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Quantum Gravity 2: A Second Oxford Symposium

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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.

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.

These days, a fierce battle is raging between two camps of physicists - strings theory developers and proponents of loop quantum gravity (seems like Amazon is going to be another battlefield :-)) . So far there are no experiments to confirm or reject any of these two theories and it is therefore inevitable to conduct the research in both directions. Rovelli's book seen from this perspective is a major attempt to present a consistent picture of how the LQG fits into the framework of our present knowledge about the world around us. This thoughts provoking book explains the path to the present state of LQG, its current problems and is a great reading not only for active developers but for anyone eager to understand the broader impact of these new physical ideas.

Be warned ! This is not an usual textbook ... You should have a fairly deep understanding of QM, QFT and GR - only then will Rovelli show you how to re-think these ideas from a new point of view.

Still, the rest of us (:-)) can benefit greatly from the crystal clear reasoning and philosofical implications he presents throughout the book.

Note: If you like John Baez books and style - you are going to love this one too ...

I have to start by saying that I think the title is very deceptive. This is hardly a book on quantum gravity, more accurately it's a book on one approach to quantum gravity, namely loop quantum gravity. No other approaches to quantum gravity are seriously considered. Even the current leading candidate for providing a quantum theory of gravity, i.e. string theory, is only presented as a straw man to show how poorly it fares (in the author's mind) compared to loop quantum gravity.

The book begins with a brief discussion on general issues in quantum gravity and by presenting some background in general relativity.

He contends that it is wrong to approach quantum gravity by treating general relativity as just another field theory. Two central themes of his approach to quantum gravity seem to be that one should not ignore the fact that general relativity is a theory of spacetime and the correct way to approach finding the quantum theory of gravity (although I don't believe he uses these exact words) is to quantize spacetime. This will lead to spacetime having a discrete structure and will provide a cutoff that will remove the ultraviolet divergences of quantum field theory (this is somewhat different from the way they are removed in string theory). While I agree both of these ideas have a lot of intuitive appeal, it's clear that the jury is still out.

The treatment of general relativity focuses mainly on things that will be useful for developing loop quantum gravity. This includes formulating it in terms of connections (instead of the metric) and presenting it in the Hamiltonian form. I found it a bit odd that he included discussions of "Newton's bucket" and Mach's principle(s), while they have some historical importance, it seems unlikely (to me anyway) that these will provide any important insights going forward.

After providing some background in quantum mechanics and quantum field theory he goes on to develop loop quantum gravity. The presentation is clear, the most up-to-date I've come across.

One of the results that is of most interest is his outline of the calculation of black hole entropy by counting states. The degrees of freedom are given by quantum fluctuations of the horizon. The result, up to an undetermined multiplicative constant, is the Bekenstein entropy. This is presented as an impressive accomplishment of loop quantum gravity. The string theory calculation is dismissed (in a footnote) as having only been done for the unphysical case of an extremal black hole. Rather than just taking the author's word for it, I'd suggest reading the string theory derivation, for example in Polchinski's book "String Theory" Volume II chapter 14. Then decide which, if either, is more impressive, but there are a couple of things to note. One the string theory calculation that the author refers to gets the multiplicative factor right. Another, which the author ignores or is unaware of, is that Polchinski gives a qualitative argument that string theory gets the entropy of the Schwarzschild black hole correct to within an undetermined multiplicative factor.

I thought the appendix that covered the history of quantum gravity great.

One could argue that anybody that has a realistic chance of understanding the material in the book would need a fairly strong background in general relativity and quantum field theory. Such readers would easily recognize that this book hardly provides a balanced perspective. Even so, I wish the book had a more appropriate title. As a book on loop quantum gravity I think it's pretty good and rates about four stars. As a book on quantum gravity I don't see how it could rate more than one or two stars.

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.

It is ironic that in his 30+ paragraph attack on Carlo Rovelli, as in his relentless attacks on the small community of physicists and mathematicians still interested in alternatives to the morass of culture, conjecture and ideology which now goes by the meaningless name "M theory," Lubos Motl has the audacity to dismiss him, and them, as dogmatists!

If the book really was as useless and silly as Motl pretends, then he would not be so afraid of it.

I seriously doubt that anyone will go to the trouble of writing a line-by-line rebuttal of Motl's screed. But I think that open minded readers would be interested to see the very terse, very positive review from Alain Connes, available at the book's CUP webpage.

Quite simply : the most extraordinary book I have never read. It will deeply change your vision of physics. You think you understand quantum mechanics and general realtivity ? You will be suprised. A must buy.

It could be that LQG isn't as popular in the physics community as it deserves to bebecause a lot of people don't appreciate an important aspect of classical GR (covered in part I of Rovelli's book). Below I give a quick and easy argument whichuses only the very basics of GR making it accessible to anyone and also rather difficult to dismiss. The hope is that by giving this argument the browser will see that there may actually be something to LQG and have a look at this book.

Ok. In 1912, while developing GR, Einstein realised something he found rather alarming. Here's one version of the argument: it starts with an utterly straightforward mathematical observation. Here is written the SHO differential equation twice Eq(1) d^2 f(x) / dx^2 + f(x) = 0 and Eq(2) d^2 g(y) / dy^2 + g(y) = 0 except in Eq(1) the independent variable is x and in Eq(2) the independent variable is y. Once we find out that a solution to Eq(1) is f(x) = cos x, we immediately know that g(y) = cos y solves Eq(2). This observation combined with general covariance has profound implications for GR. Assume pure gravity first. Say we have two coordinate systems, x-coordinates and y-coordinates. General covariance demands the equations of motion have the same form in both coordinate systems, that is, we have exactly the same differential equation to solve in both coordinate systems except in one the independent variable is x and in the other the independent variable is y. Once we find a metric function g_{ab}(x) that solves the EQM in the x-coordinates we immediately know (by exactly the same reasoning as above) that the same function written as a function of y solves the EOM in the y-coordinates. As both metric functions have the same functional form but belong to different coordinate systems, they impose different spacetime geometries. Thus we have generated a second DISTINCT solution! Now comes the problem. Say the two coordinate systems coincide at first, but at some point after t=0 we allow them to differ. We then have two solutions, they both have the same initial conditions yet they impose different spacetime geometries. The conclusion is that GR does NOT determine the proper-time between spacetime points! Bummer! The argument I have given (or rather a refinement of it) is what's known as Einstein's hole argument. It is straightforward to include matter - we have a larger set of differential equations but they still have the same form in all coordinates systems, the same argument applies and again we obtain two solutions with the same initial conditions which impose different spacetime geometries. It is very important to note that we could not have generated these extra distinct solutions if spacetime were fixed and non-dynamical, and so the resolution (background independence) only comes about when we allow spacetime to be dynamical. We can interpret these extra distinct solutions as follows. For simplicity we first assume there is no matter. Define a metric function g'_{ab} whose value at P is given by the value of g_{ab} at P_0, i.e. g'_{ab}(P) = g_{ab}(P_0). Now consider the coordinate system which assigns to P the same coordinate values that P_0 has in the x-coordinates. We then have g'_{ab} (y_0=u_0,y_1=u_1, y_2=u_2, y_3=u_3) = g_{ab} (x_0=u_0,x_1=u_1, x_2=u_2 , x_3=u_3), where u_0,u_1,u_2,u_3 range over the permissible coordinate values. But this is precisely the condition that the two metric functions have the same functional form! We see that the new solution is generated by dragging the original metric function over the spacetime manifold while keeping the coordinate lines `attached' (it is important to realise that we are not performing a coordinate transformation here). This is what's known as an active diffeomorphsm (coordinate transformations are called passive diffeomorphisms). It should be easy to see that when we have matter present, simultaneously performing an active diffeomorphism on the gravitational and matter fields generates the new distinct solution.

It was only in 1915 when Einstein finally resolved the hole argument that GR was born. The resolution (mainly taken from Rovelli's book) is: as GR does not determine the distance between spacetime points, how the gravitational and matter fields are located over spacetime, and so the values they take at spacetime points, can have no physical meaning. What GR does determine are the mutual relations that exist between the gravitational field and the matter fields (i.e. the value the gravitational field takes where the matter field takes such and such value). From these mutual relations we can form a notion of matter being located with respect to the gravitational field and vice-versa, (see Rovelli's book for exposition). What Einstein discovered was that physical entities are located with respect to one another only and not with respect to the spacetime manifold. This is what background independence is! And what Einstein was referring to when he made his remark "beyond my wildest expectations". We learnt from SR that position and motion only have meaning relative to an inertial frame; GR teaches us that there are no background geometric reference systems at all, position and motion have become completely relative! LQG people regard background independence as a central tenet in their approach to quantizing gravity - a classical symmetry that ought to be preserved by the quantum theory if we are to be truly quantizing geometry(=gravity). One immediate consequence is that LQG is UV-finite because small and large distances are gauge equivalent. A less immediate consequence is that the theory can be formulated at a level of rigour of mathematical physics, which is nothing to sneeze at in the absence of experimental guidance.

Perturbative string theory (as well as a number of non-perturbative developments) is not background independent, the scattering matrix they calculate is not invariant under active diffeomorphisms. Of the end of 2005 Rovelli et al have put together the formulism to calculate background independent scattering amplitudes (this is no easy task!). Rovelli has obtained Newton's law from the fully non-perturbative quantum theory. However it is still early days and this result is not yet convincing established.

To finish off, we should see this book on more shelves and in more book stores!! Also, look out for another LQG book by Thomas Thiemann and Peter Woit's book in April.

I'm a new comer to quantum gravity. Although I only have some background in classical mechanics and relativity , I thought the books is quite approachable as most of the terms are explained cearly following the logical reasioning. A side note: Besides string and loop quantum gravity, the book also mentioned differernt version of theoretical framework such as tiwster theory and Euclidean quantum gravity. Its quite disappointing that the book didn't go into detail of each theory and possibly give a comparison between different theory.

It is guaranteed that many people who have not seen the book - and who have no idea about physics (for example, linux evangelists) - will write a lot of irrelevant "reviews", but I am sure that those who are able to think will know which reviews are serious and which reviews are not.

I am among those who really enjoy searching for errors in physics books. While the good books usually have a couple of hundreds of typos and roughly five errors of a slightly conceptual flavor, it may be easier to enumerate the correct and valuable statements in Rovelli's book rather than the errors. The problems with this book are rather serious, numerous, and fundamental. It's mainly because the chosen point of view on the topic is problematic and perhaps obsolete.

First of all, the title is inappropriate and overly ambitious. The book does not cover quantum gravity (QG) - this term includes, among many other insights, semiclassical gravity, the calculation of particle production and Hawking radiation, the black hole information puzzle and black hole thermodynamics; holography, the AdS/CFT correspondence and other mechanisms of appearance and disappearance of spacetime dimensions; physics at the Planck scale, physics of spacetime singularities and application of quantum theory to cosmology; graviton scattering; quantum corrections to geometry, geometric dualities such as T-duality and mirror symmetry, topology changes and other topics arising in string theory; quantum effects influencing locality, causality, and the arrow of time; the origin of gravity and its interplay with other forces and particles. Rovelli's book does not really explain either of these topics.

The author starts with some terminological issues. For example, he redefines the word "relativistic" in such a way that the special theory of relativity is "nonrelativistic". It's not just a matter of unusual language: Rovelli repeatedly contradicts the fact that a theory XY must locally reduce to special relativity if XY should be called "general relativity" (GR). The models he presents probably do not respect the laws of special relativity - they are not relativistic in the usual meaning of the word - and consequently they're problematic from the viewpoint of experimental validity as well as according to the very purpose of GR: the only reason why Einstein had to look for a new theory of gravity was the required compatibility with his special theory of relativity. Because the adjective "relativistic" has a positive flavor in it, Rovelli decided to redefine it so that his promoted theory can be called "relativistic" even though it is not (according to the usual understanding of the word).

The initial chapter seems as an idiosyncratic account of reasoning about QG before anything sound about QG was known. This chapter also presents another adjective that is very popular in the LQG community: "background-independent". Rovelli correctly points out some complications that quantized geometry adds to the usual concepts of quantum field theory such as the operator product expansions. Many of these problems are simply solved by writing the metric as the sum of a nonzero classical background (a vacuum expectation value) plus a quantum, operator-valued fluctuation. This is the usual approach in particle physics and string theory and the main target of Rovelli's attacks. He does not say that such a decomposition is necessary for the concepts like the S-matrix to make sense. He does not say that such a decomposition does not eliminate the general covariance of the physical results (which really means decoupling of the unphysical modes). He does not say that even Newton's laws can be written in a background-independent fashion. Finally, he does not say that being "background independent" is vacuous unless one can show that the theory predicts many different geometric backgrounds (which is not likely in the case of LQG). He also hides the fact that there are different Hamiltonian LQG theories for different spacetime topologies; unlike string theory, LQG does not allow topology change.

The space limitations are forcing me to truncate the review; see http://schwinger.harvard.edu/~motl/rovelli.html for the full text.