We close with a short historical note on the development of the theory presented, and a few considerations on the main problems that remain open.

Brief historical note

The quantum theory of gravity presented in this book is the result of a long path to which many have contributed. We collect here a few notes about this path and a few references for the orientation of the reader. A comprehensive bibliography would be impossible.

The basic ideas are still those of John Wheeler (Wheeler 1968) and Bryce DeWitt (DeWitt 1967) concerning the state space, and those of Charles Misner (Misner 1957), Stephen Hawking (Hawking 1980), Jonathan Halliwell (Halliwell and Hawking 1985), and Jim Hartle (Hartle and Hawking 1983) concerning the sum over geometries. The theory would not have developed without Abhay Ashtekar's introduction (Ashtekar 1986) of his variables and grew out of the “loop space representation of quantum general relativity,” which was conceived in the late 1980s (Rovelli and Smolin 1988, 1990; Ashtekar et al. 1991). Graphs were introduced by Jurek Lewandowski in (Lewandowski 1994); spin networks and discretization emerged from loop quantum gravity in the 1990s (Rovelli and Smolin 1995a,b; Ashtekar and Lewandowski 1997). The geometrical picture used here derives from Bianchi et al. (2011b). The idea of using spinfoams for quantum gravity derives from the work of Hirosi Ooguri (Ooguri 1992), who generalized to four-dimensional the Boulatov model (Boulatov 1992), which in turn is a generalization to three-dimensional of the two-dimensional matrix models.

In order to find a quantum theory for the gravitational field, we can apply the general structure devised in Chapter 2 to the theory defined in Chapter 3, discretized as explained in Chapter 4. Since the resulting theory includes a number of technical complications, in this chapter we first complete a simpler exercise: apply this same strategy to euclidean general relativity in three spacetime dimensions, recalled in Section 3.5. This exercise allows us to introduce a number of ideas and techniques that will then be used in the physical case, which is the lorentzian theory in four spacetime dimensions.

There are major differences between general relativity in three dimensions and four dimensions. The foremost is that the 3d theory does not have local degrees of freedom: it is therefore infinitely simpler than the 4d theory. This is reflected in properties of the quantum theory that do not hold in the 4d theory. In spite of these differences, the exercise with the 3d theory is illuminating and provides insight into the working of quantum gravity, because in spite of its great simplicity the 3d theory is a generally covariant quantum theory of geometry.

Quantization strategy

To define the quantum theory we need two ingredients:

1. • A boundary Hilbert space that describes the quantum states of the boundary geometry.

2. • The transition amplitude for these boundary states. In the small ћ limit the transition amplitude must reproduce the exponential of the Hamilton function.

In the two previous chapters we have discussed applications of loop gravity to physically relevant, but specific situations – black holes and early cosmology. How to extract the entire information from the theory systematically, and compare it with the usual way of doing high-energy physics?

In conventional field theory, knowledge of the n-point functions

W(x1,…,xn) = 〈0|ø(xn)…ø(x1)|0〉,

amounts to the complete knowledge of the theory, as emphasized by Arthur Wightman in the 1950s (Wightman 1959). From these functions we can compute the scattering amplitudes and everything else. Can we recover the value of all these functions, from the theory we have defined in this book? This, for instance, would allow us to compare the theory with the effective perturbative quantum theory of general relativity, which, although non-renormalizable is nevertheless usable at low energy. More generally, it would connect the abstruse background-independent formalism needed for defining quantum gravity in general with the tools of quantum field theory that we are used to, from flat space physics.

The answer is yes, the value of the n-point functions can be computed from the theory we have defined in this book. This requires a careful understanding of how the information about the background around which the n-point functions are defined is dealt with in the background-independent theory. This is done in this chapter.

This book introduces the reader to a theory of quantum gravity. The theory is covariant loop quantum gravity (covariant LQG). It is a theory that has grown historically via a long, indirect path, briefly summarized at the end of this chapter. The book does not follow the historical path. Rather, it is pedagogical, taking the reader through the steps needed to learn the theory.

The theory is still tentative for two reasons. First, some questions about its consistency remain open; these will be discussed later in the book. Second, a scientific theory must pass the test of experience before becoming a reliable description of a domain of the world; no direct empirical corroboration of the theory is available yet. The book is written in the hope that some of you, our readers, will be able to fill these gaps.

This first chapter clarifies what is the problem addressed by the theory and gives a simple and sketchy derivation of the core physical content of the theory, including its general consequences.

The problem

After the detection at CERN of a particle that appears to match the expected properties of the Higgs [ATLAS Collaboration (2012); CMS Collaboration (2012)], the demarcation line separating what we know about the elementary physical world from what we do not know is now traced in a particularly clear-cut way.

This book is an introduction to loop quantum gravity (LQG) focusing on its covariant formulation. The book has grown from a series of lectures given by Carlo Rovelli and Eugenio Bianchi at Perimeter Institute during April 2012 and a course given by Rovelli in Marseille in the winter of 2013. The book is introductory, and assumes only some basic knowledge of general relativity, quantum mechanics, and quantum field theory. It is simpler and far more readable than the loop quantum gravity text Quantum Gravity (Rovelli 2004), and the advanced and condensed “Zakopane lectures” (Rovelli 2011), but it covers, and in fact focuses on, the momentous advances in the covariant theory developed in the last few years, which have lead to finite transition amplitudes and were only foreshadowed in Rovelli (2004).

There is a rich literature on LQG, to which we refer for all the topics not covered in this book. On quantum gravity in general, Claus Kiefer has a recent general introduction Kiefer (2007). At the time of writing, Ashtekar and Petkov are editing a Springer Handbook of Spacetime, with numerous useful contributions, including that of John Engle's article on spinfoams.

A fine book with much useful background material is that of John Baez and Javier Munian (1994). See also Baez (1994b), with many ideas and a nice introduction to the subject. An undergraduate-level introduction to LQG is provided by Rodolfo Gambini and Jorge Pullin (Gambini and Pullin 2010).