In this chapter I shall consider one of the most important questions concerned with the long-term future of the universe and, indeed, one of the most important questions in physics. The question is whether or not the proton is stable. Until recently it had been assumed by physicists that the proton was indeed stable, that is, a proton left to itself would last forever. Recently, however, some theories of elementary particles have been put forward which imply that the proton is unstable, with a very long lifetime. In this chapter we shall try to see in what way these theories arise, and what are the consequences of proton decay. Before we can understand where the new theories fit, we shall have to know something about the theory of elementary particles, in much more detail than we considered in Chapter 4. To remind the reader I may repeat some of the points made earlier.

Every since the time of the ancient Greeks, people have wondered what is the ultimate nature of matter. They have wondered about the ultimate constituents of matter and about the manner in which these constituents affect each other or interact with one another. The Greek physical philosopher Democritus, who was born in the fifth century bc, speculated that all matter was made of atoms, which were eternal, indivisible and invisible. In the past hundred years or so and particularly in the last three or four decades a tremendous effort has gone into the investigation of this problem.

In quantum mechanics it turns out that phenomena which are forbidden in classical physics (such as particles escaping from a black hole) have a small, but real chance of happening by a mechanism called tunneling, whereby a particle crosses a ‘classical’ barrier. By a classical barrier we mean one that would be a barrier if only the laws of classical physics operated. Thus an electron which does not have sufficient energy to surmount the barrier produced by an electrical field bounces off the barrier and cannot penetrate it according to the laws of classical physics, as shown in the upper sketch in Fig. 10.1. The wavelike properties of matter in quantum mechanics, however, give the electron a small chance of getting through (see lower sketch in Fig. 10.1). This phenomenon of tunneling is important in radioactive decay of a heavy nucleus such as a uranium or a radium nucleus and also in some processes in electronics. Since quantum effects are essentially microscopic effects, it is difficult to give an example of the phenomenon of tunneling in terms of every day happenings, but presently we shall try to explain radioactivity in such terms.

We shall see that the phenomenon of quantum tunneling causes some slow and subtle changes in the remaining pieces of matter after all the black holes have gone, or even before the black holes disappear. These processes would not be possible according to classical physics, since the latter implies that the matter in the form of white dwarfs, neutron stars and other smaller pieces of matter would stay in the same form forever.

In this chapter we shall discuss briefly how stars are born and how they evolve during their life, and then we shall consider in some detail how they eventually die, that is, reach the three final states of white dwarf, neutron star and black hole. We shall also discuss the phenomenon of supernova, a phenomenon which is relevant to the final state of some stars.

The precise manner in which stars are formed is not clearly understood. The region between the stars is not empty but consists of gas clouds, consisting mostly of hydrogen, and dust grains of various kinds. The material between the stars is not uniformly distributed in space but is spread in a patchy fashion. In most places the density of gas is very low, a typical density being 10−19 kg/m3, that is about a hundred million (108) hydrogen atoms per cubic meter. Now the gravitational force between two portions of matter is inversely proportional to the square of the distance between them (for example, if the distance is doubled, the force becomes a quarter) and directly proportional to the product of the masses (if the masses of both portions are doubled, the force becomes four times as strong). Thus in a gas cloud, the higher the density of the gas, the stronger the gravitational attraction of different parts for each other.

Occasionally a cloud will become sufficiently dense and massive for gravitational attraction to draw it close together (this could conceivably happen also in the neighbourhood of a supernova explosion, as will be explained later).

As mentioned in the last chapter, when the mass of the star is greater than about three times the mass of the Sun, even the neutron Fermi pressure and other outward forces exerted by neutrons are not sufficient to withstand the force of gravity. There are no known forces in nature that can balance the force of gravity under these circumstances, and the star collapses and collapses until it reaches a very small volume with very high density. The precise nature of the final form that matter takes in this case is not known. What is indicated by Einstein's theory of gravitation is that when all the matter in the star goes within a certain small volume, no further communication is possible with the matter inside this volume, since any rays of light (which are the fastest possible signals) leaving it are pulled back towards the central region by the strength of gravity. The star then becomes a ‘black hole’. It is called ‘black’ because no radiation of any kind comes from within it. If the star has no rotation initially, the black hole quickly settles down to a spherical shape, the radius of the hole (from within which nothing can come out) being dependent on the total mass. For a mass M, the radius of the black hole, known as the Schwarzschild radius (after the German astronomer Karl Schwarzschild (1873–1916), who first found the solution of Einstein's equations in 1916 corresponding to a black hole) is 2GM/c2, where G is the Newtonian gravitational constant and c is the velocity of light.

Once the theory of free quantum fields in curved spacetime had been worked out, the most natural extension was to include the effects of non-gravitational self and mutual interactions. Although this topic is still being developed, the basic framework is well established, and in this final chapter we outline the formal steps necessary for the computation of particle creation effects and the renormalization of 〈Tµν〉.

Two questions immediately spring to mind once interactions are included. The first is to what extent interactions can stimulate or inhibit particle creation by gravity over and above the free field case. Of course, interactions can lead to non-gravitational creation too, but we are more interested in processes that would be forbidden in Minkowski space, such as the simultaneous creation of a photon with an electron–positron pair.

The second question concerns renormalization theory. Will a field theory (e.g. Q.E.D.) that is renormalizable in Minkowski space remain so when the spacetime has a non-trivial topology or curvature? This question is of vital importance, for if a field theory is likely to lose its predictive power as soon as a small gravitational perturbation occurs, then its physical utility is suspect. It turns out to be remarkably difficult to establish general renormalizability, and significant progress has so far been limited to the so-called λϕ4 theory.

A third issue of great interest concerns black hole radiance. Is the Hawking flux precisely thermal even in the presence of field interactions? If not, a violation of the second law of thermodynamics seems possible.

In January 1974, Hawking (1974) announced his celebrated result that black holes are not, after all, completely black, but emit radiation with a thermal spectrum due to quantum effects. This announcement proved to be a pivotal event in the development of the theory of quantum fields in curved spacetime, and greatly increased the attention given to this subject by other workers. In devoting an entire chapter to the topic of quantum black holes, we are reflecting the widespread interest in Hawking's remarkable discovery.

With the presentation of all the major aspects of free quantum field theory in curved spacetime complete, we here deploy all the various techniques described in the foregoing chapters. The basic result – that the gravitational disturbance produced by a collapsing star induces the creation of an outgoing thermal flux of radiation – is not hard to reproduce. The wavelength of radiation leaving the surface of a star undergoing gravitational collapse to form a black hole is well known to increase exponentially. It therefore seems plausible that the standard incoming complex exponential field modes should, after passing through the interior of the collapsing star and emerging on the remote side, also be exponentially redshifted. It is then a simple matter to demonstrate that the Bogolubov transformation between these exponentially redshifted modes and standard outgoing complex exponential modes is Planckian in structure. This implies that the ‘in vacuum’ state contains a thermal flux of outgoing particles.

Having invested so much effort in mastering curved space quantum field theory, the reader may be dismayed to return to the topic of flat spacetime. Flat spacetime does not, however, imply Minkowski space quantum field theory.

We consider three main topics in which the general curved spacetime formalism must be applied to achieve sensible results, even though the geometry is flat. This enables some non-trivial geometrical effects to be explored within the considerable simplification afforded by a flat geometry. In particular, we are able to discuss 〈Tµν〉 in some special cases without employing the full theory of curved space regularization and renormalization to be developed in chapter 6.

The first case examines the effects of a non-trivial topology. We do not treat particle creation at this stage, but limit the discussion to 〈Tµν〉, which is nonzero even for the vacuum. This topic is one of the few in our subject which makes contact with laboratory physics, for the disturbance to the electromagnetic vacuum induced by the presence of two parallel conducting plates is actually observable. The force of attraction that appears is called the Casimir effect, and has been extensively discussed in the literature.

The treatment of boundary surfaces leads naturally to a very simple, yet extremely illuminating, system that is well worth studying in detail. This is the case of the ‘moving mirror’, in which a boundary at which the quantum field is constrained moves about.

This short chapter presents some explicit examples of the theory of regularization and renormalization discussed in chapter 6. The number of spacetimes for which one may compute 〈Tµν〉 in terms of simple functions is extremely limited, and we think it probable that all such cases have been included either here, in chapter 6, or in our references.

Special importance is attached to the Robertson–Walker models, both because of their cosmological significance, and also because, being conformally flat, they provide a good illustration of conformal anomalies at work. However, precisely because of their simplicity, these models do not display the full non-local structure of the stress-tensor, and in §7.3 we turn briefly to the less elegant but more realistic example of an anisotropic, homogeneous cosmological model.

Although the primary subject of this book is the theory of quantum fields propagating in a prescribed background spacetime, the motivation for much of this work rests with its possible application to cosmological and astrophysical situations, where the gravitational dynamics must be taken into account. Many cosmologists, for example, believe that the back-reaction of quantum effects induced by the background gravitational field could have a profound effect on the dynamical evolution of the early universe, such as bringing about isotropization. We do not dwell in detail on this important extension of the theory, but note that the results presented here constitute the starting point for such investigations. A short discussion of the wider cosmological implications is given in §7.4.

The basic formalism of quantum field theory is generalized to curved spacetime in this chapter, in a straightforward way. The discussion is preceded by a very brief summary of pseudo-Riemannian geometry. The treatment is in no way intended to be complete, and we refer the reader to Weinberg (1972), Hawking & Ellis (1973), or Misner, Thorne & Wheeler (1973) for further details. Readers unfamiliar with conformal transformations and Penrose conformal diagrams are advised to read §3.1 carefully, however.

The basic generalization of the particle concept to curved spacetime is readily accomplished. What is not so easy is the physical interpretation of the formalism so developed. There has, in fact, been a certain amount of controversy over the meaning – and meaningfulness – of the particle concept when a background gravitational field is present. In some cases, such as for static spacetimes, the concept seems well defined, while in others (e.g. spacetimes that admit closed timelike world lines or do not everywhere possess Cauchy surfaces) the notion of particle can seem hopelessly obscure. We restrict consideration to ‘well-behaved’ spacetimes, and do not embark upon a philosophical discourse about the meaning of particles. Instead we relate the formalism directly to what an actual particle detector might be expected to register in the particular quantum state of interest. It is in this concrete operational sense that we define particles in curved spacetime. Although this approach has been studied before, we give the most developed treatment of particle detectors so far.

The subject of quantum field theory in curved spacetime, as an approximation to an as yet inaccessible theory of quantum gravity, has grown tremendously in importance during the last decade. In this book we have attempted to collect and unify the vast number of papers that have contributed to the rapid development of this area. The book also contains some original material, especially in connection with particle detector models and adiabatic states.

The treatment is intended to be both pedagogical and archival. We assume no previous acquaintance with the subject, but the reader should preferably be familiar with basic quantum field theory at the level of Bjorken & Drell (1965) and with general relativity at the level of Weinberg (1972) or Misner, Thorne & Wheeler (1973). The theory is developed from basics, and many technical expressions are listed for the first time in one place. The reader's attention is drawn to the list of conventions and abbreviations on page ix, and the extensive references and bibliography.

In preparing this book we have drawn upon the material of a very large number of authors. In adapting certain published material (including that of the authors) we have gratuitously made what we consider to be corrections, occasionally without explicitly warning the reader that our use of that material differs from the original publications.

The last decade has witnessed remarkable progress in the construction of a unified theory of the forces of nature. The electromagnetic and weak interactions have received a unified description with the Weinberg–Salam theory (Weinberg 1967, Salam 1968), while attempts to incorporate the strong interaction as described by quantum chromodynamics into a wider gauge theory seem to be achieving success with the so-called grand unified theories (Georgi & Glashow 1974, for a review see Cline & Mills 1978).

The odd one out in this successive unification is gravity. Not only does gravity stand apart from the other three forces of nature, it stubbornly resists attempts to provide it with a quantum framework. The quantization of the gravitational field has been pursued with great ingenuity and vigour over the past forty years (for reviews see Isham 1975, 1979a, 1981) but a completely satisfactory quantum theory of gravity remains elusive. Perhaps the most hopeful current approaches are the supergravity theories, in which the graviton is regarded as only one member of a multiplet of gauge particles including both fermions and bosons (Freedman, van Nieuwenhuizen & Ferrara 1976, Deser & Zumino 1976; for a review see van Nieuwenhuizen & Freedman 1979).

In the absence of a viable theory of quantum gravity, can one say anything at all about the influence of the gravitational field on quantum phenomena? In the early days of quantum theory, many calculations were undertaken in which the electromagnetic field was considered as a classical background field, interacting with quantized matter.

In this chapter we shall summarize the essential features of ordinary Minkowski space quantum field theory, with which we assume the reader has a working knowledge. A great deal of the formalism can be extended to curved spacetime and non-trivial topologies with little or no modification. In the later chapters we shall follow the treatment given here.

Most of the detailed analysis will refer to a scalar field, but the main results will be listed for higher spins also. This restriction will enable the important features of curved space quantum field theory to emerge with the minimum of mathematical complexity.

Much of the chapter will be familiar from textbooks such as Bjorken & Drell (1965), but the reader should take special note of the results on the expectation value of the stress–energy–momentum tensor and vacuum divergence (§2.4), as these will play a central role in what follows. Special importance also attaches to Green functions, treated in detail in §2.7. The reader may be unfamiliar with thermal Green functions and metric Euclideanization. As these will be essential for an understanding of the quantum black hole system, an outline of this topic is given here.

Finally, although we shall not develop a lot of our formalism using the Feynman path-integral technique, we do make use of the basic structure of the path integral in the work on renormalization in chapter 6, and again on interacting fields in curved space in chapter 9.

Since the book first went to press, there have been several important advances in this subject area. The topic of interacting fields in curved space has been greatly developed, especially in connection with the phenomenon of symmetry breaking and restoration in the very early universe, where both high temperatures and spacetime curvature are significant. A direct consequence of this work has been the formulation of the so-called inflationary universe scenario, in which the universe undergoes a de Sitter phase in the very early stages. This work has focussed attention once more on quantum field theory in de Sitter space, and on the calculation of 〈φ2〉. A comprehensive review of the inflationary scenario is given in The Very Early Universe, edited by G.W. Gibbons, S.W. Hawking and S.T.C. Siklos (Cambridge University Press, 1983).

Further results of a technical nature have recently been obtained concerning a number of the topics considered in this book. Mention should be made of the work of M.S. Fawcett, who has finally calculated the quantum stress tensor for a Schwarzschild black hole (Commun. Math. Phys., 81 (1983), 103), and of W.G. Unruh & R.M. Wald, who have clarified the thermodynamic properties of black holes by appealing to the effects of accelerated mirrors close to the event horizon (Phys. Rev. D, 25 (1982), 942; 27 (1983), 2271). Interest has also arisen over field theories in higher-dimensional spacetimes, in which Casimir and other vacuum effects become important.

In previous chapters the production of quanta by a changing gravitational field was studied in detail. It was pointed out that only in exceptional circumstances does the particle concept in curved space quantum field theory correspond closely to the intuitive physical picture of a subatomic particle. In general, no natural definition of particle exists, and particle detectors will respond in a variety of ways that bear no simple relation to the usual conception of the quantity of matter present.

For some purposes it is advantageous to study the expectation values of other observables. Part of the problem with the particle concept concerns the fact that it is defined globally, in terms of field modes, and so is sensitive to the large scale structure of spacetime. In contrast, physical detectors are at least quasi-local in nature. It therefore seems worthwhile to investigate physical quantities that are defined locally, i.e., at a spacetime point, rather than globally. One such object of interest is the stress–energy–momentum (or stress) tensor, Tµν(x), at the point x. In addition to describing part of the physical structure of the quantum field at x, the stress-tensor also acts as the source of gravity in Einstein's field equation. It therefore plays an important part in any attempt to model a self-consistent dynamics involving the gravitational field coupled to the quantum field. For many investigators, especially astrophysicists, it is this back-reaction of the quantum processes on the background geometry that is of primary interest.