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.

Review of the Intellectual State of the Art

In this book, the concepts and theories behind spacecraft–environment interactions were examined. Each chapter is illustrated with selected examples. On the basis of these examples and theories, a reader should be able to identify and construct simple, first-order estimates of the principal interactions of importance to a specific spacecraft. The discipline of spacecraft–environment interactions represented by this process is, however, continuing to evolve both in its intellectual underpinnings and in its value to the spacecraft designer. In this final chapter, the current state of the art in spacecraft–environment interactions is reviewed and the future direction of progress in the field is predicted.

Neutrals

The primary interaction concerns for the neutral atmosphere are drag, atomic oxygen erosion, glow, and contamination. Although the overall processes associated with spacecraft drag in LEO or polar orbits are reasonably well understood, the detailed effects are often hard to predict accurately. The neutral environment for the Earth and its reaction to the Sun are, in principle, moderately well understood. Indeed, statistical models have been built that offer reasonable accuracy. However, there are still outstanding questions to be answered as to how solar and geophysical activity couple to the atmosphere and how to model the often almost impulsive atmospheric responses to sudden changes in these parameters. Errors as high as factors of 10 to 100 in predicting the density along an orbit are not unusual. Once the neutrals strike the spacecraft surface, the accommodation of the neutrals on the surface becomes an issue.

α Cygni variables

According to the nomenclature of the GCVS, luminous variable B and A supergiants are called α Cygni variables, and are classified among the pulsating variables. The class also includes massive O and late type stars, since these belong to the same evolutionary sequence. In the MK spectral-classification system, they have luminosity classes Ib, Iab, Ia and Ia+ (in increasing order of luminosity). The most luminous supergiants are also called ‘hypergiants’ - these are, in fact, Luminous Blue Variables (LBVs). Ia supergiants are pre-LBV objects, therefore we also refer to Section 2.1 for all details that are related to both groups of variables. All OBA supergiants are variable (Rosendhal & Snowden 1971, Maeder & Rufener 1972, Sterken 1977). The amplitudes of the most luminous supergiants resemble the microvariations observed in LBVs during quiescence, the level of variability increases towards higher luminosities for all spectral classes.

Pulsational instability accounts to some extent for the semi-regular variations (Leitherer et al. 1985, Wolf 1986) - it should also be noted that the β Cep instability strip widens into the supergiant region. The amplitudes of the variations seem to increase with the time scales at which they occur.

HD 57060 = UW CMa and HD 167971 are two interesting cases of microvariations. HD 57060 (Fig. 3.1) is a binary consisting of an O8 supergiant star and an O or B type main-sequence star in synchronous revolution with a period of 4d39.

Variable stars

All stars display variations of brightness and colour in the course of their passage through subsequent stages of stellar evolution. As a rule, however, a star is called variable when its brightness or colour variations are detectible on time scales of the order of the mean life time of man. The variations may be periodic, semi-periodic or irregular, with time scales ranging from a couple of minutes to over a century. It is this kind of variable star which is the topic of this book. The typical time scale, the amplitude of the brightness variations, and the shape of the light curve can be deduced from photometric observation, and those quantities place the star in the appropriate class. For example, a star of the UV Ceti type typically has brightness variations (the so-called flares) of several magnitudes in an interval of time as short as a few minutes, whereas a Cepheid shows periodic variations of about one magnitude in a time span of several days. However, spectral type, luminosity class and chemical composition are complementary important spectroscopic parameters that are needed for classifying variable stars according to the origin of their variations.

At the very beginning of the space age, spacecraft designers learned that the effects of the space environment on a spacecraft's systems would be vital factors in spacecraft design and operation. Since those early years, the topic of spacecraft–environment interactions has developed into a multidisciplinary field involving engineers and scientists from all over the world. Traditionally, engineers have been interested in spacecraft design and operational issues, and scientists have concentrated on the fundamental physics and chemistry associated with the interactions. These diverse interests have led to numerous books and conferences. The field has grown substantially in the past decade with the advent of the Shuttle and the ability to perform repeatable, in-situ experiments. The authors therefore concluded that, with the growth of the field and the expanding interest in it, it was timely to prepare a comprehensive book summarizing the many recent discoveries. In particular, since the field has evolved in a way that has been driven by mission and spacecraft requirements rather than as a specific discipline, a book would be a valuable step in integrating the field intellectually. Such a book would also serve as an introduction to the discipline for graduate students and professionals. For specific applications, these individuals could then turn to one of the handbooks or collections of conference papers referenced throughout the book.

This book is the direct outgrowth of courses that the authors have taught.