Introduction

In this chapter we will introduce holonomies and some associated concepts which will be important in the description of gauge theories to be presented in the following chapters. We will describe the group of loops and its infinitesimal generators, which will turn out to be a fundamental tool in describing gauge theories in the loop language.

Connections and the associated concept of parallel transport play a key role in locally invariant field theories like Yang–Mills and general relativity. All the fundamental forces in nature that we know of may be described in terms of such fields. A connection allows us to compare points in neighboring fibers (vectors or group elements depending on the description of the particular theory) in an invariant form. If we know how to parallel transport an object along a curve, we can define the derivative of this object in the direction of the curve. On the other hand, given a notion of covariant derivative, one can immediately introduce a notion of parallel transport along any curve.

For an arbitrary closed curve, the result of a parallel transport in general depends on the choice of the curve. To each closed curve γ in the base manifold with origin at some point o the parallel transport will associate an element H of the Lie group G associated to the fiber bundle. The parallel transported element of the fiber is obtained from the original one by the action of the group element H.

Introductiong

Since the unification of the electromagnetic and weak interactions through the Glashow–Salam–Weinberg model [75], Yang–Mills theories [76] have been widely accepted as correctly describing elementary particle physics. This belief was reinforced when they proved to be renormalizable [77, 78]. Moreover, the discovery of color symmetry as the underlying gauge invariance associated with strong interactions raised the possibility that all interactions of nature could possibly be cast as Yang–Mills theories. This spawned interest in grand unified models and some partial successes were achieved in this direction.

A crucial ingredient in the description of elementary particle physics through gauge theories is the maintenance of the gauge invariance of physical results and the underlying theory and this is also crucial in order to be able to prove renormalizability.

The success of the electroweak model is yet to be achieved by the quark model of strong interactions. The reason is that perturbative techniques, which were adequate for the electroweak model, are only appropriate in the high energy regime of strong interactions. This motivated the interest in non-perturbative techniques, especially to prove the existence of a confining phase. A great effort took place in the late 1970s and suggestive arguments were put forward but a rigorous proof of quark confinement is still lacking.

In several of these attempts the use of loops played an important role. Loops were used in a variety of contexts and approaches including the one we are focusing on in this book, the loop representation.

Introduction

Continuing with the idea of describing gauge theories in terms of loops, we will now introduce a set of techniques that will aid us in the description of loops themselves. The idea is to represent loops with a set of objects that are more amenable to the development of analytical techniques. The advantages of this are many: whereas there is limited experience in dealing with functions of loops, there is a significant machinery to deal with analytic functions. They even present advantages for treatment with computer algebra.

Surprisingly, we will see that the end result goes quite beyond our expectations. The quantities we originally introduced to describe loops immediately reveal themselves as having great potential to replace loops altogether from the formulation and go beyond, allowing the development of a reformulation of gauge theories that is entirely new. This formulation introduces new perspectives with respect to the loop formulation that have not been fully developed yet, though we will see in later chapters some applications to gauge theories and gravitation.

The plan for the chapter is as follows: in section 2.2 we will start by introducing a set of tensorial objects that embody all the information that is needed from a loop to construct the holonomy and therefore to reconstruct any quantity of physical relevance for a gauge theory. In section 2.3 we will show how the group of loops is a subgroup of a Lie group with an associated Lie algebra, the extended loop group.

Introduction

Having cast general relativity as a Hamiltonian theory of a connection, we are now in a position to apply the same techniques we used to construct a loop representation of Yang–Mills theories to the gravitational case. We should recall that we are dealing with a complex SU(2) connection. However, we can use exactly the same formulae that we developed in chapter 5 since few of them depend on the reality of the connections. Whenever the presence of a complex connection introduces changes, we will discuss this explicitly.

As we have seen, we can introduce a loop representation either through a transform or through the quantization of a non-canonical algebra. The initial steps are exactly the same as those in the SU(2) Yang–Mills case. The differences arise when we want to write the constraint equations. In the Yang–Mills case the only constraint was the Gauss law and one had to represent the Hamiltonian in terms of loops. In the case of gravity one has to impose the diffeomorphism and Hamiltonian constraints in terms of loops. In order to do so one can either use the transform or write them as suitable limits of the operators in the T algebra. We will outline both derivations for the sake of comparison. As we argued in the Yang–Mills case both derivations are formal and in a sense equivalent, although the difficulties are highlighted in slightly different ways in the two derivations.

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.