**Review of “Covariant Loop Quantum Gravity, an elementary introduction to quantum gravity and spinfoam theory” by Carlo Rovelli and Francesca Vidotto**

One of the central problems of contemporary physics is finding a theory that allows for describing the quantum behavior of the gravitational field. This book is a remarkable update on one of the most promising approaches for the treatment of this problem: loop quantum gravity. It places special emphasis on the covariant techniques, which provide with a definition of the path integral, an approach known as spin foams. It is a field that has undergone quite a bit of development in the last two decades. The book gives an overview of this area, discussing a series of results that are presented with great clarity. Both students and established researchers will benefit from the book, which provides a dependable introduction and reference material for further studies. Only a basic knowledge of general relativity, quantum mechanics and quantum field theory is assumed. The conceptual aspects and key ideas are discussed in the main body of the book and the complements discuss many of the formal aspects of the theory and are needed to acquire a good command of the subject.

The first chapter puts in perspective the problem of quantizing gravity. In order to do this, one must abandon the notion that physics describes matter in terms of quantum fields in a given space-time. The first step in this direction was given by general relativity, which for the first time treats space-time as another dynamical system that interacts with the rest of matter and represents the gravitational field. To quantize gravity therefore does not mean to consider quantum states in a space-time but to identify the quantum states of space-time. In order to achieve this goal, the notion of quanta of space is introduced by considering certain geometrical properties that can be quantized. This follows a brief historical introduction to the quantum notion of space-time and the discovery of the physical role of the Planck scale.

One of the great virtues of the book is the introduction through simple examples of concepts that are needed in the quantization process. For instance, the Hamilton function is defined for simple mechanical systems like the particle. This is used to introduce the associated concept of transition amplitude in terms of path integrals and their physical interpretation. The parameterized description of the same systems leads in a simple and natural way to the treatment of generally covariant systems. Particular detail is paid to the description without a notion of time of classical and quantum physics.

The tetradic formulation of gravitational dynamics is presented in a very economical and effective way, with special emphasis on the Holst action and the introduction of the Barbero-Immirzi parameter. Although classically the action leads to the same equations of motion as Palatini’s, it has an additional term that plays a substantial role in the quantum theory. Holst’s action is the starting point for the quantization of gravity in four dimensions in loop quantum gravity. The action depends linearly on the momentum conjugate to the connection. This quantity cannot be freely specified; its components satisfy a set of relations called the linear simplicity constraints that plays a very important role in covariant loop quantum gravity. The discussion of the transition to the Hamiltonian version of the theory in metric or Ashtekar variables is, in spite of its brevity, very clear and is accompanied by valuable observations that are not commonly found in other texts. The emphasis on the conceptual aspects of the quantization process is a constant throughout the book.

The quantum theory is introduced gradually in the following four chapters. It starts with a discussion of the classical discretization of Yang-Mills field theory and Regge calculus. The discretization used in the covariant description of quantum gravity is a combination of both. Before tackling the four dimensional theory, the quantization of the Euclidean version of gravity in three dimensions is discussed. That allows the gradual introduction of ideas and techniques starting with a considerably simpler theory that does not have local degrees of freedom. The transition amplitude for this model was first written by Ponzano and Regge in the 1960’s, and is a concrete implementation of the formal definition of path integral defined as a sum over geometries. Although it is the quantization of a topological theory, the transition amplitude has divergences that must be regularized and renormalized. For this reason, the book does a careful analysis of the origin of the divergences. It also discusses the extension of the quantum treatment of three-dimensional gravity to the case with positive cosmological constant, which was first discussed by Turaev and Viro in the 1990’s and is rigorously defined in a finite way.

Although illuminated by the simple cases treated in the previous chapters, the presentation of the covariant formulation of gravity in chapter 7, and the study of its classical limit in the next chapter, are the sections that are most demanding of the reader. Nevertheless, the general tone of the presentation is very simple and reproduces the stages of the previous models with the additional requirement of imposing the simplicity constraints. To this aim, a discretization of the classical four-dimensional theory is first introduced. Then the Hilbert space associated with the boundary variables restricted by the simplicity constraints is identified with a basis of states of the gravitational field provided by spin networks that encode the quanta of space associated with areas and volumes. The final step is the definition of the transition amplitudes extending to the four-dimensional Lorentz invariant case the description introduced in three dimensions. The classical limit is introduced like in other quantum systems, studying the coherent states from the Hilbert space of the theory, constructing the transition amplitude in terms of those states and studying its long distance behavior.

After a succinct discussion of how to introduce matter, i.e. fermions, scalars and gauge bosons, three applications of covariant loop quantum gravity are presented: the study of the thermal properties of black holes; the early universe using what is known as spin foam cosmology, developed in the last few years by Bianchi and others; and the calculation of the graviton propagator to first order around flat space. Although the applications studied are useful to show how the developed formalism allows us to tackle physical situations, it is an area where there is much left to be done, and that is evolving rapidly.

Summarizing, the book succeeds in presenting a simple introduction to covariant loop quantum gravity emphasizing the physical and conceptual aspects and providing enough elements to enter this interesting area of research. Although the bibliography presented is not exhaustive it allows the reader to find reference material for further study.

**Covariant Loop Quantum Gravity, an elementary introduction to quantum gravity and spinfoam theory**

by Carlo Rovelli and Francesca Vidotto

Cambridge University Press

265 pages | Hardback

9781107069626

Published: November 2014

**Price: £37.50**

This work is licensed under a Creative Commons Attribution 3.0 Unported License.