A First Course in Loop Quantum Gravity

Many non-experts who are interested in physics know that the current theory of gravity, Einstein's general theory of relativity, stands apart from the theories of the other three fundamental forces, the electromagnetic, weak nuclear, and strong nuclear forces, in that the former is a classical theory whereas the latter are quantum theories. Physicists like to unify things, so they would like to have the theory of gravity be a quantum theory as well, so they can then have a "theory of everything". In addition to being satisfying on a philosophical and esthetic level, this would also potentially be able to explain what's going on at the center of black holes and at the "beginning" of the Big Bang, for which general relativity gives a singularity, an unphysical point of infinite energy density indicating that general relativity is no longer properly describing the physical reality in these situations.One approach to a quantum theory of gravity is the famous (perhaps infamous) "string theory". Another approach, the approach discussed in this book, is loop quantum gravity, in which a more traditional approach to quantizing the gravitational field is employed. In a nutshell, loop quantum gravity is a canonical quantization of a modification of the Hamiltonian formulation of general relativity originally developed by Arnowitt, Deser, and Misner in the early 1960s ("ADM formalism"), said modification being the replacement of the position and momentum variables used by ADM with a new set of variables introduced by Ashtekar in the 1980s. Canonical quantization refers to the quantization procedure based on the Hamiltonian originally developed by Dirac in the 1920s and further developed by him in the 1950s. Loop quantum gravity uses a "loop representation" which has as a key result that there is a minimum allowed length, area, and volume in the universe, i.e, these quantities are quantized. This in turn leads to the result that the universe has a minimum size, and if we play the movie of cosmic expansion backwards, we reach this minimum size before we reach the singularity, and the universe starts to expand again, the so-called "Big Bounce". Also, the relation between the surface area and entropy of a black hole reduces to a relationship between the number of area quanta which make up the surface area and the entropy.This book is directed at the advanced undergraduate level, and aims to fill a gap in the pedagogy of this subject, which requires much more math and physics background to really understand than any undergraduate would have. As a result, it takes a very simplistic approach and should be considered no more than a very preliminary introduction to the topic.The first part of the book covers all the basic background concepts needed to understand the program of loop quantum gravity, including general relativity, the generalized Hamiltonian with constraints, and canonical quantization. It then goes on to discuss the ADM formalism, Ashtekar variables, the loop representation, and applications of loop quantum gravity to cosmology and black hole thermodynamics. It ends with a very honest evaluation of the limitations and incompleteness of the theory at the present time. Loop quantum gravity is still a work in progress, and although there have been many important advances in terms of formulating a theory, there is as yet no validation experimentally, nor any obvious path to such a validation.You will not really understand loop quantum gravity when you finish this book, only understand a very superficial version of it "dumbed down" for the intended readership of this book, but still extremely formidable in terms of its demands on the reader mathematically speaking. If you haven't already had at least an introductory exposure to general relativity, quantum mechanics, Lagrangian and Hamiltonian mechanics, and quantum field theory, you're in for some very tough sledding indeed. If you have this background, after you finish this book you will be prepared to delve more deeply into the topic if you have the time and inclination. Included are references to more advanced texts as well as important review papers to guide this further exploration.I give this book 5 stars for identifying a pedagogical need, clearly defining the intended readership and the goals, and carefully describing the inevitable limitations and gaps in trying to present such a sophisticated, difficult topic to a relatively unprepared audience.

This book contains a nice overview of the Physics involved in the more traditional approach to quantizing gravity, that is Loop Quantum Gravity (LQG). It is important to say though that all the topics treated are only superficially presented which I suppose it is ok considering the audience it is target at, that is, advanced undergraduates who have not yet had a real exposure to topics like Classical Field Theory, Quantum Field Theory, Dirac's theory of Constraints and General Relativity. Having said this I must say that the book gives you not so much formulas but the really important (and necessary), the rest is very nicely explained with words, so it reads much as a story. It starts by revising special relativity and the formulation of Maxwell's electrodynamics both first through the usual approach and then using relativistic tensorial formulation. Then it proceeds to present General Relativity first giving the basics of curved geometry, i.e. vectors, tangent spaces and the jumping straight to curvature. After this, Einsteins equations are presented along with two of its exact solutions: the Schwarszchild and Robertson-Walker spacetimes, also a word on diffeomorphisms and the Lie derivative are given. Then the famous 3+1 decomposition of GR is given, this is, the ADM (Arnowitt-Deser-Misner) formulation and the metric is also decomposed in triads which will be essentially used later as one of the canonically variables of the reformulation of GR. After this the theory of Hamiltonian mechanics with constraints is given and then Yang-Mills theories. Then the authors reminds about quantum mechanics, and the generalization of this to field theories, quantum field theories, a brief introduction to interacting field theories is given explaining how divergences appear in perturbative series where also Feynman diagrams are illustrated. Finally a word on renormalization is also given. After this the reformulation of General Relativity in terms of the new Ashtekar's variables is presented. All the grace of this new change of variable is that now GR can be written much similar to a usual Yang Mills theory and originally there was hope that since YM were known on as how to quantize them, the same would happen here but unfortunately the Hamiltonian Constraint is either complex (and simple) or is real and so complicated that it has not been solved until this day. Toward the end of the book are presented loop variables, the concept of spin networks, mentioned the result that in loop quantum gravity there is an Area and Volume operator and so area and volume in this theory are quantize, there is a chapter on loop quantum cosmology and finally the last chapter touches upon some known facts about black holes like Hawking Radiation and black hole entropy, spin foams which are basically the many spin networks intertwined together as you move from one geometry to another in the path integral for GR, some possible experiments that could be done to test this theory and the Loved by me and intriguing "Problem of Time" which basically consists in that as the Hamiltonian of GR is just a sum of constraints and on the other part constraints vanish weakly, the whole thing (i.e. the entire Hamiltonian) vanishes and by hamilton's equations this implies that there is no evolution in time of any dynamical variable, in other words, the whole system is frozen!, well I like the book and give to it four stars because sometimes is just a little too sloppy on some of the material presented although when it is, it does gives you a proper reference to look at for further study, all in all, nice book!