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There have been many different attempts to provide a quantum description of gravitational phenomena. Although there is at present no immediate experimental evidence of quantum effects of the gravitational field, it is expected on general grounds that at sufficiently high energies quantum effects may be relevant. The fact that quantum field theories in general involve virtual processes of arbitrarily high energies may suggest that an understanding of quantum gravity may be needed to provide a complete picture of quantum fields. Ultraviolet divergences arise as a consequence of an idealization in which one expects the field theory in question to be applicable up to arbitrarily high energies. It is generally accepted that for high energies gravitational corrections could play a role. On the other hand, classical general relativity predicts in very general settings the appearance of singularities in which energies, fields and densities become intense enough to suggest the need for quantum gravitational corrections.
In spite of the many efforts invested over the years in trying to apply the rules of quantum mechanics to the gravitational field, most attempts have remained largely incomplete due to conceptual and technical difficulties. There are good reasons why the merger of quantum mechanics and gravity as we understand them at present is a difficult enterprise. We now present a brief and incomplete list of the issues involved.
In this chapter we will study the quantization of the free Maxwell theory. Admittedly, this is a simple problem that certainly could be tackled with more economical techniques, and this was historically the case. However, it will prove to be a very convenient testing ground to gain intuitive feelings for results in the language of loops. It will also highlight the fact that the loop techniques actually produce the usual results of more familiar quantization techniques and guide us in the interpretation of the loop results.
We will perform the loop quantization in terms of real and Bargmann  coordinates. The reason for considering the complex Bargmann coordinatization is that it shares many features with the Ashtekar one for general relativity. It also provides a concrete realization of the introduction of an inner product purely as a consequence of reality conditions, a feature that is expected to be useful in the gravitational case.
The Maxwell field was first formulated in the language of loops by Gambini and Trias . The vacuum and other properties are discussed in reference  and multiphoton states are discussed in referece . The loop representation in terms of Bargmann coordinates was first discussed by Ashtekar and Rovelli .
The organization of this chapter is as follows: in section 4.1 we will first detail some convenient results of Abelian loop theory, which will simplify the discussion of Maxwell theory and will highlight the role that Abelian theories play in the language of loops.
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.
Since the unification of the electromagnetic and weak interactions through the Glashow–Salam–Weinberg model , Yang–Mills theories  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.
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.
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.
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 , 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.
Now in paperback, this text provides a self-contained introduction to applications of loop representations and knot theory in particle physics and quantum gravity. Loop representations (and the related topic of knot theory) are of considerable current interest because they provide a unified arena for the study of the gauge invariant quantization of Yang-Mills theories and gravity, and suggest a promising approach to the eventual unification of the four fundamental forces. This text begins with a detailed review of loop representation theory. It then goes on to describe loop representations in Maxwell theory, Yang-Mills theories as well as lattice techniques. Applications in quantum gravity are then discussed in detail. Following chapters move on to consider knot theories, braid theories and extended loop representations in quantum gravity. A final chapter assesses the current status of the theory and points out possible directions for future research.
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 . 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.
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.
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.
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.
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