Introduction

1 Introduction

The loop approach to quantum gravity is more than twenty years old.¹ Today, it forms a wide research area around a well-defined tentative theory of quantum spacetime. The approach provides a candidate theory of quantum gravity, 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 and the physics of the Big Bang.

We still do not know if the theory is physically correct. Direct or indirect experimental support is lacking. This is the case, unfortunately, for all present approaches to quantum gravity. The reason, of course, is the minuteness of the scale at which (presumably) quantum properties of spacetime manifest. Waiting for experiments, a theory must be evaluated and compared with alternatives only in terms of its consistency with what we do know about Nature, internal coherence, and its capacity to produce unambiguous novel predictions. But sound scientific standards demand that no definitive conclusion be drawn.

Although fairly well developed, loop quantum gravity (or “loop gravity”) is not yet a complete theory, nor has its full consistency with classical general relativity been rigorously established yet. The sector of the theory, which has not yet solidified, is the dynamics, which exists in several variants that are presently under investigation. The strength of the theory is its compelling capacity to describe quantum spacetime in a background-independent and nonperturbative fashion, and, especially, its genuine attempt to synthesize the conceptual novelties introduced by quantum mechanics with the ones introduced by general relativity: loop quantum gravity offers a possible conceptual framework in which general relativity and quantum field theory make sense together and consistently.

The other large research program for a quantum theory of gravity besides loop 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 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 in 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 for its relevance, and the present state of our knowledge.
Section 3, “The Development of Loop Gravity”, is a short overview of the historical development of the theory.
Section 4, “Resources”, contains pointers to introductory literature, institutions at which loop gravity is studied, web pages, and other information that may be of use to students and researchers.
Section 5, “Ideas and Physical Inputs”, discusses the physical and mathematical ideas on which loop quantum gravity is based.
The actual theory is introduced in Section 6, “Formalism”, at a simple, technical level.
Section 7, “Physical Results”, is a list of the main physical results that have been derived from the theory.
Section 8, “Open Problems and Current Lines of Investigation”, illustrates the main open problems and some currently active lines of research.
Section 9, “Short Summary and Conclusion”, summarizes very briefly the state and the results of the theory.

At the cost of several repetitions, the structure of this review is modular: to a large extent sections are independent of one another, have different style, 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 and nonspecialists 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, 5 and 8 stimulating. I will not enter into technical details. 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 can correct errors and omissions.