Sunday, October 07, 2007

Topological geometrodynamics

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
time scales"

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

23 comments:

Matti Pitkanen said...

Hi,

thank you for comments on TGD. I will represent my comments to morrow. Just now I am too exhausted. Have been writing and rewriting for 12 hours!

Cheers,
Matti

Matti Pitkanen said...

Dear Javier,

sorry for not being able to respond more rapidly. You were troubled of various things which you did not understand.

1. How TGD gives standard model.

2. What the notion of "many-sheeted" means.

3. What fractality means.

4. How TGD relates to Wheeler-de-Witt equation.

5. How p-adic numbers emerge.



Consider question 1 first.

a) TGD is not a manner to quantize GRT but completely different theory. In TGD the abstract manifold geometry is replaced with much richer sub-manifold geometry and this leads to the unification of gravitation with other interactions.

b) First of all this brings in the isometry group of compact space S=CP_2 identifiable SU(3), color group of strong interactions.

c) Second point is that submanifold geometry brings in besides induced metric also induced spinor structure of CP_2. The induced gauge potentials are simply projections of components CP_2 spinor connection. It has as a holonomy group SU(2)_LxU(1) having identification as electroweak gauge group. This symmetry is automatically broken. Components of color gauge potentials correspond to projections of SU(3) Killing vector fields much like in Kaluza-Klein approach.

d) These gauge fields are classical and do not yet correspond to elementary particles which actually corresponds to 3-dimensional light-like surfaces, orbits of 2-D partons. The handle number of 2-D partonic surface classifies various fermion families. Thus there is no need to add anything to standard model gauge group.



From the emergence of symmetries of standard model it is a long path to demonstrating that one indeed obtains standard model particle multiplets and definite differences from standard model are predicted. For instance, color is not spin like quantum number but corresponds to CP_2 partial wave in well-defined sense. Super-conformal invariance due to the metric 2-dimensionality of light-like 3-surfaces is also absolutely essential. One very beautiful prediction is that lepton and quark numbers are conserved separately so that there is no problem about decay of proton.

Matti Pitkanen said...

Dear Javier,

You also wondered what the notions of "many-sheeted" and "fractality" might mean.

The origin of the notion is what I call topological field quantization. Gravitational field, and classical electroweak and color fields correspond to induced metric and induced gauge fields in TGD framework. The fundamental field space becomes essentially compact CP_2 whereas for ordinary gauge potentials the field space is non-compact affine space.

This means that the imbedding of, say, constant magnetic field as induced gauge fields cannot be global. What happens that field region splits into topological field quanta, say magnetic flux quanta. such that at boundaries of the quantum the imbedding cannot be continued anymore. The great idea is that all structures we see around us correspond to this kind of miniuniverses with outer boundary defining the boundary of corresponding space-time.


These space-time sheets are basic building blocks of TGD space-time and by glueing them together by using wormhole contacts one can obtain fractal hierarchy of space-time sheets with increasing size and typically weaker gauge fields.

For instance, at level of cosmology this would correspond to a hierarchy of magnetic field starting from cosmic strings for which M^4 projection is 2-D string orbit and CP_2 projection complex 2-surface of CP_2, and continuing to thicker and thicker magnetic flux tubes with 4-D M^4 projection.
Recall that magnetic fields appear in all astrophysical length scales and are one of the mysterious of astrophysics and cosmology.
Topological field quanta come in several varieties. CP_2 type extremals have Euclidian signature of metric and have 1-D light-like random curve as M^4 projection: their small deformations provide a model for elementary particles. 1-D lightlike random curve gives by the way rise to Virasoro conditions.

Later it became clear that partons can be identified as light-like 3-surfaces, typically 3-D light-like throats of wormhole contacts between space-time sheets analogous to blackhole horizons, and possessing by their metric 2-dimensionality extended conformal symmetries, are the proper model for partons and can be seen as fundamental objects in TGD Universe.

You wondered about whether TGD gives Virasoro symmetries: these symmetries are of course prerequisites of Super-Kac-Moody type symmetries and symmetries that I have christened super-canonical. 4-D character of space-time is absolutely essential for extened super-conformal symmetries so that TGD works only for the physical space-time dimension.



There are also "topological light rays", "massless extremals" which represent precisely targeted radiation beam which does not suffer dispersion (signal form is preserved) and propagates with light velocity. These solutions are not possible in Maxwell electrodynamics which is clear also from the fact that they can carry light-like gauge currents.


All this gives naturally rise to a fractal structure in very general sense. Nothing to do with Mandelbrot fractals which are very restricted form of fractal. p-Adic physics gives a quantitative characterization of fractality but I shall discuss it in separate posting.

Matti Pitkanen said...

Dear Javier,

How Wheeler-de-Witt equation relates to TGD was also you question. The answer is that the failure of TGD counterpart of WdW was crucial for ending up with the basic philosophy behind quantum TGD.

WdW emerges in the canonical quantization of general relativity. Same procedure can be tried also in the case of TGD and I of course did it for 27 years ago or so. It turned out that the procedure fails completely due to the extreme nonlinearity of the theory: this occurs for any YM type action except four-volume which is however a non-physical choice.

For Kahler action which is Maxwell action for induced Kahler form of CP_2 and which is the unique physical choice, the situation is especially acute since all 4-surfaces which have 2-D (in general) Lagrangian submanifold of CP_2 as CP_2 projection are vacuum extremals and quantization fails completely around these. In particular, perturbative approach around flat M^4 fails completely for any YM action since the necessary kinetic term vanishes identically for perturbations. In practice this means that time derivatives of CP_2 coordinates cannot be expressed in terms of canonical momentum densities and one cannot derive the counterpart of WdW.

Also path integral quantization around flat M^4 fails for same reasons: now failure comes from the identical vanishing of kinetic terms around vacuum extremals of Kahler action.


This forced completely new geometric approach to quantum TGD based on the notion of "world of classical worlds" (WCW) consisting of lightlike 3-surfaces and possessing metric, Kahler structure, and spinor structure with modes of configuration space spinor fields identified as Fock states and anticommutation relations of fermionic oscillator operators interpreted in terms of anticommutations of complexified gamma matrices of the WCW.

The challenge is to provide this space with Kahler geometry determined by Kahler function which is essentially Kahler action for a preferred 4-D extermal of Kahler action assigned to 3-D lightlike 3-surface. This means that general coordinate invariance assigns to 3-surface a unique space-time and classical field dynamics. Spacetime can be regarded as a generalized Bohr orbit. Classical Bohr orbitology becomes an exact part of quantum theory, not an artifact of stationary phase approximation. Path integral is replaced with mathematically well-defined functional integral. Kahler function is the counterpart for the effective action of quantum field theories and codes for radiative corrections in space-time geometry among other things.

The existence of Riemann connection forces already in the case loop spaces a unique Kahler metric. Same is expected to occur now and is expected also to force imbedding space (M^4xCP_2 by physical constraints) to be highly unique. Essentially a union of infinite dimensional symmetric spaces for which all points (3-surfaces) are metrically equvalent, is forced. This cosmological principle at the level of WCW makes the theory also calculable.

Best,
Matti

Matti Pitkanen said...

Dear Javier,

you asked also about emergence of p-adic numbers in TGD Universe. The story goes as follows.

Around 1990 or so I started to play with the possibility that p-adic numbers might relate in some manner to physics. Somehow I ended up with an observation that the ratios of weak boson mass scale, hadron mass scale, and electron mass scale correspond to the ratios of Mersenne primes M_89, M_107, M_127; here one has M_n=2^n-1, n some prime (square roots of them)

This led to the idea of p-adic thermodynamics in which one replaces Boltzmann weight exp(-E/kT) with a power p^(E/T). p-Adic existence requires that E/T is integer so that very strong constraints on the Hamiltonian result. Harmonic oscillator satisfies the constraints. Energy can be replaced with scaling generator in conformally symmetric theory. Particle mass squared is indeed essentially scaling generator L_0 of conformal symmetries with integer spectrum so that T=1/n is the necessary quantization of p-adic temperature.

A highly predictive theory results. Partition function is completely fixed by conformal symmetry and the p-adic powers series representing expectation value of mass square converges extremely rapidly for physical Mersenne primes which are large. Two lowest terms give practically exact result. Hence the model is testable.


The mysterious ratio of Planck mass to proton mass thus reduces to number theory. Even more, the calculations reproduce elementary particle masses successfully assuming what I call p-adic length scale hypothesis. Physically favored p-adic primes p are near powers of 2: p=about 2^k, k integer and preferredly prime. The interpretation is that evolution which has occurred already at elementary particle scales has selected these p-adic length scales as the fittest ones.

The notion of p-adic scale becomes a competely new element of physics and would involve also p-adic fractality: this is in sharp contrast to the prevailing Planck length scale reductionism. p-Adic length scale hypothesis allows to get rough overall view about physics in all length scales based on the identification of the p-adic length scale characterizing given system.

The prediction is hierarchy of scaled variants of say standard model physics and quantization of Planck constant brings new element to this picture. The empirical support comes from the fact that neutrino mass scales are known to differ widely on different experiments. Also electron's effective mass in condensed matter systems varies as much as by a factor of 100. TGD based model for light hadrons assumes that the p-adic mass scale of quark varies and depends on hadron.

%%%%%%%%%%%%%%%%

This does not yet say anything about origin of p-adic physics and how it is realized at space-time level. After having worked with the problem for more than decade and having constructed TGD inspired theory of consciousness during the same period, I dare claim that p-adic physics is the physics of cognition and intentionality.

What this requires is a generalization of number concept. Reals and p-adic number fields are integrated to a larger structure by "gluing" them along common rationals and possibly also along common algebraic numbers to single super structure. Different number fields would be like pages of book and rationals and common algebraics like the back of this big book. Also algebraic extensions of p-adics must be included.

The notion of manifold must be generalized so that also manifold has this kind of book structure. In particular the 8-D imbedding space has this book structure and space-time surfaces have both real and p-adic space-time sheets meeting along common rationals (algebraics). If p-adic space-time sheet and real spacetime sheet have many common points, real space-time sheet receives effective p-adic topology and this explains the success of p-adic mass calculations.

p-Adic space-time sheets would the "mind stuff" of Descartes and interpreted as correlates of cognition and intentionality. The success of p-adic mass calculations forces to conclude that cognition and intentionality are present already at elementary particle level.

One really fascinating implication is based on the properties of p-adic norm: two p-adic numbers differing by very large power of p are near to each other. Thus p-adic infinitesimals correspond to infinitely large distances in real sense and p-adic space-time sheets have literally infinite size: cognition would be a cosmic phenomenon, our cognitive bodies would have size of cosmos and infinite temporal duration in real sense, in this sense we would be gods. The intersections with real space-time sheets are however always discrete which correlates with the fact that all cognitive representations (say numerical calculations) are always discrete.

Best,
Matti

Javier said...

Ok, thanks for the long explanations.

I guess that with them everybody reading the entry have the bare minimun to read your own blog and try to follow it. At least the most physics/gravity oriented part. Or at least I hope so.

One of the purposes of my blog is to provide some guidance to the diferent approachs to quantum gravity. I know positively that some of my eventual readers are students or graduate physicians not working in this area or people with an strong interest in science and/or scince fiction who don´t accept the supremacy of string theory (and wouldn´t accept it even if I would say the opposite xD).

Be sure, anyway, that there are things that I am far to understand properly, but I will answer them in your blog acording as I go processing the information.

Matti Pitkanen said...

Hi,


I just want to emphasize that TGD is a theory which is developing. Like a many-dimensional crossword puzzle with basic problems related to the interpretation, conceptualization, and attempts to formulate intuitions mathematically. This involves a lot of trial and error and conjectures.

This is really something very different from what theories like QCD in which the problem is just how to calculate efficiently using Feynmann graphs.

Concerning maths. There is a lot of background mathematics involved, but basically at conceptual level.

I try to give a list about the most relevant math concepts. I hasten to emphasize that I can handle this math only at conceptual level and I think always as a physicist: I am not a mathematician producing formulas like machine gun.


1. The basic ideas of submanifold geometry (I learned them from Eisenhart's classic) relying on notion of induced metric, are easily generalized to the induction of spinor structure is basic ingredient needed. Ability to imagine general coordinate invariant actions for surface dynamics and ability to write field equations for them is easy to learn.

2. The notions of symmetric space and Kahler metric are needed. CP_2 is basic example. This involves basic Lie group theory.

3. Some basic facts about topology are needed but the topology involved in applications is very simple: for instance, just 2-dimensional topology which one can directly visualize, is needed in order to understand family replication phenomenon.


3. Second more abstract aspect is the notion of infinite-D geometry and here Dan Freed's thesis on loop spaces was my starting point.
The great philosophical idea is that infinite-D Kahler geometry and thus physics is unique from the mere existence requirement.

4. A late-comer are Von Neumann algebras known has hyperfinite factors of type II_1 automatically related to the spinors of the world of classical worlds. I wish I would understand them in more profound level. The inclusions of these algebras lead to a beautiful formulation of quantum measurement theory with measurement resolution appearing at fundamental level. Non-commutative mathematics and quantum groups appear naturally and Connes tensor product is crucial and might fix S-matrix almost completely.

Von Neumann algebras inspired also the generalization of imbedding space to describe hierarcy of Planck constants: probably brane constructions are somewhat similar but I really do not know. In any case, almost copies of imbedding space are glued together like pages of book.

5. p-Adic mathematics leads to number theory: algebraic extensions of p-adic numbers etc. TGD forces generalization of number concept by gluing reals and various p-adics together along common rationals and perhaps also common algebraics. This gives to a book like structure with various number fields representing the pages of book glued together along common points. This leads at manifold level to the notion of p-adic space-time sheet. This is also a notion that mathematicians should formulate rigorously.

6. Number theory helps also, in particular basic facts about classical number fields. I believe TGD as a generalized number theory program is a very powerful paradigm.

Adosgel said...

¿No habrás por casualidad una traducción al castellano de esta teoría o lo que sea?

No se ingles y me muero de curiosidad.

Saludos Sauron.

Javier said...

Hola Adosegel, bienvenido al blog.

No tengno noticia de que haya nad asobre la TGD en Español, y dudo mucho que pueda haberlo. La única persona que parece trabajar en esa teoria es Pitkanen y el no habla Español.

De todos modos no te agobies. Lo mio con esta teoria es más un caso de curiosidad que otra cosa. Mientras no lea una expoición clara de sus fundamentos, bien desarrollada y todo lo demás, es eso, una curiosidad, pero no la propondria como una teoria seria. Pero bueno, la curisidad no creo que este de más en la ciencia.

Adosgel said...

Pues como sigo sin saber ingles, tampoco sé de que va este otro trabajito que talvez ya conozcas; pero por seacaso

http://www.arxiv.org/PS_cache/arxiv/...711.0770v1.pdf

No sé que te parecerá.

Saludos.

Adosgel said...

Perdón, voy a omitir el http para que quepa

www.arxiv.org/PS_cache/arxiv/...711.0770v1.pdf

Saludos

Adosgel said...

Esta si, que la he probado

http://www.arxiv.org/PS_cache/arxiv/pdf/0711/0711.0770v1.pdf

Adosgel said...

http://www.arxiv.org/PS_cache/arxiv/pdf/0711/0711.0770v1.pdf

A ver esta vez si la he copiado bien.

Mejor que borres los demás intentos, ¡que desastre!

Saludos.

Javier said...

Estoo, adosgel, que lo que me vinculas es el artículo de Garret Lisi sobre su ToE con E8. No tiene nada que ver con la TGD. ¿Seguro que no hay algún error?

Adosgel said...

No pretendía vincularlo, solo saber si lo conocías y tienes alguna opinión formada; pero he ojeado el foro de migui y he podido comprobar que lo estais tratando.
Por mi parte, me tendré que quedar a dos velas en mi burbuja del castellano.

Saludos, te sigo leyendo.

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