Quantum Gravity (Cambridge Monographs on Mathematical Physics) 1st Edition

This book gives a detailed overview of that "other theory" of quantum gravity called `loop quantum gravity'. String theory has been viewed by many as a promising theory of quantum gravity and there are many reasons to believe this is the case. String theory though requires a formidable amount of mathematics for its construction, and has yet to have any experimental verification. Therefore, it is not surprising that other alternative theories of quantum gravity would be constructed, these needing a minimal amount of formalism and staying as close as possible to what can be observed. Loop quantum gravity is one of these, and is the most popular alternative to string theory as a theory of quantum gravity.

The initial five chapters of the book motivate the need for quantum gravity and also phrase the theories of general relativity and relativistic quantum physics in a language that will be used to formulate the theory of loop quantum gravity. General relativity (GR) is formulated as a dynamical system defined by the Hamilton-Jacobi equation for a functional defined on the space G of three-dimensional SU(2) connections. The equation is invariant under internal gauge transformations and 3D diffeomorphisms.

The quantization of GR is obtained in terms of complex-valued Schrodinger wave functionals on G. The derivatives of the Hamilton-Jacobi functional are replaced by derivative operators in order to obtain the quantum dynamics. The Hamilton-Jacobi equation then becomes the Wheeler-DeWitt equation governing the quantum dynamics of spacetime. The quantum (kinematical) state space is defined letting G be the space of smooth 3D real connections defined everywhere on a 3D surface S. The functionals are defined in terms of an ordered collection of smooth oriented paths L (essentially a lattice) on S, and are called "cylindrical functions" by the author. Scalar products are defined, which when completed gives the (kinematical) Hilbert space K. Lest the reader believe that this is nothing more than a quantum Yang-Mills theory on a lattice, the author is careful to note that a continuous theory over all possible lattices in S is being considered. The space K is nonseparable, but factoring out the diffeomorphisms gives a separable one. It has an orthonormal basis, and contains a subspace K0 of states invariant under local SU(2) gauge transformations. The ubiquitous spin network states form a orthonormal basis for K0. Again the author cautions against viewing this as ordinary lattice gauge theory, since diffeomorphism invariance makes it different from the latter. The spin networks are graphs L with links and nodes. The nodes are joined by links and the curves in L overlap only at the nodes. Each node has a multiplicity that measures the number of links going in and out of it. The author shows explicitly how to construct the spin network states, which are an orthornormal basis for K0.

So what about the observables of loop quantum gravity? The connection and its momentum are the field variables in the canonical theory, and are replaced by field operators. The momentum operator has a curious operation in this theory: the author describes it as "grasping" a path. The momentum operator though is not gauge invariant on K0, and so the author defines a new gauge-invariant (and self-adjoint) operator associated to S and has a straightforward operation on spin network states. This operator represents the physical area of S, and its spectrum, interestingly, can be interpreted as a quantized area. This result is related to the derivation of the entropy of black holes in the book, and is considered to be one of the significant results given by loop quantum gravity. A similar construction is done with the volume, giving a volume operator, which also has a discrete spectrum, but only has contributions from the nodes of a spin network state. Loop quantum gravity therefore truly gives a "quantized geometry." Each node of a spin network represents a quantum of volume, giving a collection of "chunks" separated from each other by surfaces, the areas of which are governed by the area operator. The graph L of the spin network determines the adjacency relation between these chunks, and is interpreted as the graph dual to a cellular decomposition of physical space. Spin network states therefore determine a discrete quantized three-dimensional metric.

The dynamics of loop quantum gravity requires the construction of the Hamiltonian operator. As in quantum field theories, this involves regularization, and the author shows how the Hamiltonian operator acts only on the nodes of the spin network, and gives a detailed discussion of the background independence of the theory. The latter is one of the most important features of loop quantum gravity, and is frequently advertised as one of its virtues. The author also discusses the extent to which the Hamiltonian operator is unique, outlining in the process several alternatives. When matter fields are considered, the author shows that the total Hamiltonian is finite, again pointing to the background-independence of loop quantum gravity. Loop quantum gravity reduces to classical general relativity as Planck's constant goes to zero, but the author lists many issues that have not been settled by this theory. One of these of course concerns the observable consequences, the lack of which it shares with string theory. Loop quantum gravity also does not attempt to unify the different interactions in nature in a single theory, as does string theory. But loop quantum gravity does give some interesting predictions, one of these being that the size of the universe is quantized. It also predicts an inflationary phase in the expansion of the early universe, as numerical solutions of the Wheeler-DeWitt equation indicate. By far the most interesting consequence of loop quantum gravity, is that it makes more reasonable the Bekenstein-Hawking interpretation of the entropy of a black hole. In fact the Bekenstein-Hawking entropy can be derived in loop quantum gravity, up to a factor called the Immirzi parameter. These are all impressive achievements, considering the status of quantum gravity now as compared to three decades ago.

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I'm experiencing some hesitation as this is hands down the most advanced book that I have ever 'reviewed' and, in fact, ever read. I have only completed Part 1 -"Relativistic Foundations" as I have to go deeper into Differential Geometry before proceeding into Part II - "Loop Quantum Gravity." Part 1 is a pretty amazing, often philosophical introduction both describing the problem that QG is trying to approach (the contradictions between QFT and GR)and laying the foundation for LQG. It becomes clear (slowly) that our notions of space and time need serious overhauling before we can understand what LQG is all about. String theory doesn't have this problem as it more or less uses our 20th century notions of space and time as its framework. Sure, it adds a few dimensions and curls some up but it's pretty much still the same old space and time. LQG does not use this framework and rather seems to work towards a physics without time. Rovelli does a masterful job in Part 1 of slowly, clearly and precisely helping the reader to make this transition. Most memorable to me is his discusson of the ten meanings of time where he demonstrates that we have already stripped time of many of its seemingly inherent properties ('flowing', 'measured' to name 2). He just proposes stripping off a few others! I'm looking forward to Part II.

This is a excellent book. Dr. Rovelli is attempting a rare exercise in that he is trying to substantially change a physicists view of the world. This book is equal parts philosphy and mathematics and tries to instill in the reader an intuition that most books never achieve. The book is not for someone who likes to be given an equation or model and then is shown how to "turn the crank" to obtain an answer or those who think the theory is finished as long as you constrain yourself to stay away from the pathologies. (Think Standard Model and a single photon that has the energy density of a black hole. ) Most present day theories have these pathologies, patches and inconsistencies precisely because they were built from previous approximate theories and then were modified when problems showed up. This book starts with a clean sheet of paper and asks, "What should a relativistic quantum theory of gravity look like?' "What are the mathematical structures that are needed and how do they fit together to give us a consistent view of the world?" Indeed the book goes back through classical mechanics and quantum just to show what the mathematical structures do in each of these early theories. The philosophical basis, the "why is this important", for the combination of quantum ideas and the philosophy of relativism is very well laid out in this book. That is what I was seeking, "the why". Theories that throw another constraint on the space of solutions, or add more symmetry just to fix problems that pop up means that they have serious flaws in their basic foundations that need to be addressed differently. Dr. Rovelli shows that loop quantum gravity has the same clean sparseness as it's foundations and is very appealing. The discrete quantization of volume and area is a major sucess. If you like having your world view changed, a new paradigm, I highly recommend this book.