My intention for this post was to make an exposition of Horava-Witten theory and the phenomenological and cosmological theories related to them (brane universes, ekptyroptic, etc).
But a confluence of factors has decided me to make a brief post about a diferent topic. In a previous entry I talked about conformal gravity. That was a nice theory which could presumably reproduce general relativity in the appropiate limits, and it was renormalizable. I haven´t head nothing new about it since that post, but I neither have had time to search so maybe there is something going on and simply a am not aware of it.
Since them I have had notice of ¡3! diferent claims about new viable paths to quantum gravity. I´ll explain a bit about them.
The first is the effort of Martin Reuter in triying to prove that classical pure (i.e. no matter) general relativity can have an UV fixed point. I am not still very expert in the insights of Wilson and Polchinsky in the renormalization group, but the idea goes as follows. In any QFT loops usually give rise to infinite terms. To get around them you do a cutt-off and you separate the integral in a finte and an infinte part. Them you add a new term to the lagraingian (a counterterm) which generates an aditional diagram whcih cancels the divergent part of the integral. Inthis way ou can deal with any theory, renormalizable or not. But in renormalizable theories the counterterms have the same form as the original terms of the lagrangian so you can cancel the divergences by apropiate redefinitions of parameters of the lagrangian (coupling constants, masses, wavefunctions, etc). If the neccessary counterterms are not of the same form that the orignial lagrangian this resoults into the adition of new parameters of the theory. You can have by this procedure a finite theory,, but with an infinite number of free parameters, i.e., with no predicitive power. General relativity is of this type. But there is a possible scape to this problem. It could be that there are aditional aspects which constraint the values of all that necessary infinte number of parameters so all of them have the same value and the predictive power of the theory is recovered. Theories with such behaviour are said to have an UV (ultra violete) fixed point. (Wilson-Polchinsky renormalization group theory consist in many other aspects, but this is all I need of renormalization group theory to explain what´s going on here).
Martin Reuter is triying to probe that pure general relativy actually has an UV fixed point. To do so he must do some desompositoins which allow to do some numerical analisys. The results of the numerical analisys strongly suggest the existence of the UV fixed point. He also can uses that results to get some results incosmology. I havent readed too mcuh about the details so I´ll not say anything about them. What I must say is that there is a problem with all this. If matter is added to the theory it can be shown that you are adding wht are called relevant operators and that you will go out of the UV fixed point so semmengly all that effort would be poinless. In fact some people in LQG (canonical as well as spin foam versions) are triying to recover matter as some kind of topological deffects, or other, very bizarre, constructons out of pure gravity. If they would be succesfull ther would be no need to add matter and the UV fixed point would be kept. That would be very good for them because the wouold have on one hand a way to do perturbative quantum gravity from conventional Einstein gravity and on the other hand a non perturbative formulation which allows them to prove things about diferent interesting questions (pity that they don´t know how to recover conventional gerneral relativity from that non perturbvative formulation).
Well, very recenly I have had news about a theory made by Krasnov kown as "non metric quantum gravity". You can read about it here.
The presentation sounds very promising. Using Plebansky action (a very familar for LQG people alternative formulation of general relativity equivalent to it, under addecquate constraints, in the classical limit). They quantize it in somethnig resembling the well known background field method . This method was introduced in the fifties by de Witt to obtain for the first time a (quasi)covariant quantization of general relativity which gave a more rigurous tratement of the previous work of Feynman who had obtained Feynman graphs (obvious xD) for general relativity. The theory is based in separating the field to be quantized in two parts, one, the background, is not quantized and only the fuctuations about it are quantized. Ideally the nonquantizd part would be a limit in which the quantum fluctuations of the quantized part are irrelevant. It is interesting thatthis formalism later impulsed the technology wich would allow to quantize yang mills theories (in fact later better ways were found to quantize them and in the QFT textbooks usually there is no reference to the background field method). In fact there is nothing terribly special about the background field method, it is justa diferent way to do perturbative quantization. So one could ask how this would render Plebansky theory renormalizable. The asnwer is ¡it doesn´t !. If you read the article you find that contrary to the claims the theory is not renormalizable. But beofre wondering if you would make a demand to the authros about fake publicity continuate reading -if you have arrived untill here a litle more effort doesn´t matter ;)-. The trick is that they claim that they can prove that the theory has an UV fixed point (yeah, again). The advantage that I see of this aproach over the Reuter´s one is that there are not numerical calculus and artificial algoritmic constructions involved. But still the same drawback applies. They are theories of pure gravity and the UV would dissapear if matter is added (well, I guess it would be so, I cant say for sure).
There are aditional papers exploring the properties of the theory. Seemengly it can explain dark matter, quasars anomalous red shifts and such that. Pitty it can´t explain ordinary, vissible, matter.
If I must decide among these thre theories, conformal gravity, reuters approach, and kranosv non metric gravity I clearly prefer conformal gravity. It is a renormalizbel theory, without need to use fixed points, and it allows the introductoin of matter. I have readed in Jackes Distler blog some claism which maybe would be a problem for this theory, but I am not totally sure about it.
And to end I´ll say a few words about the new baby quantum gravity, or I would better say a TOE (theory of everything). It´s author is Garret Lissi. You can find his paper here. Sabine Hossfander makes a good blog post about it here so I invite to read that entry. I just will quote a phrase of Sabbine:
He neither can say anything about the quantization of gravity, renormalizability, nor about the hierarchy problem
Well, the requirements for a TOE seem to have drop a bit since the last time I watched ;).
Of courseI again reiterate that I don´t consider myself nothin remotely similar to an authority in this questions and that I invite to the reader to form his own conclusions.