I collect in this appendix some simple reflections on scientific methodology, and on the content of scientific theories, relevant for quantum gravity. In particular, I try to make more explicit the methodological assumptions at the root of some of the research described in this book, and to give it some justification.

I am no professional philosopher, and what follows has no ambition in that sense. I am convinced, however, of the utility of a dialog between physics and philosophy. This dialog has played a major role during the other periods in which science has faced fundamental problems. I think that most physicists underestimate the effect of their own epistemological prejudices on their research. And many philosophers underestimate the effect – positive or negative – they have on fundamental research. On the one hand, a more acute philosophical awareness would greatly help the physicists engaged in fundamental research. As I have argued in Chapter 1, during the second half of the twentieth century fundamentals were clear in theoretical physics and the problems were technical, but today foundational problems are back on the table, as they were at the time of Newton, Faraday, Heisenberg and Einstein. These physicists couldn't certainly have done what they have done if they weren't nurtured by (good or bad) philosophy. On the other hand, I wish contemporary philosophers concerned with science would be more interested in the ardent lava of the fundamental problems science is facing today. It is here, I believe, that stimulating and vital issues lie.

The aim of this chapter is to introduce the general ideas on which this book is based and to present the picture of quantum spacetime that emerges from loop quantum gravity, in a heuristic and intuitive manner. The style of the chapter is therefore conversational, with little regard for precision and completeness. In the course of the book the ideas and notions introduced here will be made precise, and the claims will be justified and formally derived.

In its conventional formulation, mechanics describes the evolution of states and observables in time. This evolution is governed by a hamiltonian. This is also true for special-relativistic theories, where evolution is governed by a representation of the Poincaré group, which includes a hamiltonian. This conventional formulation is not sufficiently broad because general-relativistic systems – in fact, the world in which we live – do not fit into this conceptual scheme. Therefore we need a more general formulation of mechanics than the conventional one. This formulation must be based on notions of “observable” and “state” that maintain a clear meaning in a general-relativistic context. A formulation of this kind is described in this chapter.

The conventional structure of conventional nonrelativistic mechanics already points rather directly to the relativistic formulation described here. Indeed, many aspects of this formulation are already utilized by many authors. For instance, Arnold identifies the (presymplectic) space with coordinates (t, qi, pi) (time, lagrangian variables and their momenta) as the natural home for mechanics. Souriau has developed a beautiful and little-known relativistic formalism. Probably the first to consider the point of view used here was Lagrange himself, in pointing out that the most convenient definition of “phase space” is the space of the physical motions. Many of the tools used below are also used in hamiltonian treatments of generally covariant theories as constrained systems, although generally within a rather obscure interpretative cloud.

In this book I have tried to present a compact and unified perspective on quantum gravity, its technical aspects and its conceptual problems, the way I understand them. There are a great number of other aspects of quantum gravity that I would have wished to cover, but my energies are limited. Just to mention a few of the major topics I have left out: 2 + 1 gravity, the Kodama state and related results, quantum gravity phenomenology, supergravity in LQG, coherent states … Here I briefly summarize the physical picture that emerges from LQG and the solution it proposes to the characteristic conceptual issues of quantum gravity. I conclude with a short summary of the main open problems and the main results of the theory.

Lev Landau has called GR “the most beautiful” of the scientific theories. The theory is first of all a description of the gravitational force. Nowadays it is very extensively supported by terrestrial and astronomical observations, and so far it has never been questioned by an empirical observation.

But GR is far more than that. It is a complete modification of our understanding of the basic grammar of nature. This modification does not apply solely to gravitational interaction: it applies to all aspects of physics. In fact, the extent to which Einstein's discovery of this theory has modified our understanding of the physical world and the full reach of its consequences have not yet been completely unraveled.

This chapter is not an introduction to GR, nor an exhaustive description of the theory. For this, I refer the reader to the classic textbooks on the subject. Here, I give a short presentation of the formalism in a compact and modern form, emphasizing the reading of the theory which is most useful for quantum gravity. I also discuss in detail the physical and conceptual basis of the theory, and the way it has modified our understanding of the physical world.

Classical mechanics admits two different kinds of formulations: hamiltonian and lagrangian (I never understood why). Feynman realized that so does quantum mechanics: it can be formulated canonically, with Hilbert spaces and operators, or covariantly as a sum-over-paths. The two formulations have different virtues, and calculations that are simple in one can be hard in the other. Generically, the lagrangian formalism is simpler, more transparent and intuitive and keeps symmetries and covariance manifest. The hamiltonian formalism is more general, more powerful, and, in the case of quantum theory, far more rigorous. The ideal situation, of course, is to be able to master a theory in both formalisms.

So far I have discussed the hamiltonian formulation of LQG. In this chapter I discuss the possibility of a lagrangian sum-over-paths formulation of the same theory. This formulation has been variously called sum-over-surfaces, state-sum and goes today under the name of “spinfoam formalism.” The spinfoam formalism can be viewed as a mathematically well-defined and possibly divergence-free version of Stephen Hawking's formulation of quantum gravity as a sum-over-geometries.

The spinfoam formalism is less developed than the hamiltonian version of loop theory. Also, while the general structure of the spinfoam models matches nicely with the hamiltonian loop theory, the precise relation between the two formalisms (each of which exists in several versions) is not entirely clear. But research is currently moving fast in this direction.

The aim of the spinfoam formalism is to provide an explicit tool to compute transition amplitudes in quantum gravity. These are expressed as a sum-over-paths. The “paths” summed over are “spinfoams.”

In the previous chapter, I constructed the Hilbert space and the basic operators of LQG. In this chapter I discuss the dynamical aspects of the theory. For this, we must write a well-defined version of the Wheeler–DeWitt equation (6.3) in order to construct the hamiltonian operator H.

The construction of the hamiltonian operator of LQG has taken a long time, proceeding through a number of steps. The first was to realize that the operator can be defined via a simple regularization and that any loop state Ψα solves HΨα = 0, provided that α is a loop without self-intersections. This observation opened the door to LQG. The result remains true in all subsequent definitions of H. With this first simple regularization, however, H diverges on intersections.

The second step was to realize that a finite operator H could be obtained provided that (i) its action is defined on diffeomorphism-invariant states and (ii) its density character is appropriately dealt with. By a sort of magic, the action of the operator with the correct density weight on a diff-invariant state converges trivially, in the limit in which the cut-off is removed. This result is the major pay-off of the background-independent approach to QFT. It is a manifestation of the relation between background independence and the absence of UV divergences.

In this appendix I sketch the main lines of development of the research in quantum gravity, from the first explorations in the early 1930s (the thirties) to nowadays.

I have no ambition of presenting complete references to all the important works on the subject; some of the references are to original works, others to reviews where references can be found. Errors and omissions are unfortunately unavoidable and I apologize for these. I have made my best efforts to be balanced, but in a field that has not yet succeeded in finding consensus, my perspective is obviously subjective. Trying to write history in the middle of the developments is hard. Time will pass, dust will settle, and it will slowly become clear if we are right, if some of us are right, or – a possibility never to disregard – if we all are wrong.

I am very much indebted to the many friends that have contributed to this historical perspective. I am particularly grateful to John Stachel, Augusto Sagnotti, Gary Horowitz, Ludwig Faddeev, Alejandro Corichi, Jorge Pullin, Lee Smolin, Joy Christian, Bryce DeWitt, Cecile DeWitt, Giovanni Amelino-Camelia, Daniel Grumiller, Nikolaos Mavromatos, Stanley Deser, Ted Newman and Gennady Gorelik.

In this chapter I briefly mention some of the most successful applications of LQG to concrete physical problems. I have no ambition of completeness, and I will not present any detailed derivation. For these, I refer to original papers and review articles. I only illustrate the main ideas and the main results.

The two traditional applications of quantum gravity are early cosmology and blackhole physics. In both these fields, LQG has obtained interesting results. In addition, a certain number of tentative calculations concerning other domains where Planck-scale physical effects could perhaps be observable have also been performed.