Introduction

At the beginning of the 1970s gauge theories and in particular Yang–Mills theories appeared as the fundamental theories that described particle interactions. Two main perturbative results were established: the unification of electromagnetic and weak interactions and the proof of the renormalizability of Yang–Mills theory. However, the advent of proposals to describe strong interactions in terms of gauge theories — and in particular the establishment of QCD and the quark model for the hadrons — required the development of new non-perturbative techniques. Problems such as that of confinement, chiral symmetry breaking and the U(1) problem spawned interest in various non-perturbative alternatives to the usual treatment of quantum phenomena in gauge theories. Both at the continuum and lattice levels various attempts were made [44, 48, 12, 49, 50] to describe gauge theories in terms of extended objects as Wilson loops and holonomies. Some of these treatments started at a classical level [44], with the intention of completely reformulating and solving classical gauge theories in terms of loops. Other proposals were at the quantum mechanical level; for instance, trying to find a Schwinger–Dyson formulation in order to obtain a generating functional for the Green functions of gauge theories using the Wilson loop. Among these latter proposals we find the loop representation [5, 34], based on constructing a quantum representation of Hamiltonian gauge theories in terms of loops.

Introduction

As we mentioned in the previous chapter, the definition of Yang–Mills theories in the continuum in terms of lpops requires a regularization and the resulting eigenvalue equations are, in the non-Abelian case, quite involved. Lattice techniques appear to be a natural way to deal with both these difficulties. First of all since on a lattice there is a minimum length (the lattice spacing), the theory is naturally regularized. An important point is that this is a gauge invariant regularization technique. Secondly, formulating a theory on a lattice reduces an infinite-dimensional problem to a finite-dimensional one. It is set naturally to be analyzed using a computer.

Apart from these technical advantages, the reader may find interest in this chapter from another viewpoint. In terms of lattices one can show explicitly in simple models many of the physical behaviors of Wilson loops that we could only introduce heuristically in previous chapters.

Lattice gauge theories were first explored in 1971 by Wegner [104]. He considered a usual Ising model with up and down spins but with a local symmetry. He associated a spin to each link in the lattice and considered an action that was invariant under a spin-flip of all the spins associated with links emanating from a vertex. He noted that this model could undergo phase transitions, but contrary to what happens with usual Ising models, his model did not magnetize. The absence of the magnetization posed him with the problem of distinguishing the phases of the theory.

For about twenty years after its invention, quantum electrodynamics remained an isolated success in the sense that the underlying ideas seemed to apply only to the electromagnetic force. In particular, its techniques did not seem to be useful in dealing with weak and strong interactions. These interactions seemed to lie outside the scope of the framework of local quantum field theory and there was a wide-spread belief that the best way to handle them would be via a more general, abstract S-matrix theory. All this changed dramatically with the discovery that non-Abelian gauge theories were renormalizable. Once the power of the gauge principle was fully recognized, local quantum field theory returned to the scene and, by now, dominates our thinking. Quantum gauge theories provide not only the most natural but also the only viable candidates we have for the description of electroweak and strong forces.

The basic dynamical variables in these theories are represented by non-Abelian connections. Since all the gauge invariant information in a connection is contained in the Wilson loops variables (i.e., traces of holonomies), it is natural to try to bring them to the forefront. This is precisely what is done in the lattice approaches which are the most successful tools we have to probe the non-perturbative features of quantum gauge theories. In the continuum, there have also been several attempts to formulate the theory in terms of Wilson loops.

In this book we have attempted to present in a structured fashion the various aspects of the use of loops in the quantization of gauge theories and gravitation. The discussion mixed historical and current developments and we rewrote many results in a more modern language. In this chapter we would like to concentrate on the outlook arising from the material presented and focus on current developments and on possible future avenues of work. We will divide the discussion into gauge theories and gravity, since the kinds of developments in these two fields follow naturally somewhat disjoint categories.

Gauge theories

Overall, the picture which emerges is satisfying in the sense that the bulk of the techniques developed can be applied systematically to the construction of loop representations for almost any theory based on a connection as the main canonical variable, either free or interacting with various forms of matter. In this respect we must emphasize the developments listed in chapters 1, 2 and 3 which are the main mathematical framework that we used to understand the physical applications. Many of these aspects, as we have mentioned, have been studied with mathematical rigor by various authors in spite of the fact that the presentation we have followed here is oriented towards physicists.

The main conclusion to be drawn from this book is that loop techniques are at present a practical tool for the analysis of the quantum mechanics of gauge theories.

Loops have been used as a tool to study classical and quantum Yang–Mills theory since the work of Mandelstam in the early 1960s. They have led to many insights concerning the non-perturbative dynamics of the theory including the issue of confinement and the lattice formulation. Since the inception of the Asthekar new variables, loop techniques have also found important applications in quantum gravity. Due to the diffeomorphism invariance of the theory they have led to surprising connections with knot theory and topological field theories.

The intention in this book is to present several of these results in a common framework and language. In particular it is an attempt to combine ideas developed some time ago in the context of Yang–Mills theories with the recent applications in quantum gravity. It should be emphasized that our treatment of Yang–Mills theories only covers a small part of all results obtained with loops: that which seems of most relevance for applications in gravity.

This book should allow people from outside the field to gain access in a pedagogical way to the current state of the art. Moreover, it allows experts within this wide field with heterogeneous backgrounds to learn about specific results outside their main area of expertise and as a reference volume. It should be well suited as an introductory guide for graduate students who want to get started in the subject.

Introduction

In the previous two chapters we developed several aspects of the loop representation of quantum gravity. One of the main consequences of these developments is a radically new description of one of the symmetries of the theory: because of diffeomorphism invariance wavefunctions in the loop representation must be invariant under deformations of the loops, they have to be knot invariants. This statement is much more than a semantical note. Knot invariants have been studied by mathematicians for a considerable time and recently there has been a surge in interest in knot theory. Behind this surge of interest is the discovery of connections between knot theory and various areas of physics, among them topological field theories. We will see in this chapter that such connections seem to play a crucial role in the structure of the space of states of quantum gravity in the loop representation. As a consequence we will discover a link between quantum gravity and particle physics that was completely unexpected and that involves in an explicit way the non-trivial dynamics of the Einstein equation. Such a link could be an accident or could be the first hint of a complete new sets of relationships between quantum gravity, topological field theories and knot theory.

We will start this chapter with a general introduction to the ideas of knot theory. We will then develop the notions of knot polynomials and the braid group.

Introduction

In the previous chapter we discussed the basics of the loop representation for quantum gravity. We obtained expressions for the constraints at both a formal and a regularized level and discussed generalities about the physical states of the theory. In this chapter we would like to discuss several developments that are based on the loop representation. We will first discuss the coupling of fields of various kinds: fermions using an open path formalism, Maxwell fields in a unified fashion and antisymmetric fields with the introduction of surfaces. These examples illustrate the various possibilities that matter couplings offer in terms of loops. We then present a discussion of various ideas for extracting approximate physical predictions from the loop representation of quantum gravity. We discuss the semi-classical approximation in terms of weaves and the introduction of a time variable using matter fields and the resulting perturbation theory. We end with a discussion of the loop representation of 2 + 1 gravity as a toy model for several issues in the 3 + 4 –1 theory.

Inclusion of matter: Weyl fermions

As we did for the Yang–Mills case, we now show that the loop representation for quantum gravity naturally accommodates the inclusion of matter. In the Yang–Mills case, in order to accommodate particles with Yang–Mills charge one needed to couple the theory to four-component Dirac spinors. A Dirac spinor is composed of two two-component spinors that transform under inequivalent representations of the group.

THE RESTLESS UNIVERSE

From ancient Hindu mythology comes this story about the Pole Star: King Uttanapada had two wives. The favourite, Suruchi, was haughty and proud, while the neglected Suniti was gentle and modest. One day Suniti's son Dhruva saw his co-brother Uttama playing on their father's lap. Dhruva also wanted to join him there but was summarily repulsed by Suruchi, who happened to come by. Feeling insulted, the five-year-old Dhruva went in search of a place from where he would not have to move. The wise sages advised him to propitiate the god Vishnu, which Dhruva proceeded to do with a long penance. Finally Vishnu appeared and offered a boon. When Dhruva asked for a place from where he would not have to move, Vishsnu placed him in the location now known as the Pole Star – a position forever fixed.

Unlike other stars and planets, the Pole Star does not rise and set; it is always seen in the same part of the sky. This immovability of the Pole Star has proved to be a useful navigational aid to mariners from ancient to modern times. Yet, a modern-day Dhruva could not be satisfied with the Pole Star as the ultimate position of rest. Let us try to find out why.

The Pole Star does not appear to change its direction in the sky because it happens to lie more or less along the Earth's axis of rotation. As the Earth rotates about its axis, other stars rise over the eastern horizon and set over the western horizon.

FUSION

It is often argued that man's growing energy needs will be met if he succeeds in making fusion reactors. In a fusion reactor, energy is generated by fusing together light atomic nuclei and converting them into heavier ones. The primary fuel for such a fusion reactor on the Earth would be heavy hydrogen, whose technical name is deuterium. Through nuclear fusion, two nuclei of deuterium are brought together and converted to the heavier nucleus of helium, and in this process nuclear energy is released.

The following is the recipe for a fusion reactor. First, heat a small quantity of the fusion fuel, deuterium, above its ignition point – to a temperature of some 100 million degrees Celsius. Second, maintain this fuel in a heated condition long enough for fusion to occur. When this happens, the energy that is released exceeds the heat input, and the reactor can start functioning on its own. The third and final part of the operation involves the conversion of the excess energy to a useful form, such as electricity.

The primary fuel for this process, the heavy hydrogen, is chemically similar to but a rarer version of the commonly known hydrogen. An atom of ordinary hydrogen is made up of a charged electrical particle called the proton at the nucleus with a negatively charged particle, the electron, going round it. The nucleus of heavy hydrogen carries an additional particle called the neutron in its nucleus. The neutron has no electric charge so the total charge of the nucleus of heavy hydrogen is the same as that of ordinary hydrogen.

It is often said that modern theoretical physics began with Newton's law of gravitation. There is a good measure of truth in this remark, especially when we take into account the aims and methods of modern physics – to describe and explain the diverse and complex phenomena of nature in terms of a few basic laws.

Gravity is a basic force of the Universe. From the motions of ocean tides to the expansion of the Universe, a wide range of astronomical phenomena are controlled by gravity. Three centuries ago Newton summed up gravity in his simple inverse-square law. Yet, when asked to say why gravity follows such a law, he declined to hazard an opinion, saying ‘Non fingo hypotheses’ (I do not feign hypotheses). A radically new attempt to understand gravity was made in the early part of this century by Einstein, who saw in it something of deeper significance that linked it to space and time. The modern theoretical physicist is trying to accommodate it within a unified theory of all basic forces. Yet, gravity remains an enigma today.

In this book I have attempted to describe the diversity, pervasiveness, and importance of this enigmatic force. It is fitting that I have focused on astronomical phenomena, because astronomy is the subject that first provided and continues to provide a testing ground for the study of gravity. These phenomena include the motions of planets, comets, and satellites; the structure and evolution of stars; tidal effects on the Earth and in binary star systems; gigantic lenses in spaced highly dense objects, such as neutron stars, black holes, and white holes; and the origin and evolution of the Universe itself.

THE BLACK HOLE IN HISTORY AND ASTRONOMY

The oldest mention of a black hole is found not in books of physics or astronomy but in books of history. In the summer of the year 1757, Nawab Siraj-Uddaula, the ruler of Bengal in eastern India, marched on Calcutta to settle a feud with the British East India Company. The small garrison stationed in Fort William at Calcutta was hardly a match for the Nawab's army of 50 000. In the four-day battle that ensued, the East India Company lost many lives, and a good many, including the company's governor, simply deserted. The survivors had to face the macabre incident now known as the Black Hole of Calcutta.

The infuriated Nawab, whose army had lost thousands of lives in the battle, ordered the survivors to be imprisoned in what came to be known as the Black Hole, a prison cell in Fort William. In a room 18 feet by 14 feet, normally used for housing three or four drunken soldiers, the 146 unfortunate survivors were imprisoned. The room had only two small windows (see Figure 7–1). During the 10 hours of imprisonment, from 8 p.m. on 20 June to 6 a.m. on 21 June in the hottest part of the year, 123 prisoners died. Only 22 men and 1 woman lived to tell the tale.

Apart from its macabre aspect, the Black Hole of Calcutta did bear some similarity to its astronomical counterpart, involving as it did a large concentration of matter in a small space from which there was no escape.

WHEN NEWTON AND EINSTEIN AGREE

Einstein's general theory of relativity and Newton's law of gravitation offer radically different interpretations of the phenomenon of gravity. Yet, in practical terms, the difference between their predictions seem to be very small. In Chapter 5 we saw two examples of observations in the solar system: the precession of the orbit of Mercury and the bending of light rays from a distant star by the Sun. In both cases the differences in the predictions of Newton and Einstein are very small and are measurable only with very patient and sophisticated astronomical observations. Is it just a coincidence that these two approaches give almost the same answer?

A mathematical analysis of Einstein's equations tells us that the agreement between the two approaches is not coincidental. It can be shown that, in all phenomena of weak gravitational effects and where the gravitating bodies are moving slowly compared to light, the two theories must almost agree. In our discussion of the escape speed in Chapter 3, we saw how to measure the relative strength of gravity. We use the criterion of the escape speed in the present context to understand the difference between ‘weak’ and ‘strong’ gravity. The rule is simple: compare the escape speed V with the speed of light c. If the ratio V/c is very small compared to 1, the gravitational effects are weak. If the ratio is comparable to 1, say between 0.1 and 1, the gravitational effects are strong (see Figure 6–1). Referring back to Table 3–2, we see that the gravitational effects are weak in all cases except on the surface of neutron stars.

THE PROBLEMS OF LARGE-SCALE STRUCTURE

More than seven decades have elapsed since Friedmann proposed his mathematical models that describe the expanding Universe. As we saw in Chapter 9, these models lead to the conclusion that the Universe was created some 10–15 billion years ago in a big explosion (the so-called big bang) after which it has been expanding but more and more slowly because of brakes applied by gravity. This model also tells us that the Universe was very hot to begin with, and dominated by radiation, but with expansion it has cooled down and the temperature of the radiation background today is 2.73 kelvin (see Figure 9–10) as measured by the COBE satellite and other groundbased detectors. And one other set of relics of the hot era, namely the light nuclei like deuterium, helium, etc., are found in the right amount all over the Universe. Thus, we concluded the last chapter with a fair degree of confidence in the big–bang scenario.

However, over the last quarter of a century astronomical observations have become more sophisticated and the views of the largescale structure of the Universe they present go well beyond the simplified assumptions of a ‘homogeneous and isotropic Universe’. We shall see, for example, in Figure 10–1 how galaxies are distributed over the sky in depth. The dots in the figure represent galaxies and their distribution is clearly not smooth, as a homogeneous Universe would have us believe.

WAS THERE REALLY A BIG BANG?

The big-bang cosmology described in the last two chapters has a large following amongst the astronomical community. The models of Friedmann are able to account for the observed expansion of the Universe, for the smooth background of microwave radiation, and for the abundance of light nuclei that cannot be generated inside stars. Are these not good enough reasons for believing in the overall picture?

Playing the devil's advocate in this chapter, let me voice a few dissenting views. First, a scientific theory, howsoever successful it may be, must always be vulnerable to checks of facts and conceptual consistency. Even a well established theory like Newton's had to give way to Einstein's when it was found wanting under these checks (see Chapter 5). The formidable facade of big-bang cosmology is likewise developing cracks that can no longer be plastered over.

The first crack has actually been there right from the beginning and may have been noticed by the reader of Chapter 9. He or she may ask the questions, ‘What preceeded the big bang? How did the matter and radiation in the Universe originate in the first place? Does it not contradict the law of conservation of matter and energy?’

These questions cannot be answered within the framework of Einstein's general theory of relativity, which was used to construct the Friedmann models. The moment of ‘big bang’ is a singular epoch, according to the theory, just as the end of a collapsing object, described in Chapter 7, is in a singularity.

Our discussion of gravity began with the falling apple and has taken us from ocean tides to the planets, comets, and satellites of the solar system, to the different stages in the evolution of a star, to the curved spacetime of general relativity, to the illusions of gravitational lensing, to the weird effects associated with black holes and white holes, and finally to the large-scale structure of the Universe itself. None of the other basic forces of physics has such a wide range of applications. Although gravity is by far the weakest of the four known basic forces, its effects are the most dramatic.

Indeed, it would be an amusing exercise to speculate on the state of the world if there were no gravity at all! Would atoms and molecules be affected? As far as we know, the presence or absence of gravity does not play a crucial role in the existence and stability of the microworld. The strong, weak, and electromagnetic forces are the main forces at this level. Even at the macroscopic level of the objects we see around us in our daily lives, gravity does not appear to play a crucial role in their constitution or equilibrium. After all, even astronauts have demonstrated that they can live in simulated conditions of weightlessness. Neither the astronauts nor their spacecraft come apart in such circumstances. The basic binding force at this level is the force of electricity and magnetism.

But we can go no further in dispensing with gravity. If we eliminate gravity on a bigger scale, disasters lie in store.

WHY DID THE APPLE FALL?

Apples have played a prominent role in many legends, myths, and fairytales. It was the forbidden apple that became the source of temptation to Eve and ultimately brought God's displeasure upon Adam. It was the apple of discord that led to the launching of a thousand ships and the long Trojan War. It was a poisoned apple that nearly killed Snow White, and so on.

For physicists, however, the most important apple legend concerns the apple that fell in an orchard in Woolsthorpe in Lincolnshire, England, in the year 1666. This particular apple was seen by Isaac Newton, who ‘fell into a profound meditation upon the cause which draws all bodies in a line which, if prolonged, would pass very nearly through the centre of the earth.’

The quotation is from Voltaire's Philosophie de Newton, published in 1738, which contains the oldest known account of the apple story. This story does not appear in Newton's early biographies, nor is it mentioned in his own account of how he thought of universal gravitation. Most probably it is a legend.

It is interesting to consider how rare it is to see an apple actually fall from a tree. An apple may spend a few weeks of its life on the tree, and after its fall it may lie on the ground for a few days. But how long does it take to fall from the tree to the ground? For a drop of, say, 3 metres, the answer is about three-quarters of a second.

THE MASS OF THE EARTH

Although with Newton's pioneering discoveries, gravity was the first basic force of nature to be described and studied quantitatively, it is the weakest of all known basic forces of nature. The other basic forces are the forces of electricity and magnetism and the forces of ‘strong’ and ‘weak’ interaction which act on subatomic particles. It is a measure of the success achieved to date that physicists are able to explain all observed natural and laboratory phenomena in terms of these four basic forces. As we shall see in later chapters, many physicists hope that one day they will be able to bring all the basic forces under the umbrella of one unified force.

Although atomic physicists consider gravity to be the weakest of the four known basic forces of nature, for astronomers gravity is the most dominant force in the celestial environment. How do we assess the strength of gravity in any given situation? We will try to answer this question with a few examples in this chapter.

All of us on the Earth are conscious of gravity. The feeling of weight that we have results from the gravitational pull the Earth exerts on us. Newton's inverse-square law of gravitation described in Chapter 2 tells us how strong this force is on any given body on the Earth's surface. Let m be the mass of the body and M the mass of the Earth. The distance between the body and the Earth is denoted by d.