While I orginize ideas to post about more conventional ideas in physics (or science in general)I am goint to post today about topological geometrodynamics.
Ther first time I got knowledge of these thoery was as a consecuence of seraching for p-adic numbers in the net. If I don´t remember bad I did that rsearch because I was triying to understand some chapters in that monument to abstract mathemathics writen by Jean Dieudenot which respond to the name "panoramic of pure mathemthics" (bac translation to english of the spanish title, maybe it is not quite exact).
At last I found a very good introduction to the subject of p-adic numbers in a physics report article about p-adic strings so I didn´t go too far with topological geometrodynamics. One reason for that was that the theory depends on mini-superspaces and in that time my knowledge on the Wheleer-de Witt equation was too limited to even do the try.
Well, nowadays the author of TGD, topolgoical geometrodynamics, Matti Pitkanen, is a blogger, you can read him here. I have some idea that somewhere in internet there is a listing about theories of quantum gravity and that TGD is included as a candidate. Well, in all those years a¡I never have found the time to read some of the papers of Pitkanen, at most I had readed some entries of his blog or some comments of him on another blogs. Now, while around 10 hours of studying string theory I decided to do a few breaks to try to get a bried idea about the basics of TGD. For that I followed the links in the Pitkanen blog and went here: http://www.helsinki.fi/~matpitka/powerpoint.html
Concretely I readed the three main pdf´s. Also I readed the brief presentation, kind of guided tour for very quick ideas for the 400 book on the subject presented here: http://www.helsinki.fi/~matpitka/tgdppt/tgdbiogeneralweb.mht!tgdbiogeneralweb_files/frame.htm
Well, be sure that with that background my knowledge of the subject is really poor, to say the best, but anyway I am going to tray to present a few concepts.
What is these TGD? Well, aparently in part it is a way to save the energy problem in general relativity. This problem consists in that there is not a good notion of the energy of an space time. You can give a definition of energy using pseudotensors (magnittudes which are tensors only under more restircted transformations thtat the genral diffeomorphism group), the most famous one bein the Landau one. Another thing that you can do is to define it for certain special space times with special conditions (basically poincaré invariance) at infninity. This is a definition for the global spacetime and not a local one. The most famous of this definitions is the ADM (arnowitz-desser-misner) mass. This problem is closely related to the lack of a hamiltonian formulation for general relativity. I have commented, in spanish, the problem while I introduced LQG. Anyway, the key point is that genralrelativity is a fully constrained system, that is the hamiltonian is all it constraints. In the no LQG canonical gravity the most important of this constraints is the above mentioned Wheler-de witt equation.
Hw does TGD afront the energy problem. Well, it represnt spacetime as a surface embeded in a higer dimensional space, concretely H=M4xCP2. M is the Minkowsky space time and CP2 goes for the comple projective plane (I guess). At this time of the presentation Pitkanen says a few things whose reason I don´t understand. for example it relates this choice to particle physics, or at least quantum numbers, well, be sure that I don´t see areaosn for it in that exposition.
Later he says:
"Simple topological considerations lead to the notion of manysheeted
space-time and general vision about quantum TGD. In particular,
already classical considerations strongly suggests fractality meaning infinite
hierarchy of copies of standard model physics in arbitrarily long length and
I neither understand this. The "manyshheted" part maybe could be related to the fact that in canonical quantum gravity you must do foliation of spacetime. The "fractality", well, I supose it referes to the well known fractals discovered and popularized by Mandelbroit, but, what the hell do they here? I mean, he says infinite copies of the standar model at arbitrary length. Certianly that looks like a fractal (selfsimilarity), but How did we go to an statement about the solution of the energy problem going to an upper dimension spacetime to all of this?
The only thing with I see with a certian logic is the reference to the Wheelers superspace. A superspace is. lossely speaking, a selection of a restricted metrics addecuated for a particular problem in general relativity (mainly for cosmological problems). There is a deduction of the Wheler-de witt eqution as a WKB aproximation to the euclidean path integral aproach to quantum gravity, so this could be understood as semiclassical quantum gravity in an upper dimensional spacetime, But, where the standar modell appears in here?.
Later he speaks about p-adics and p-adics mass and imbeddings. P-adic numbers are a very curious concept. In the construction of real numbers from fractionary ones one uses Cauchy sequences. If one makes the same construction, but instedad of using the usual norm for the Cauchy sequence one uses a diferent one based on prime descompostions (very loosely speaking)by a certain prime number you get p-adic numbers. In fact the prime number p is arbitrary, so you have as many p-adic numbers as prime numbers. There is some completion of the primes by something called adelic numbers (I am studiying now algebraic geometry I try to keep number theory as far of me as I can, so don´t ask me the details, please xD).
Well, once agian, where did p-adic numbers appear?. In p-adic strings it is argued that in real experiments you never obtain as a resoult of a measure a "pure" real number, that is, you always get an enteire or a fractionary number. Maybe Pitkanen tries to argue something similar to this, but once agian in the order of preesentation of his pdf, concretely in here: http://www.helsinki.fi/~matpitka/tgdppt/absTGD.pdf this is the order wollowed, and I guess that the possible readers will agree that it is really hard to disguish a logic in the derivation of the asserted results.
Later he presents a second part of the theory where he claims that he uses three dimenional lighlike hypersurfaces which plays the role of (super)string theries, they reproduce super kac-moody algebras (in fact I would think that more representative of string theory would be virasoro algebras, but well, who knows?). To me this lightlike hypersurface stuff remember me twistor theory and not string theory, but for sure, once again, a 6 pdf paper is a very bad way toget a proper idea of a 400 hundred book . One thing that I must say here is if these hypersurfaces resemble string theory they must have associated states which could be identified with ordinary fields in minkowsky space and It would be here where I would expect to see the appearence of the standard model and ot in the embeding of ordinary spacetime in an upper spacetime.
Well, a lot of question undoubtly. I have made this post like an invitation to pitkanen to discuss a bit his theory if he is interested. I am tired to see his journal with almost no comments to posts where he speaks about scientific facts, and instead seeing entires of very well known physicians talking about topics more appropiate for "peopple magazine" or similars (i.e. absoluttely unrelated to science) full of comments so I try somehow to equilibrate that balance.
UPDATE: Matti Pitkanen has writen an updated introdution to TGD. He has given in a blog entry a brief review of it. You can read it here