I would like to know how you could apply quantum gravity for Minkowskian or F.R.W metric...that is how you can get the quantum version of the metric Gab for these two problems is there a solution?..What is the method used?..thanks.
So would I. Unfortunately, there is no definite theory of gravity. There are situations where people claim to have a quantum theory of gravity (like in string theory), but this is misleading. String theory has not really described the structure of spacetime at the small scale...in fact current string theory relies on a background spacetime (like flat Minkowski e.g.) to live upon. To answer your question, there is no "qauntum version of the Minkowski metric".
The longer answer is as follows:
For low energy physics, weak gravitational curvatures, we do believe that you can describe gravitation by a qauntum field theory, where the Minkowski metric is a *classical* solution to Einstein's equations and serves as a background for the quantum field theory (as we do in QED, etc). The field that can be quantized is then the fluctuations of the metric about this background. This is called linearized gravity and is therefore not a complete quantization of general relativity. Therefore, perhaps we are not surprised in hindsight that such a qauntum field theory of linearized gravity is non-renormalizable and couldn't describe the true nature of spacetime on the small scale, or for large curvatures, like in the case of black holes e.g. However, we do expect that at low energy/small curvature situations quantum gravity takes the form of a field theory with gravitons as the quanta.
After that attempt, a program was begun to understand how to do quantum mechanics for a background independent theory, which general relativity is (in the sense that it doesn't sit on a background spacetime or any other fields for that matter). From that "canonical quantization" program became what is now known as "quantum geometry", since in this theory, space comes in smallest bits (areas and volumes). Time can be described as the discrete jumps from one quantum geometrical configuration to another. It remains to be shown how one can obtain smooth spacetime geometry, like that described by the Minkowski metric, from this quantum geometry.
I would be interested in any comments you might have on it.
I think it is a major book and that he is one of the most creative
people working in quantum gravity. I am especially impressed by the level of philosophical sophistication about foundations issues and the depth of philosophical questioning.
The difficulties that have been revealed in attempting to bring general relativity and quantum mechanics together into a single theory may be rooted in some unresolved differences at the foundations level. His exploration of this brings both GR and QM into sharper focus for me.
I would appreciate hearing from other readers of Rovelli's book.
Certainly Carlo is one of the important players in the development of the loop formulation of qg. I had started my PhD work with a friend of his, Lee Smolin, but he left after a couple of years for a new institute in Waterloo, Ontario, so that ended that. Then I went over to the "dark side": the string/supergravity camp.
Consequently, I am familiar with Smolin's development of loop formulation of qg (which he worked on with Roveli). But after some time, I realized that this formulation was taking a back seat and something borne out of it was taking the center stage: spin networks. Some people still use the term "loop qg" to decribe the non-perturbative qg program, but that term doesn't properly title the program...it played a role, though.
There are currently two attempts at quantization of pure gr. One is the "canonical" quantization program, which uses spin networks as a basis for a space on which operators (like the area and volume operators) act. This network then represents a (quantum) state of space. There are a number of constraints in the theory to take into account, and all but one was. The remaining thorny issue is how these networks should advance in time; that is, how is the Hamiltonian constraint applied?
The second method is the "covariant" quantization, which is the analog of Feynman's sum over histories in point particle quantum mechanics, represented by "spin foams". These are like "world-foams", the time advancement of a spatial graph. But how exactly the spin networks (which are graphs) in the canonical formulation relate, if at all, to the spin foams is unknown.
Check out Abhay Ashtekar's review articles on the arXiv for some of the story. John Baez has written popular stuff on spin foams.
You mentioned that there are currently two attempts to
quantize general relativity------probably there should be one general category like "QGR", quantum general relativity, that includes both.
Of the two current lines of research that you mentioned: the "spin foam" and the "canonical" approach using spin networks,
an interesting off-shoot of the latter is some recent work by
Rodolfo Gambini and Jorge Pullin on "discrete quantum gravity"
Here is a recent paper by them, one of several, that has references to earlier papers
they seem to have duplicated a result of Martin Bojowald (a postdoc at Penn State) removing the big bang singularity but by a somewhat different method.
It strikes me that you must be personally acquainted with many of the people now most active in QGR, whose results I have been following with considerable interest
you recommended Abhay Ashtekar's survey articles. My favorite
is one he wrote last year
"Quantum Geometry in Action: Big Bang and Black Holes"
he has impressive communication skill
I assume that you must have been at Penn State
when Smolin was your research advisor (before your
going over to the "dark side"). An exceptional bunch
of creative people to have associated with!
BTW that survey stresses the recent
result of the removal of the big bang singularity by Bojowald and
later summarized by a paper Ashtekar and Lewandowski wrote with Bojowald this year ("Mathematical Structure of Loop Quantum Cosmology" gr-qc/0304074)
I'd like to hear more of your impressions of that group of quantum gravity people.