1 Introduction

The loop approach to quantum gravity is ten years old. The first announcement of this approach was given at a conference in India in 1987 [182]. This tenth anniversary is a good opportunity to attempt an assessment of what has and what has not been accomplished in these ten years of research and enthusiasm.

During these ten years, loop quantum gravity has grown into a wide research area and into a solid and rather well-defined tentative theory of the quantum gravitational field. The approach provides a candidate theory of quantum gravity. It provides a physical picture of Planck scale quantum geometry, calculation techniques, definite quantitative predictions, and a tool for discussing classical problems such as black hole thermodynamics.

We do not know whether this theory is physically correct or not. Direct or indirect experimental corroboration of the theory is lacking. This is the case, unfortunately, for all present approaches to quantum gravity, due, of course, to the minuteness of the scale at which quantum properties of spacetime (presumably) manifest themselves. In the absence of direct experimental guidance, we can evaluate a theory and compare it with alternative theories only in terms of internal consistency and consistency with what we do know about Nature.

Long standing open problems within the theory (such as the lack of a scalar product, the incompleteness of the loop basis and the related difficulty of dealing with identities between states, or the difficulty of implementing the reality conditions in the quantum theory) have been solidly and satisfactorily solved. But while it is fairly well developed, loop quantum gravity is not yet a complete theory. Nor has its consistency with classical general relativity been firmly established yet. The sector of the theory which has not yet solidified is the dynamics, which exists in several variants presently under intense scrutiny. On the other hand, in my opinion the strength of the theory is its compelling capacity to describe quantum spacetime in a background independent nonperturbative manner, and its genuine attempt to synthesize the conceptual novelties introduced by quantum mechanics with the ones introduced by general relativity.

The other large research program for a quantum theory of gravity, besides loop quantum gravity, is string theory, which is a tentative theory as well. String theory is more ambitious than loop gravity, since it also aims at unifying all known fundamental physics into a single theory. In section 2.3, I will compare strengths and weaknesses of these two competing approaches to quantum gravity.

This ``living review'' is intended to be a tool for orienting the reader to the field of loop gravity. Here is the plan for the review:

• Section 2, ``Quantum Gravity: Where are We?'', is an introduction to the problem, the reason of its relevance, and the present state of our knowledge.
• Section 3, ``History of Loop Quantum Gravity'', is a short overview of the historical development of the theory.
• Section 4, ``Resources'' contains pointers to introductory literature, institutions where loop gravity is studied, web pages, and other information that may be of use to students and researchers.
• Section 5, ``Main Ideas and Physical Inputs'', discusses, at a rather technical level, the physical and mathematical ideas on which loop quantum gravity is based.
• The actual theory is described in detail in section 6, ``The Formalism'' .
• Section 7, ``Results'', is devoted to the results that have been derived from the theory. I have divided results in two groups. First, the ``technical'' results (7.1), namely the ones that are important for the construction and the understanding of the theory itself, or that warrant the theory's consistency. Second, the ``physical'' results (7.2): that is, what the theory says about the physical world.
• In section 8, ``Open Problems and Current Lines of Investigation'', I illustrate what I consider to be the main open problems and currently active lines of research.
• In section 9, ``Short Summary and Conclusion'', I summarize very briefly the state and the results of the theory, and present (necessarily very preliminary!) conclusions.

At the cost of several repetitions, the structure of this review is very modular: Sections are to a large extent independent of each other, have different styles, and can be combined according to the interest of the reader. A reader interested only in a very brief overview of the theory and its results can find this in section 9 . Graduate students or persons of general culture may get a general idea of what goes on in this field and its main ideas from sections 2 and 7 . If interested only in the technical aspects of the theory and its physical results, one can read sections 6 and 7 alone. Scientists working in this field can use sections 6 and 7 as a reference, and I hope they will find sections 2, 3, 5 and 8 stimulating.

I will not enter into technical details. However, I will point to the literature where these details are discussed. I have tried to be as complete as possible in indicating all relevant aspects and potential difficulties of the issues discussed.

The literature in this field is vast, and I am sure that there are works whose existence or relevance I have failed to recognize. I sincerely apologize to the authors whose contributions I have neglected or under-emphasized, and I strongly urge them to contact me to help me make this review more complete. The ``living reviews'' are constantly updated, and I will be able to correct errors and omissions in the future.

Loop Quantum Gravity Carlo Rovelli http://www.livingreviews.org/lrr-1998-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de