). Such considerations suggest that general covariance, as a technically crucial but physically contentless feature of general relativity, simply cannot be the source of any significant conceptual or physical problems.

Before we, as philosophers, take a look at string theory I want to mention that more than one person has suggested to me that it is still too early for philosophical and foundational studies of string theory. Indeed, the suggestion emphasizes, since string theory is still in the process of development, and its physical and mathematical principles are not completely formulated, there is, in a sense, no theory for the philosopher to analyse. And I must admit that I think there is something to this suggestion. In a sense I hope I will make clear, there does not yet exist a precise mathematical formulation for string theory as there is Hilbert space for (elementary) quantum theory, and Riemann spacetime for general relativity. Because these latter formulations exist, we can ask precise questions and prove precise theorems about their interpretation. The Kochen–Specker theorem about non-contextualist hidden variable theories, the Fine–Brown proof of the insolvability of the quantum measurement problem, and the current determinism–hole argument debate are some examples. Without a clearly formulated mathematical structure, I don't think we can expect to get analogous distinctly stringy results.

This suggests a related worry. String theory, at least in the first quantized theory, is a relativistic quantum theory of strings (one-dimensional extended objects). One may well agree that of course all of the standard philosophical and foundational issues of quantum theory and relativity are still there, but be sceptical about whether string theory will either shed any light on these old problems, or give rise to stringy problems.

Introduction

At present, our physical worldview is deeply schizophrenic. We have, not one, but two fundamental theories of the physical universe: general relativity, and the Standard Model of particle physics based on quantum field theory. The former takes gravity into account but ignores quantum mechanics, while the latter takes quantum mechanics into account but ignores gravity. In other words, the former recognizes that spacetime is curved but neglects the uncertainty principle, while the latter takes the uncertainty principle into account but pretends that spacetime is flat. Both theories have been spectacularly successful in their own domain, but neither can be anything more than an approximation to the truth. Clearly some synthesis is needed: at the very least, a theory of quantum gravity, which might or might not be part of an overarching ‘theory of everything’. Unfortunately, attempts to achieve this synthesis have not yet succeeded.

Modern theoretical physics is difficult to understand for anyone outside the subject. Can philosophers really contribute to the project of reconciling general relativity and quantum field theory? Or is this a technical business best left to the experts? I would argue for the former. General relativity and quantum field theory are based on some profound insights about the nature of reality. These insights are crystallized in the form of mathematics, but there is a limit to how much progress we can make by just playing around with this mathematics.

Introduction

Prologue

Any branch of physics will pose various philosophical questions: for example about its concepts and general framework, and the comparison of these with analogous structures in other branches of physics. Indeed, a thoughtful consideration of any field of science leads naturally to questions within the philosophy of science.

However, in the case of quantum gravity we rapidly encounter fundamental issues that go well beyond questions within the philosophy of science in general. To explain this point, we should first note that by ‘quantum gravity’ we mean any approach to the problem of combining (or in some way ‘reconciling’) quantum theory with general relativity. An immense amount of effort has been devoted during the past forty years to combining these two pillars of modern physics. Yet although a great deal has been learned in the course of this endeavour, there is still no satisfactory theory: rather, there are several competing approaches, each of which faces severe problems, both technical and conceptual.

This situation means that there are three broad ways in which quantum gravity raises philosophical questions beyond the philosophy of science in general.

1. Each of the ‘ingredient theories’ – quantum theory and general relativity – poses significant conceptual problems in its own right. Since several of these problems are relevant for various topics in quantum gravity (as Chapter 8 of this volume bears witness), we must discuss them, albeit briefly. We will do this in Section 2.2, to help set the stage for our study of quantum gravity. But since these problems are familiar from the literature in the philosophy, and foundations, of physics, we shall be as brief as possible.

2. […]

Introduction

This chapter addresses issues raised by Christian (Chapter 14), Penrose (Chapter 13), and others in this volume: What is the physical significance, if any, of general covariance? Norton (1993) has given a valuable historical survey of the tangled history of this question. His subtitle ‘Eight decades of dispute’ is very apt! Many people attribute the difficulties inherent in the quantization of gravity to the general covariance of Einstein's general theory of relativity, so clarification of its true nature is important.

Christian gives a clear account of what may be called the current orthodoxy with regard to the status of general covariance, and I disagree with little of what he says. However, in my view that merely draws attention to a problem – it does not say how the problem is overcome. I believe general relativity does resolve the problem, but that this has escaped notice. If this view is correct – and most of this chapter will be devoted to arguing that it is – then I believe it has important implications for the problem that Penrose is trying to solve in Chapter 13. I think he is trying to solve a problem that has already been solved. Nevertheless, I do feel all attempts to construct viable models of physical collapse are valuable, since they constitute one of the few alternatives to many-worlds interpretations of quantum mechanics.

Introduction

The world of classical, relativistic physics is a world in which the interactions between material bodies are mediated by fields. The ‘black body catastrophe’ provided the first indication that these fields (in particular the electromagnetic field) should be ‘quantized’. Modern field theory contains quantum field–theoretic descriptions of three of the four known interactions (forces) – all except gravity. It is characteristic of the theories of these three forces that the values of the fields carrying the forces are subject to the Heisenberg uncertainty relations, such that not all the field strengths at any given point can be specified with arbitrary precision.

Gravity, however, has resisted quantization. There exist several current research programmes in this area, including superstring theory and canonical quantum gravity. One often comes across the claim that the gravitational field must be quantized, and that quantization will give rise to a similar local uncertainty in the gravitational field. Here we will examine this claim, and see how the very things that make general relativity such an unusual ‘field’ theory not only make the quantization of the theory so technically difficult, but make the very idea of a ‘fluctuating gravitational field’ so problematic.

What is a field?

Maxwell's theory of electromagnetism describes the interaction of electrically charged matter (consisting of ‘charges’) and the electromagnetic field.

In recent years it has sometimes been difficult to distinguish between articles in quantum gravity journals and articles in philosophy journals. It is not uncommon for physics journals such as Physical Review D, General Relativity and Gravitation and others to contain discussion of philosophers such as Parmenides, Aristotle, Leibniz, and Reichenbach; meanwhile, Philosophy of Science, British Journal for the Philosophy of Science and others now contain papers on the emergence of spacetime, the problem of time in quantum gravity, the meaning of general covariance, etc. At various academic conferences on quantum gravity one often finds philosophers at physicists' gatherings and physicists at philosophers' gatherings. While we exaggerate a little, there is in recent years a definite trend of increased communication (even collaboration) between physicists working in quantum gravity and philosophers of science. What explains this trend?

Part of the reason for the connection between these two fields is no doubt negative: to date, there is no recognized experimental evidence of characteristically quantum gravitational effects. As a consequence, physicists building a theory of quantum gravity are left without direct guidance from empirical findings. In attempting to build such a theory almost from first principles it is not surprising that physicists should turn to theoretical issues overlapping those studied by philosophers.

But there is also a more positive reason for the connection between quantum gravity and philosophy: many of the issues arising in quantum gravity are genuinely philosophical in nature.

Introduction

One of the few propositions about quantum gravity that most physicists in the field would agree upon, that our notion of spacetime must, at best, be altered considerably in any theory conjoining the basic principles of quantum mechanics with those of general relativity, will be questioned in this chapter. We will argue, in fact, that most, if not all, of the conceptual problems in quantum gravity arise from the sort of thinking on display in the preceding quotation.

Introduction

In 1976, J.S. Bell published a paper on ‘How to teach special relativity’ (Bell 1976). The paper was reprinted a decade later in his well-known book Speakable and unspeakable in quantum mechanics – the only essay to stray significantly from the theme of the title of the book. In the paper Bell was at pains to defend a dynamical treatment of length contraction and time dilation, following ‘very much the approach of H.A. Lorentz’ (Bell 1987, p. 77).

Just how closely Bell stuck to Lorentz's thinking in this connection is debatable. We shall return to this question shortly. In the meantime, we briefly rehearse the central points of Bell's rather unorthodox argument.

Bell considered a single atom modelled by an electron circling a more massive nucleus, ignoring the back-effect of the field of the electron on the nucleus. The question he posed was: what is the prediction in Maxwell's electrodynamics (taken to be valid in the frame relative to which the nucleus is initially at rest) as to the effect on the electron orbit when the nucleus is set (gently) in motion in the plane of the orbit? Using only Maxwell's field equations, the Lorentz force law and the relativistic formula linking the electron's momentum and its velocity – which Bell attributed to Lorentz – he concluded that the orbit undergoes the familiar longitudinal (‘Fitzgerald’ (sic)) contraction, and its period changes by the familiar (‘Larmor’) dilation.

Introduction

Our basic ideas about physics went through several upheavals early this century. Quantum mechanics taught us that the classical notions of the position and velocity of a particle were only approximations of the truth. With general relativity, spacetime became a dynamical variable, curving in response to mass and energy. Contemporary developments in theoretical physics suggest that another revolution may be in progress, through which a new source of ‘fuzziness’ may enter physics, and spacetime itself may be reinterpreted as an approximate, derived concept (see Fig. 5.1). In this article I survey some of these developments.

Let us begin our excursion by reviewing a few facts about ordinary quantum field theory. Much of what we know about field theory comes from perturbation theory; perturbation theory can be described by means of Feynman diagrams, or graphs, which are used to calculate scattering amplitudes. Textbooks give efficient algorithms for evaluating the amplitude derived from a diagram. But let us think about a Feynman diagram intuitively, as Feynman did, as representing a history of a spacetime process in which particles interact by the branching and rejoining of their worldlines. For instance, Fig. 5.2 shows two incident particles, coming in at a and b, and two outgoing particles, at c and d. These particles branch and rejoin at spacetime events labelled x, y, z, and w in the figure.

According to Feynman, to calculate a scattering amplitude one sums over all possible arrangements of particles branching and rejoining.

The incomplete revolution

Quantum mechanics (QM) and general relativity (GR) have profoundly modified our understanding of the physical world. However, they have left us with a general picture of the physical world which is unclear, incomplete, and fragmented. Combining what we have learned about our world from the two theories and finding a new synthesis is a major challenge, perhaps the major challenge, in today's fundamental physics.

The two theories have opened a major scientific revolution, but this revolution is not completed. Most of the physics of this century has been a series of triumphant explorations of the new worlds opened by QM and GR. QM leads to nuclear physics, solid state physics, and particle physics; GR leads to relativistic astrophysics, cosmology, and is today leading us towards gravitational astronomy. The urgency of applying the two theories to increasingly larger domains, and the momentous developments and the dominant pragmatic attitude of the middle of the twentieth century have obscured the fact that a consistent picture of the physical world, more or less stable for three centuries, has been lost with the advent of QM and GR. This pragmatic attitude cannot be satisfactory, or productive, in the long run. The basic Cartesian–Newtonian notions such as matter, space, time, and causality, have been deeply modified, and the new notions do not stay together. At the basis of our understanding of the world reigns a surprising confusion.

It has now been 25 years since Hawking (Hawking 1974, 1975, Bardeen, Carter, and Hawking 1973) first surprised the world of physics with his analysis of quantum fields near black holes. Black holes, as their name implies, were believed to be objects into which things could fall, but out of which nothing could come. They were the epitome of black and dark objects. However, Hawking's analysis predicted that black holes should radiate, the radiation should be continuous and thermal, and the temperature should be inversely proportional to the mass of the black hole. Since black holes can also be said to have an energy proportional to their mass, this result led to opening of a whole new field of black hole thermodynamics.

That black holes could behave like thermodynamic objects had been intimated by results over the the previous five years. Christodolou (1970), Hawking and Ellis (1973, especially Lemma 9.2.2), Misner, Thorne, and Wheeler (1973) and Bekenstein (1973, 1974) had shown that there were certain formal similarities between black holes and thermodynamic objects. In particular, if one assumed positive energy for matter (an uncontested assumption), then – as Hawking most clearly showed – the area of a black hole horizon does not decrease. However, this formal similarity with entropy, which also does not decrease for an isolated system, did not seem to have any real relation with thermodynamics. The entropy of a body does not decrease only if the body is isolated, and not in interaction with any other system.