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note: because important websites are frequently "here today but gone tomorrow", the following was
archived from http://home.netcom.com/~mthorn/tgd.htm on July 12, 2009. This is NOT an
attempt to divert readers from the aforementioned website. Indeed, the reader should only read
this back-up copy if the updated original cannot be found at the original author's site.
Topological GeometroDynamics
Matti Pitkänen
Postal address:
Köydenpunojankatu 2 D 11
10940, Hanko, Finland
E-mail: [email protected]
URL-address: http://tgdtheory.com
(former address: http://www.helsinki.fi/~matpitka )
"Blog" forum: http://matpitka.blogspot.com/
ISBN 978-0-9551170-8-4
Topological GeometroDynamics is a modification of General Relativity inspired by the conceptual
problems related to the definitions of inertial and gravitational energy in Relativity.
1
Topological GeometroDynamics can be also seen as a generalization of superstring models.
Physical space-times are seen as 4-dimensional surfaces in certain 8-dimensional space. The choice of
this space is fixed by symmetries of the Standard Model of Particle Physics so that geometrization of
known classical fields and elementary particle quantum numbers result.
The notion of "many-sheeted space-time" allows re-interpretation of the structures of the perceived
World in terms of Macroscopic space-time topology. The generalization of the number concept based
on fusion of real numbers and p-adic number fields implies a further generalization of the space-time
concept allowing to identify space-time correlates of cognition and intentionality.
Quantum measurement theory extended to a quantum theory of Consciousness becomes an organic
part of theory. A highly non-trivial prediction is the existence of a fractal hierarchy of copies of
Standard Model particle physics with dark matter identified in terms of Macrocosmic quantum phases
characterized by dynamical and quantized Planck constant.
The book is a comprehensive overview and analysis of Topological GeometroDynamics as a
mathematical and physical theory.
http://freelance-quantum-gravity.blogspot.com/2007/10/topological-geometroheadcache.html
Sunday, October 07, 2007
Topological GeometroDynamics
Publicado por Javier en Sunday, October 07, 2007
Etiquetas: 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 on 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" (back 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 Topolgoical GeometroDynamics (TGD) -- Matti Pitkanen -- is a
blogger. You can read him here. I have some idea that somewhere on the Internet there is a listing
about theories of Quantum Gravity and that TGD is included as a candidate. Well, in all those years 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 to http://www.helsinki.fi/~matpitka/powerpoint.html
(also
mhtml:http://tgd.wippiespace.com/public_html/TGDstartweb.mht!TGDstartweb_files/slide0008.htm).
2
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 page book on the subject presented here:
http://www.helsinki.fi/~matpitka/tgdppt/tgdbiogeneralweb.mht!tgdbiogeneralweb_files/frame.htm (also
http://tgd.wippiespace.com/public_html/TGDbrief.pdf).
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 try to present a few concepts.
What is this 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 a space-time. You
can give a definition of energy using pseudo-tensors (magnitudes which are tensors only under more
restircted transformations thtat the general diffeomorphism group), the most famous one being 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 space-time 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 (loop quantum gravity). Anyway, the
key point is that General Relativity 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 abovementioned Wheeler-deWitt equation.
Hw does TGD afront the energy problem? Well, it represents space-time as a surface embeded in a
higher 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
reasons 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 a reaosn 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 "many-sheeted" part may 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
(self-similarity). 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 Wheeler's superspace. A
superspace is, loosely 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-deWitt
eqution as a WKB aproximation to the Euclidean path integral aproach to quantum gravity. So this
could be understood as semi-Classical quantum gravity in an upper dimensional space-time. But where
the Standard Model appears in here?
3
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 instead of using the usual norm for the Cauchy sequence one uses a
diferent one based on prime decompostions (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
entire 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 . Tis is the order wollowed and I guess that
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 3-dimenional light-like
hypersurfaces which play 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 light-like hypersurface stuff reminds me of twistor theory and not string
theory. But for sure, once again, a 6-page pdf paper is a very bad way toget a proper idea of a 400-page
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 space-time in an upper space-time.
Well, a lot of question undoubtly. I have made this post as 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 "People 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 .
2:40 PM ,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 the Standard Model.
2. What the notion of "many-sheeted" means.
3. What fractality means.
4. How TGD relates to the Wheeler-deWitt equation.
5. How p-adic numbers emerge.
Consider Question #1 first …
4
(a) TGD is not a manner to quantize GR but a completely different theory. In TGD, the abstract
manifold geometry is replaced with much richer sub-manifold geometry. This leads to the
unification of gravitation with other interactions.
(b) First of all, this brings in the isometry group of compact space S=CP2 identifiable SU(3), color
group of strong interactions.
(c) The second point is that submanifold geometry brings in besides induced metric also induced
spinor structure of CP2. The induced gauge potentials are simply projections of components
CP2 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 KaluzaKlein 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 the Standard Model, it is a long path to demonstrating that one
indeed obtains Standard Model particle multiplets. Definite differences from Standard Model are
predicted. For instance, color is not spin like a 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 3surfaces 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.
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 fiel, and
classical electroweak and color fields correspond to induced metric and induced gauge fields in TGD
framework. The fundamental field space becomes essentially compact CP2 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 is 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 mini-universes 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 M4 projection is 2-D string orbit and CP2 projection complex 2-surface of
CP2 and continuing to thicker-and-thicker magnetic flux tubes with 4-D M4 projection.
Recall that magnetic fields appear in all astrophysical length scales and are one of the mysterious of
Astrophysics and Cosmology.
5
Topological field quanta come in several varieties. CP2-type extremals have Euclidian signature of
metric and have 1-D light-like random curve as M4 projection. Their small deformations provide a
model for elementary particles. 1-D light-like 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 the 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 "supercanonical". 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 also clear from the fact
that they can carry light-like gauge currents.
All this gives naturally rise to a fractal structure in a 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.
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 Y-M type action except 4volume which is, however, a non-physical choice.
For Kahler action which is Maxwell action for induced Kahler form of CP2 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 CP2 as CP2 projection are vacuum extremals and quantization fails
completely around these. In particular, perturbative approach around flat M4 fails completely for any YM action since the necessary kinetic term vanishes identically for perturbations. In practice, this means
that time derivatives of CP2 coordinates cannot be expressed in terms of canonical momentum densities
and one cannot derive the counterpart of WdW.
Also, path integral quantization around flat M4 fails for same reasons: now failure comes from the
identical vanishing of kinetic terms around vacuum extremals of Kahler action.
This forced a completely new geometric approach to Quantum TGD based on the notion of "World
of Classical Worlds" (WCW) consisting of light-like 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.
6
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 light-like 3surface. This means that general coordinate invariance assigns to 3-surface a unique space-time and
Classical field dynamics. Space-time 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 (M4xCP2 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 also makes the theory calculable.
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 M89, M107, M127.
Here one has Mn=2n-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 L0 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 2k, 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 padic fractality. Tis 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 a 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. The -GD based model for light
hadrons assumes that the p-adic mass scale of quark varies and depends on hadron.
7
This does not yet say anything about origin of p-adic physics and how it is realized at the space-time
level. After having worked with the problem for more than decade. Having constructed TGD-inspired
theory of Consciousness during the same period, I dare claim that p-adic physics is the physics of
Vognition 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 a
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. 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 already present at the 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 sens. 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.
1:12 PM , 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 scienc-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.
7:19 AM, Matti Pitkanen said...
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&error and conjectures.
8
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 a machine gun.
1. The basic ideas of submanifold geometry (I learned them from Eisenhart's classic) relying on the
notion of induced metric are easily generalized to the induction of spinor structure is basic
ingredient needed. The 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. CP2 is a 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. The 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 II1 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.
9
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