Customer Review

30 of 34 people found the following review helpful
5.0 out of 5 stars A useful overview of an alternative theory, April 24, 2005
Verified Purchase(What's this?)
This review is from: Quantum Gravity (Cambridge Monographs on Mathematical Physics) (Hardcover)
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.
Help other customers find the most helpful reviews 
Was this review helpful to you? Yes No

[Add comment]
Post a comment
To insert a product link use the format: [[ASIN:ASIN product-title]] (What's this?)
Amazon will display this name with all your submissions, including reviews and discussion posts. (Learn more)
Name:
Badge:
This badge will be assigned to you and will appear along with your name.
There was an error. Please try again.
Please see the full guidelines here.

Official Comment

As a representative of this product you can post one Official Comment on this review. It will appear immediately below the review wherever it is displayed.   Learn more
The following name and badge will be shown with this comment:
 (edit name)
After clicking the Post button you will be asked to create your public name, which will be shown with all your contributions.

Is this your product?

If you are the author, artist, manufacturer or an official representative of this product, you can post an Official Comment on this review. It will appear immediately below the review wherever it is displayed.  Learn more
Otherwise, you can still post a regular comment on this review.

Is this your product?

If you are the author, artist, manufacturer or an official representative of this product, you can post an Official Comment on this review. It will appear immediately below the review wherever it is displayed.   Learn more
 
System timed out

We were unable to verify whether you represent the product. Please try again later, or retry now. Otherwise you can post a regular comment.

Since you previously posted an Official Comment, this comment will appear in the comment section below. You also have the option to edit your Official Comment.   Learn more
The maximum number of Official Comments have been posted. This comment will appear in the comment section below.   Learn more
Prompts for sign-in
 

Comments


Sort: Oldest first | Newest first
Showing 1-1 of 1 posts in this discussion
Initial post: Dec 7, 2008 9:51:17 PM PST
Fantastic Review!

I am buying this book for myself for Christmas. As someone who left academia behind because String Theory was the only game in town it will be the greatest resource. Anyone who has managed to get through the lectures by Lee Smolin on the web will not cease to be amazed. My thesis 'Quantum Field Theory on S4' as an attempt to explore the Coleman wormhole conjecture to minimise the Cosmological constant and using renormalization group equations to run the top quark Higgs coupling up to the wormhole scale. Most of my thesis involves summation over angular momentum eigenvalues of SO(n+1), with n->4 as dimensional regularization of a 4-sphere embedded in 5 dimensional Euclidean space. The symmetry of S4 guaranteed the Wightman axioms were satisfied, and the Euclidiean formulation led to the temperature being inversely proportional to the radius etc..

So I had ( curved Euclidean space-time ) <==> ( Yang-Mills fields ) thanks to Ashketar, Smolin, Rovelli etc.. I actually think I know how to quantize the left hand side and de-Euclidianize it (ADS I guess) and see what happens... after doing my day job ;-(

As for those who criticise it as 'not mentioning the great success of String Theory', they can take Ockham's Razor and go and play in some Calabi-Yau space and leave the rest of us to actually explore a theory with a semi-classical limit with a positive cosmological constant.

Brett.
‹ Previous 1 Next ›

Review Details

Item

Reviewer

Dr. Lee D. Carlson
(VINE VOICE)    (HALL OF FAME REVIEWER)    (REAL NAME)   

Location: Baltimore, Maryland USA

Top Reviewer Ranking: 6,510