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Over the last three years, a number of fundamental physical issues were addressed in loop quantum gravity. These include: A statistical mechanical derivation of the horizon entropy, encompassing astrophysically interesting black holes as well as cosmological horizons; a natural resolution of the big-bang singularity; the development of spin-foam models which provide background independent path integral formulations of quantum gravity and `finiteness proofs' of some of these models; and, the introduction of semi-classical techniques to make contact between the background independent, non-perturbative theory and the perturbative, low energy physics in Minkowski space. These developments spring from a detailed quantum theory of geometry that was systematically developed in the mid-nineties and have added a great deal of optimism and intellectual excitement to the field. The goal of this article is to communicate these advances in general physical terms, accessible to researchers in all areas of gravitational physics represented in this conference., Comment: 24 pages, 2 figures; report of the plenary talk at the 16th International Conference on General Relativity and Gravitation, held at Durban, S. Africa in July 2001

A program was recently initiated to bridge the gap between the Planck scale physics described by loop quantum gravity and the familiar low energy world. We illustrate the conceptual problems and their solutions through a toy model: quantum mechanics of a point particle. Maxwell fields will be discussed in the second paper of this series which further develops the program and provides details., 35 pages, no figures, RevTeX4 with AMSTeX; v2: three references added. v3: Sections IV and V revised in view of discussions at the recent Schr\"odinger Institute workshop on quantum field theory in curved space-times. In particular, now this simple quantum mechanical example also serves as a (partial) introduction to the constructions and techniques used in quantum geometry

To bridge the gap between background independent, non-perturbative quantum gravity and low energy physics described by perturbative field theory in Minkowski space-time, Minkowskian Fock states are located, analyzed and used in the background independent framework. This approach to the analysis of semi-classical issues is motivated by recent results of Varadarajan. As in that work, we use the simpler U(1) example to illustrate our constructions but, in contrast to that work, formulate the theory in such a way that it can be extended to full general relativity., Clarifying remarks and three references added. To appear in CQG

Black hole mechanics was recently extended by replacing the more commonly used event horizons in stationary space-times with isolated horizons in more general space-times (which may admit radiation arbitrarily close to black holes). However, so far the detailed analysis has been restricted to non-rotating black holes (although it incorporated arbitrary distortion, as well as electromagnetic, Yang-Mills and dilatonic charges). We now fill this gap by first introducing the notion of isolated horizon angular momentum and then extending the first law to the rotating case., Comment: 31 pages REVTeX, 1 eps figure; Minor typos corrected and a footnote added

Geometrical structures intrinsic to non-expanding, weakly isolated and isolated horizons are analyzed and compared with structures which arise in other contexts within general relativity, e.g., at null infinity. In particular, we address in detail the issue of singling out the preferred normals to these horizons required in various applications. This work provides powerful tools to extract invariant, physical information from numerical simulations of the near horizon, strong field geometry. While it complements the previous analysis of laws governing the mechanics of weakly isolated horizons, prior knowledge of those results is not assumed., Comment: 37 pages, REVTeX; Subsections V.B and V.C moved to a new Appenedix to improve the flow of main arguments

Boundary conditions defining a generic isolated horizon are introduced. They generalize the notion available in the existing literature by allowing the horizon to have distortion and angular momentum. Space-times containing a black hole, itself in equilibrium but possibly surrounded by radiation, satisfy these conditions. In spite of this generality, the conditions have rich consequences. They lead to a framework, somewhat analogous to null infinity, for extracting physical information, but now in the \textit{strong} field regions. The framework also generalizes the zeroth and first laws of black hole mechanics to more realistic situations and sheds new light on the `origin' of the first law. Finally, it provides a point of departure for black hole entropy calculations in non-perturbative quantum gravity., 4 pages, RevTeX. Minor typos were corrected and the fact that, in contrast to Ref [4], isolated horizons are now allowed to have distortion and rotation was clarified

Over the past six years, a detailed framework has been constructed to unravel the quantum nature of the Riemannian geometry of physical space. A review of these developments is presented at a level which should be accessible to graduate students in physics. As an illustrative application, I indicate how some of the detailed features of the microstructure of geometry can be tested using black hole thermodynamics. Current and future directions of research in this area are discussed.

Mechanics of non-rotating black holes was recently generalized by replacing the static event horizons used in standard treatments with `isolated horizons.' This framework is extended to incorporate dilaton couplings. Since there can be gravitational and matter radiation outside isolated horizons, now the fundamental parameters of the horizon, used in mechanics, must be defined using only the local structure of the horizon, without reference to infinity. This task is accomplished and the zeroth and first laws are established. To complement the previous work, the entire discussion is formulated tensorially, without any reference to spinors., Some typos corrected, references updated. Some minor clarifications added. 20 pages, 1 figure, Revtex file

Over the past five years, there has been significant progress on the problem of quantization of diffeomorphism covariant field theories with {\it local} degrees of freedom. The absence of a background space-time metric in these theories gives rise to a host of conceptual and technical difficulties because most of the familiar methods from axiomatic, constructive and perturbative quantum field theory are no longer applicable. Perhaps the most striking examples of these problems arise in the construction of a quantum field theory of geometry. We show that these problems can be tackled using new non-perturbative methods. In particular, one can rigorously define certain geometric operators and show that their spectrum is discrete. Thus, there is a precise sense in which the geometry is quantized at the Planck scale and the continuum picture is only a coarse-grained approximation., 20 pages, To appear in the Proceedings of the March '95 Boston Conference on the Foundations of Quantum Field Theory

A set of boundary conditions defining an undistorted, non-rotating isolated horizon are specified in general relativity. A space-time representing a black hole which is itself in equilibrium but whose exterior contains radiation admits such a horizon. However, the definition is applicable in a more general context, such as cosmological horizons. Physically motivated, (quasi-)local definitions of the mass and surface gravity of an isolated horizon are introduced and their properties analyzed. Although their definitions do not refer to infinity, these quantities assume their standard values in the static black hole solutions. Finally, using these definitions, the zeroth and first laws of black hole mechanics are established for isolated horizons., Comment: 56 pages, LaTeX2e, 7 eps figures Revised version with an improved treatment of the phase space

Of the five human senses, only two can be excited essentially instantaneously by signals from distant sources: hearing and sight. In both cases, signals propagate as waves. It is not an accident therefore that we have television but not its equivalent in the realm of smell, taste or touch. Smell for example, propagates by diffusion, a slow process which is difficult to control, while taste and touch require immediate contact with the source of the signal. There is, however, a rather fundamental difference also between hearing and sight. Hearing occurs because sound excites cells in the cochlear tube of the inner ear and each little longitudinal section of the tube is sensitive to a different frequency. Consequently, we are capable of distinguishing a very large class of ‘sound profiles’. Thus, the space of sounds that human beings are capable of hearing has a very large dimension, which, for all practical purposes, can be taken to be infinite. This is in striking contrast with sight. It turns out that the space of colors that human beings are capable of seeing is only three dimensional!

In his Ph.D. thesis, Bekenstein suggested that, for a black hole in equilibrium, a multiple of its surface gravity should be identified with its temperature and a multiple of the area of its event horizon should be identified with its thermodynamic entropy [1]. In this reasoning, he had to use not only general relativity but also quantum mechanics. Indeed, without recourse to the Planck’s constant, ℏ, the identification is impossible because even the physical dimensions do not match. Around the same time, Bardeen, Carter and Hawking derived laws governing the mechanics of black holes within classical general relativity [2]. These laws have a remarkable similarity with the fundamental laws of thermodynamics. However, the derivation makes no reference to quantum mechanics at all and, within classical general relativity, a relation between the two seems quite implausible: since nothing can come out of black holes and since their interiors are completely inaccessible to outside observers, it would seem that, physically, they can only have zero temperature and infinite entropy. Therefore the similarity was at first thought to be purely mathematical. This viewpoint changed dramatically with Hawking’s discovery of black hole evaporation in the following year [3]. Using an external potential approximation, in which the gravitational field is treated classically but matter fields are treated quantum mechanically, Hawking argued that black holes are not black after all! They radiate as if they are black bodies with a temperature equal to 1/27π times the surface gravity. One can therefore regard the similarity between the laws of black hole mechanics and those of thermodynamics as reflecting physical reality and argue that the entropy of a black hole is given by 1/4-th its area. Thus, Bekenstein’s insights turned out to be essentially correct (although the precise proportionality factors he had suggested were modified).

The basic framework for a systematic construction of a quantum theory of Riemannian geometry was introduced recently. The quantum versions of Riemannian structures --such as triad and area operators-- exhibit a non-commutativity. At first sight, this feature is surprising because it implies that the framework does not admit a triad representation. To better understand this property and to reconcile it with intuition, we analyze its origin in detail. In particular, a careful study of the underlying phase space is made and the feature is traced back to the classical theory; there is no anomaly associated with quantization. We also indicate why the uncertainties associated with this non-commutativity become negligible in the semi-classical regime., 20 pages, revtex, 1 figure. Some typos corrected. References updated

A set of boundary conditions defining a non-rotating isolated horizon are given in Einstein-Maxwell theory. A space-time representing a black hole which itself is in equilibrium but whose exterior contains radiation admits such a horizon . Physically motivated, (quasi-)local definitions of the mass and surface gravity of an isolated horizon are introduced. Although these definitions do not refer to infinity, the quantities assume their standard values in Reissner-Nordstrom solutions. Finally, using these definitions, the zeroth and first laws of black hole mechanics are established for isolated horizons., Comment: 9 pages, LaTeX2e, 3 eps figures

Two recent developments suggest how familiar properties of gravity and matter may emerge from the quantum geometry that underlies loop quantum gravity.

Certain features associated with the symmetry reduction of the vacuum Einstein equations by two commuting, space-like Killing vector fields are studied. In particular, the discussion encompasses the equations for the Gowdy $T^3$ cosmology and cylindrical gravitational waves. We first point out a relation between the $SL(2,R)$ (or SO(3)) $\sigma$ and principal chiral models, and then show that the reduced Einstein equations can be obtained from a dimensional reduction of the standard SL(2,R) sigma-model in three dimensions. The reduced equations can also be derived from the action of a `generalized' two dimensional SL(2,R) sigma-model with a time dependent constraint. We give a Hamiltonian formulation of this action, and show that the Hamiltonian evolution equations for certain phase space variables are those of a certain generalization of the principal chiral model. Using these Hamiltonian equations, we give a prescription for obtaining an infinite set of constants of motion explicitly as functionals of the space-time metric variables., Comment: 17 pages, latex, reference added, a subtlety commented on; to appear in Int. J. Mod. Phys. D

A `black hole sector' of non-perturbative canonical quantum gravity is introduced. The quantum black hole degrees of freedom are shown to be described by a Chern-Simons field theory on the horizon. It is shown that the entropy of a large non-rotating black hole is proportional to its horizon area. The constant of proportionality depends upon the Immirzi parameter, which fixes the spectrum of the area operator in loop quantum gravity; an appropriate choice of this parameter gives the Bekenstein-Hawking formula S = A/4*l_p^2. With the same choice of the Immirzi parameter, this result also holds for black holes carrying electric or dilatonic charge, which are not necessarily near extremal., Revtex, 8 pages, 1 figure

It is shown that there is a precise sense in which the Heisenberg uncertainty between fluxes of electric and magnetic fields through finite surfaces is given by (one-half $\hbar$ times) the Gauss linking number of the loops that bound these surfaces. To regularize the relevant operators, one is naturally led to assign a framing to each loop. The uncertainty between the fluxes of electric and magnetic fields through a single surface is then given by the self-linking number of the framed loop which bounds the surface., Comment: 13 pages, Revtex file, 3 eps figures

A new functional calculus, developed recently for a fully non-perturbative treatment of quantum gravity, is used to begin a systematic construction of a quantum theory of geometry. Regulated operators corresponding to areas of 2-surfaces are introduced and shown to be self-adjoint on the underlying (kinematical) Hilbert space of states. It is shown that their spectra are purely discrete, indicating that the underlying quantum geometry is far from what the continuum picture might suggest. Indeed, the fundamental excitations of quantum geometry are one dimensional, rather like polymers, and the three-dimensional continuum geometry emerges only on coarse graining. The full Hilbert space admits an orthonormal decomposition into finite-dimensional subspaces which can be interpreted as the spaces of states of spin systems. Using this property, the complete spectrum of the area operators is evaluated. The general framework constructed here will be used in a subsequent paper to discuss three-dimensional geometric operators, e.g. the ones corresponding to volumes of regions.

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.

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.

3-dimensional gravity coupled to Maxwell (or Klein-Gordon) fields is exactly soluble under the assumption of axi-symmetry. The solution is used to probe several quantum gravity issues. In particular, it is shown that the quantum fluctuations in the geometry are large unless the number and frequency of photons satisfy the inequality $\N(\hbar G��)^2 << 1$. Thus, even when there is a single photon of Planckian frequency, the quantum uncertainties in the metric are significant. Results hold also for a sector of the 4-dimensional theory (consisting of Einstein Rosen gravitational waves)., 8 pages, No figures, ReVTeX

Gravitational waves with a space-translation Killing field are considered. In this case, the 4-dimensional Einstein vacuum equations are equivalent to the 3-dimensional Einstein equations with certain matter sources. This interplay between 4- and 3- dimensional general relativity can be exploited effectively to analyze issues pertaining to 4 dimensions in terms of the 3-dimensional structures. An example is provided by the asymptotic structure at null infinity: While these space-times fail to be asymptotically flat in 4 dimensions, they can admit a regular completion at null infinity in 3 dimensions. This completion is used to analyze the asymptotic symmetries, introduce the analog of the 4-dimensional Bondi energy-momentum and write down a flux formula. The analysis is also of interest from a purely 3-dimensional perspective because it pertains to a diffeomorphism invariant 3-dimensional field theory with {\it local} degrees of freedom, i.e., to a midi-superspace. Furthermore, due to certain peculiarities of 3 dimensions, the description of null infinity does have a number of features that are quite surprising because they do not arise in the Bondi-Penrose description in 4 dimensions., 39 Pages, REVTEX, CGPG-96/5-1

The asymptotic behavior of Einstein-Rosen waves at null infinity in 4 dimensions is investigated in {\it all} directions by exploiting the relation between the 4-dimensional space-time and the 3-dimensional symmetry reduction thereof. Somewhat surprisingly, the behavior in a generic direction is {\it better} than that in directions orthogonal to the symmetry axis. The geometric origin of this difference can be understood most clearly from the 3-dimensional perspective., 16 pages, REVETEX, CGPG-96/5-2

Over the last two years, the canonical approach to quantum gravity based on connections and triads has been put on a firm mathematical footing through the development and application of a new functional calculus on the space of gauge equivalent connections. This calculus does not use any background fields (such as a metric) and is thus well-suited to a fully non-perturbative treatment of quantum gravity. Using this framework, quantum geometry is examined. Fundamental excitations turn out to be one-dimensional, rather like polymers. Geometrical observables such as areas of surfaces and volumes of regions have purely discrete spectra. Continuum picture arises only upon coarse graining of suitable semi-classical states. Next, regulated quantum diffeomorphism constraints can be imposed in an anomaly-free fashion and the space of solutions can be given a natural Hilbert space structure. Progress has also been made on the quantum Hamiltonian constraint in a number of directions. In particular, there is a recent approach based on a generalized Wick transformation which maps solutions to the Euclidean quantum constraints to those of the Lorentzian theory. These developments are summarized. Emphasis is on conveying the underlying ideas and overall pictures rather than technical details., 23 pages, Plain TeX, to appear in the Proceedings of GR-14

It is shown that there is an interesting interplay between self-duality, loop representation and knots invariants in the quantum theory of Maxwell fields in Minkowski space-time. Specifically, in the loop representation based on self-dual connections, the measure that dictates the inner product can be expressed as the Gauss linking number of thickened loops., Comment: 18 pages, Revtex. No figures. To appear in Class. Quantum Grav

Autobiographical Reflections J. Stachel. Introduction A. Ashtekar, J. Renn, A. Shimony. I: Historical and Philosophical Roots of Relativity. The Prehistory of Relativity J. Eisenstaedt. Interpretations and Equations of the Michelson Experiment and its Variations H. Melcher. The Trouton Experiment, E = mc2, and a Slice of Minkowski Space-Time M. Janssen. The N-Stein Family J.D. Norton. Eclipses of the Stars: Mandi, Einstein, and the Early History of Gravitational Lensing J. Renn, T. Sauer. The Varieties of Unity: Sounding Unified Theories 1920-1930 C. Goldstein, J. Bitter. Indiscernibles, General Covariance, and Other Symmetries: The Case for Non-Reductive Relationalism S. Saunders. On Relative Orbital Rotation in Relativity Theory D.B. Malament. II: Foundational Issues in Relativity and their Advancement. The Unique Nature of Cosmology G.F.R. Ellis. Time, Structure, and Evolution in Cosmology L. Smolin. Timekeeping in an Expanding Universe J.L. Anderson. Gravitational Lensing from a Space-Time Perspective J. Ehlers, S. Frittelli, E.T. Newman. Rigidly Rotating Disk Revisted C.V. Vishveshwara. DSS 2+2 R.A. d'Inverno. Geometry, Null Hypersurfaces and New Variables D.C. Robinson. On Vacuum Twisting Type-N Again J.F. Plebanski, M. Przanowski. Quasi-Local Energy J.N. Goldberg. Space-Time Defects: Open and Closed Shells Revisited R.J. Gleiser, P.S. Letelier. Dimensionally Challenged Gravities S. Deser. A Note on Holonomic Constraints W.M. Tulczyjew. Towards an Action-at-a-Distance Concept of Spacetime D.H. Wesley, J.A. Wheeler. III: Foundational Issues in Quantum Physics and their Advancement. Inevitability, Inseparability and Gedanken Measurement M. Beller. The Concept of Quantum State: New Views on Old Phenomena M. Paty. Elementary Processes D. Ritz Finkelstein. On Quantum Non-Locality, Special Relativity, and Counterfactual Reasoning A. Shimony, H. Stein. Coherence, Entanglement and Reductionist Explanation in Quantum Physics G. Jaeger, S. Sarkar. IV: Science, History and the Challenges of Progress. Physics and Science Fiction A.I. Janis. Can We Learn From History? Do We Want To? L. Tisza. Patterns of Appropriation in the Greek Intellectual Life of the 18th Century: Case Study on the Notion of Time K. Gavroglu, M. Patiniotis. Darwin, Marx and Warranted Progress: Materialism and Views of Development in Nineteenth-Century Germany W. Lefevre . Albert Einstein and the Founding of Brandeis University S.S. Schweber. Appendix: John Stachel's Publications.

Author(s): ASHTEKAR, A; LEWANDOWSKI, J; MAROLF, D; MOURAO, J; THIEMANN, T | Abstract: In gauge theories, physical histories are represented by space-time connections modulo gauge transformations. The space of histories is thus intrinsically non-linear. The standard framework of constructive quantum field theory has to be extended to face these {\it kinematical} non-linearities squarely. We first present a pedagogical account of this problem and then suggest an avenue for its resolution.

Quantum mechanics is formulated on the true space of physical states—the projective Hilbert space. It is found that the postulates of quantum mechanics assume an intrinsically geometric form. In particular, observables are represented by real‐valued functions, the Lie bracket on the space of observables is given by a Poisson structure, and the Schrodinger evolution is equivalent to the flow along a Hamiltonian vector field.

In my talk I discussed three recent advances in loop quantum cosmology: 1) Path integral formulation and its WKB approximation; 2) Cosmological spin foams and lessons they provide; and 3) Probability of a slow roll inflationary phase compatible with the 7 year WMAP data. In addition to presenting an overview, this discussion also provides the necessary background for a number of talks in the parallel sessions.

In a quantum mechanical treatment of gauge theories (including general relativity), one is led to consider a certain completion, $\agb$, of the space $\ag$ of gauge equivalent connections. This space serves as the quantum configuration space, or, as the space of all Euclidean histories over which one must integrate in the quantum theory. $\agb$ is a very large space and serves as a ``universal home'' for measures in theories in which the Wilson loop observables are well-defined. In this paper, $\agb$ is considered as the projective limit of a projective family of compact Hausdorff manifolds, labelled by graphs (which can be regarded as ``floating lattices'' in the physics terminology). Using this characterization, differential geometry is developed through algebraic methods. In particular, we are able to introduce the following notions on $\agb$: differential forms, exterior derivatives, volume forms, vector fields and Lie brackets between them, divergence of a vector field with respect to a volume form, Laplacians and associated heat kernels and heat kernel measures. Thus, although $\agb$ is very large, it is small enough to be mathematically interesting and physically useful. A key feature of this approach is that it does not require a background metric. The geometrical framework is therefore well-suited for diffeomorphism invariant theories such as quantum general relativity., Minor corrections have been made. 54 pages, LATEX file, no figures

{\sl A Hamiltonian framework for 2+1 dimensional gravity coupled with matter (satisfying positive energy conditions) is considered in the asymptotically flat context. It is shown that the total energy of the system is non-negative, vanishing if and only if space-time is (globally) Minkowskian. Furthermore, contrary to one's experience with usual field theories, the Hamiltonian is} {\rm bounded from above}. This is a genuinely non-perturbative result. {\sl In the presence of a space-like Killing field, 3+1 dimensional vacuum general relativity is equivalent to 2+1 dimensional general relativity coupled to certain matter fields. Therefore, our expression provides, in particular, a formula for energy per-unit length (along the symmetry direction) of gravitational waves with a space-like symmetry in 3+1 dimensions. A special case is that of cylindrical waves which have two hypersurface orthogonal, space-like Killing fields. In this case, our expression is related to the ``c-energy'' in a non-polynomial fashion. While in the weak field limit, the two agree, in the strong field regime they differ significantly. By construction, our expression yields the generator of the time-translation in the full theory, and therefore represents the physical energy in the gravitational field.} \footnote{$^1$}{This is a detailed account of the results presented in the Brill-Misner symposium at the University of Maryland in May1993}, 22, CGPG-94/6-3

The goal of this article is to provide an overview of the current state of the art in loop quantum cosmology for three sets of audiences: young researchers interested in entering this area; the quantum gravity community in general; and, cosmologists who wish to apply loop quantum cosmology to probe modifications in the standard paradigm of the early universe. An effort has been made to streamline the material so that, as described at the end of section I, each of these communities can read only the sections they are most interested in, without a loss of continuity., 138 pages, 15 figures. Invited Topical Review, To appear in Classical and Quantum Gravity. Typos corrected, clarifications and references added

A canonical transformation is performed on the phase space of a number of homogeneous cosmologies to simplify the form of the scalar (or, Hamiltonian) constraint. Using the new canonical coordinates, it is then easy to obtain explicit expressions of Dirac observables, i.e.\ phase space functions which commute weakly with the constraint. This, in turn, enables us to carry out a general quantization program to completion. We are also able to address the issue of time through ``deparametrization'' and discuss physical questions such as the fate of initial singularities in the quantum theory. We find that they persist in the quantum theory {\it inspite of the fact that the evolution is implemented by a 1-parameter family of unitary transformations}. Finally, certain of these models admit conditional symmetries which are explicit already prior to the canonical transformation. These can be used to pass to quantum theory following an independent avenue. The two quantum theories --based, respectively, on Dirac observables in the new canonical variables and conditional symmetries in the original ADM variables-- are compared and shown to be equivalent., Comment: 34 pages

Quantization of the free Maxwell field in Minkowski space is carried out using a loop representation and shown to be equivalent to the standard Fock quantization. Because it is based on coherent state methods, this framework may be useful in quantum optics. It is also well-suited for the discussion of issues related to flux quantization in condensed matter physics. Our own motivation, however, came from a non-perturbative approach to quantum gravity. The concrete results obtained in this paper for the Maxwell field provide independent support for that approach. In addition, they offer some insight into the physical interpretation of the mathematical structures that play, within this approach, an essential role in the description of the quantum geometry at Planck scale., 40 pages

Holonomy algebras arise naturally in the classical description of Yang-Mills fields and gravity, and it has been suggested, at a heuristic level, that they may also play an important role in a non-perturbative treatment of the quantum theory. The aim of this paper is to provide a mathematical basis for this proposal. The quantum holonomy algebra is constructed, and, in the case of real connections, given the structure of a certain C-star algebra. A proper representation theory is then provided using the Gel'fand spectral theory. A corollory of these general results is a precise formulation of the ``loop transform'' proposed by Rovelli and Smolin. Several explicit representations of the holonomy algebra are constructed. The general theory developed here implies that the domain space of quantum states can always be taken to be the space of maximal ideals of the C-star algebra. The structure of this space is investigated and it is shown how observables labelled by ``strips'' arise naturally., Comment: 44 pages

Quantum theory of the free Maxwell field in Minkowski space is constructed using a representation in which the self dual connection is diagonal. Quantum states are now holomorphic functionals of self dual connections and a decomposition of fields into positive and negative frequency parts is unnecessary. The construction requires the introduction of new mathematical techniques involving ``holomorphic distributions''. The method extends also to linear gravitons in Minkowski space. The fact that one can recover the entire Fock space --with particles of both helicities-- from self dual connections alone provides independent support for a non-perturbative, canonical quantization program for full general relativity based on self dual variables., Comment: 14 pages