Monday, February 28, 2005

Dark Matter and Living Matter

Dark matter and living matter represent two deep mysteries of the recent world view. There however exists an amazing possibility that there might be close connection between these mysteries.

Do Bohr rules apply to astrophysical systems?

D. Da Rocha and Laurent Nottale have proposed that Schrödinger equation with Planck constant hbar replaced with what might be called gravitational Planck constant hbar_{gr}= GmM/v_0 (hbar=c=1). v_0 is a velocity parameter having the value v_0=about 145 km/s giving v_0/c=4.6\times 10^{-4}. This is rather near to the peak orbital velocity of stars in galactic halos. Also subharmonics and harmonics of $v_0$ seem to appear. The support for the hypothesis coming from empirical data is impressive. The support for the hypothesis coming from empirical data is impressive. It is surprising that findings of this caliber have not received any attention in popular journals while the latest revolutions in M-theory gain all possible publicity: also I heard from the article by accident from Victor Christianto to whom I am deeply grateful.

Is dark matter in astroscopic quantum state?

Nottale and Da Rocha believe that their Schrödinger equation results from a fractal hydrodynamics. Many-sheeted space-time however suggests that astrophysical systems are not only quantum systems at larger space-time sheets but correspond to a gigantic value of gravitational Planck constant. The gravitational (ordinary) Schrödinger equation would provide a solution of the black hole collapse (IR catastrophe) problem encountered at the classical level. The basic objection is that astrophysical systems are extremely classical whereas TGD predicts macrotemporal quantum coherence in the scale of life time of gravitational bound states. The resolution of the problem inspired by TGD inspired theory of living matter is that it is the dark matter at larger space-time sheets which is quantum coherent in the required time scale. I have proposed already earlier the possibility that Planck constant is quantized and the spectrum is given in terms of logarithms of Beraha numbers B_n= 4cos^2(pi/n): the lowest Beraha number B_3 =1 is completely exceptional in that it predicts infinite value of Planck constant. The inverse of the gravitational Planck constant could correspond a gravitational perturbation of this as 1/hbar_{gr}= v_0/GMm. The general philosophy would be that when the quantum system would become non-perturbative, a phase transition increasing the value of hbar occurs to preserve the perturbative character and at the transition n=4 --> 3 only the small perturbative correction to 1/hbar (3)=0 remains. This would apply to QCD and to atoms with Z>137 as well. TGD predicts correctly the value of the parameter v_0 assuming that cosmic strings and their decay remnants are responsible for the dark matter. The harmonics of v_0 can be understood as corresponding to perturbations replacing cosmic strings with their n-branched coverings so that tension becomes n^2-fold: much like the replacement of a closed orbit with an orbit closing only after n turns. 1/n-sub-harmonic would result when a magnetic flux tube split into n disjoint magnetic flux tubes.

Planetary system as a testing ground

The study of inclinations (tilt angles with respect to the Earth's orbital plane) leads to a concrete model for the quantum evolution of the planetary system. Only a stepwise breaking of the rotational symmetry and angular momentum Bohr rules plus Newton's equation (or geodesic equation) are needed, and gravitational Shrödinger equation holds true only inside flux quanta for the dark matter.
• During pre-planetary period dark matter formed a quantum coherent state on the (Z^0) magnetic flux quanta (spherical shells or flux tubes). This made the flux quantum effectively a single rigid body with rotational degrees of freedom corresponding to a sphere or circle (full SO(3) or SO(2) symmetry).
• In the case of spherical shells associated with inner planets the SO(3)--> SO(2) symmetry breaking led to the generation of a flux tube with the inclination determined by m and j and a further symmetry breaking, kind of an astral traffic jam inside the flux tube, generated a planet moving inside flux tube. The semiclassical interpretation of the angular momentum algebra predicts the inclinations of the inner planets. The predicted (real) inclinations are 6 (7) resp. 2.6 (3.4) degrees for Mercury resp. Venus). The predicted (real) inclination of the Earth's spin axis is 24 (23.5) degrees.
• The v_0--> v_0/5 transition necessary to understand the radii of the outer planets can be understood as resulting from the splitting of (Z^0) magnetic flux tube to five flux tubes representing Earth and outer planets except Pluto, whose orbital parameters indeed differ dramatically from those of other planets. The flux tube has a shape of a disk with a hole glued to the Earth's spherical flux shell.
• A remnant of the dark matter is still in a macroscopic quantum state at the flux quanta. It couples to photons as a quantum coherent state but the coupling is extremely small due to the gigantic value of hbar_gr scaling alpha by hbar/hbar_gr: hence the darkness. Note however that it is the entire condensate that couples to electromagnetism with this coupling, individual charged particles couple normally.

Living matter and dark matter

The most interesting predictions from the point of view of living matter are following.
• The dark matter is still there and forms quantum coherent structures of astrophysical size. In particular, the (Z^0) magnetic flux tubes associated with the planetary orbits define this kind of structures. The enormous value of h_{gr} makes the characteristic time scales of these quantum coherent states extremely long and implies macro-temporal quantum coherence in human and even longer time scales.
• The rather amazing coincidences between basic bio-rhythms and the periods associated with the states of orbits in solar system suggest that the frequencies defined by the energy levels of the gravitational Schrödinger equation might entrain with various biological frequencies such as the cyclotron frequencies associated with the magnetic flux tubes. For instance, the period associated with n=1 orbit in the case of Sun is 24 hours within experimental accuracy for v_0: the duration of day in Earth and in a good approximation also in Mars! Second example is the mysterious 5 second time scale associated with the Comorosan effect.
Indeed, the basic assumption of TGD inspired quantum biology is that the "electromagnetic bodies" associated with living systems are the intentional agents would conform with the idea that it is dark matter what makes ordinary matter living by acting as quantum controlling agent. Already now there exist a rather detailed theory about how these electromagnetic (or more generally, field-) bodies use biological body as a motor instrument and sensory receptor. For instance, the basic mechanisms of metabolisms would involve flow of matter between space-time sheets liberating energy quanta defining universal metabolic energy currencies same everywhere in Universe and having nothing to do with the details of living systems. The strange time delays of consciousness observed first by Libet suggests that the size of the field body is at least of the order of Earth size as also the frequency scale of EEG suggests (EEG would be involved with communications with magnetic body and biological body). For more details see the chapter "TGD and Astrophysics". For the notion of electromagnetic body see the relevant chapters of the book Genes, Memes, Qualia, and Semitrance.

How to Put an End to the Suffering Caused by Path Integrals

Path integrals have caused a lot of suffering amongst theoretical physicists. Lubos Motl gives a nice summary about Wick-rotation used quite generally as a trick to give some meaning to these poorly defined objects which have caused so much frustration. The idea of Wick rotation is to define path integrals of quantum field theory in Minkowskian space M^4 by replacing M^4 temporarily by Euclidian space E^4, by calculating the integral here as Euclidian functional integral having more meaning, and returning back to M^4 by analytically continuing in various parameters such as the momenta of particles. The trick has been also applied in the hope of making sense of path integral of General Relativity as well as in string models. I have never liked the trick, not because it is a trick, but just for the fact that this trick is needed at all. Something must be fatally wrong at the fundamental level. To see what this something might be, one must recall what Feynman did for long time ago.

How one ends up to path integral?

The path integral approach was abstracted by Feynman from a unitary time evolution operator by decomposing the time evolution to a product of infinite number of infinitesimally short time evolutions. After this "obvious" generalizations of the formalism lacking a real mathematical justification were made. Despite all the work done it can be safely stated, that the notion of path integral does not mathematically exist. The tricky definition of the functional integral through Wick rotation transforming it to functional (I will drop the attribute Euclidian in the sequel) integral is certainly not enough for a mathematician. I hasten to add that even the functional integrals are deeply problematic since the introduction of local interactions automatically induces infinities, and only in the case of so called renormalizable theories there exist a prescription for getting rid of these infinities.

What are the implicit philosophical ideas behind path integral formulation?

When the best brains have been unable to give a real meaning to the notion of path integral despite a work of about six decades, it is time to ask what might be behind these difficulties and whether it could relate to some cherished philosophical assumptions. a) Feynman's approach starts from Hamiltonian quantization and the notion of time is that of Newtonian mechanics. The representability of the unitary time evolution operator as sum over paths is natural in this context. No absolute time exists in the world of Special Relativity so that there are reasons to get worried. It might not be necessary nor even sensible in the Minkowskian context. c) The sexy idea about the sum of all histories with the classical physics identified as the history corresponding to the stationary phase might be simply wrong. Even Feynman could be sometimes wrong, believe or not! One can quite well consider some other, more sensible, approach to define S-matrix elements. d) Infinities are the basic problem of modern physics and are present for both path- and functional integrals. Local divergences are practically always present always as one tries to make a free theory interacting by introducing local interactions consistent with classical field theory. The basic assumption behind locality is that fundamental particles are pointlike. In string models this assumption is given up and there are indeed reasons to believe that superstrings of various kinds allow perturbation theory free of infinities. The unfortunate fact is that this perturbation series very probably does not converge to anything well-defined and is only an asymptotic series. The now-disappearing hope was that M-theory could resolve this problem by providing a non-perturbative approach to strings. d) In the perturbative approach the functional integrals give rise to Gaussian determinants, which are typically infinite formally. They can be eliminated but are aesthetically very awkward. TOE should be maximally aesthetic! These observations do not lead us very far but give some hints about what might go wrong. Perhaps the entire idea about sum over all possible paths with classical physics resulting via stationary phase approximation is utterly wrong. Perhaps the idea about space-time-local interactions is wrong and perhaps higher-dimensional fundamental objects might allow to get over the problems.

Neither Hamiltonian formalism nor path integral works in TGD

When I started to develop mathematical theory around the basic idea that space-times can be regarded as 4-dimensional surfaces in H=M^4xCP_2, I soon learned that perturbative approach fails completely. Indeed, it would be natural to construct a perturbation theory around canonically imbedded M^4 but for the only reasonable candidate for the action, Kähler action, the functional power series vanishes in the third order at M^4 so that the kinetic terms defining propagators vanish identically. For the same reason also Hamiltonian formalism fails completely. This is the case much more generally, and the enormous vacuum degeneracy (any 4-surface for which CP_2 projection belongs to at most 2-D Lagrange manifold is a non-deterministic vacuum extremal) kills all hopes about conventional quantization.

Geometrization of quantum physics as a solution to the problems

This puzzling state of affairs led to the idea that if quantization is not possible one should not quantize! This idea grew gradually to the vision that quantum states correspond to the modes of completely classical spinor fields of an infinite-dimensional configuration space CH of 3-surfaces, the world of classical worlds. This allows also the geometrization of fermionic statistics and super-conformal symmetries in terms of gamma matrices associated with the Kähler metric. The breakthrough came from the realization that general coordinate invariance in 4-dimensional sense is the key requirement. The obvious problem is that you have only 3-dimensional basic objects but you want 4-dimensional Diff invariance. Obviously the very definition of the configuration space geometry should assign to a given 3-surface X^3 a unique four-surface X^4(X^3) for 4-D general coordinate transformations to act on it. What would be the physical interpretation of this? X^4(X^3) defines the classical physics associated with X^3. Actually something more: X^4(X^3) is an analog of Bohr orbit since it is unique so that one can expect a quantization of various classical observables. Classical physics in the sense of Bohr orbitology would become part of quantum physics and of configuration space geometry. This is certainly something totally new and would mean a partial return from the days of Feynman to the good old days of Bohr when everything was still understandable and concrete. There are also other implications. Oscillator operators are the essence of quantum theory and can be geometrized only if configuration space has Kähler metric defined by so called Kähler function. Since classical physics should be coded by this Kähler function, it should be defined by a preferred extremal X^4(X^3) of some physically meaningful action principle. The so called Kähler action, which is formally Maxwell action for CP_2 Kähler form induced to space-time surface, is the unique candidate. The first guess is that X^4(X^3) could be identified as an absolute minimum of Kähler action. This is however a little bit questionable option since there is no lower bound for the value of Kahler action and if it gets negative and infinite, vacuum functional defined as the exponent of Kahler function vanishes. Indeed, it took 15 years to learn that this need not be the quite correct definition. A candidate for a more realistic identification came from a proposal for a general solution of field equations in terms of so called Kähler calibration. The magnitude of Kähler action would be minimized separately in regions where Lagrangian density L_K has a definite sign. This means that X^4(X^3) is as near as possible to a vacuum extremal. The Universe is maximally lazy energy saver! By minimizing energy of solution it might be possible to fix the time derivatives of the imbedding space coordinates at X^3 in order to find the X^4(X^3) by solving the partial differential equations as initial value problem at X^3. A considerable reduction of computational labor. This is of extreme importance, and even more so because Kähler action does not define a fully deterministic variational principle. There are indeed hopes of understanding the theory even numerically!

Generalized Feynman diagrams as computations/analytic continuations

A generalization of the notion of Feynman diagram emerging in TGD framework replaces sum over classical paths with what might be regarded as computation or analytic continuation. The first observation is that the path integral over all space-time surfaces with a fixed collection of 3-surfaces as a boundary does not make sense in TGD framework. Sum reduces to a single 3-surface X^4(X3) since classical physics in the sense of Bohr's orbitology is a quintessential part of configuration space geometry and quantum theory. Classical world is not anymore identified as a path with a stationary phase. This suggests completely different approach to the notion of Feynman diagram. It however took quite a long time before I realized how to formulate this approach more precisely. The idea came when I constructed a TGD inspired model for topological quantum computation. In topological quantum computation braids are the basic structures and quantum computation coded into the knotting and linking of the threads of the braid. This leads to a view that generalized Feynman diagrams do not represent sum over all classical paths but represent something analogous to computations with vertices representing some fundamental algebraic operations. A given computation can be carried out in very many equivalent manners and there always exists a minimal computation. In the language of generalized Feynman diagrams this would mean that diagrams with loops are always equivalent with tree diagrams. The summation over loops would be obviously multiple counting in this framework. This would be nothing but a far reaching generalization of the duality symmetry, which originally lead to string models. I have formulated this generalization in terms of Hopf (ribbon-) algebras here and in a different manner here. That there are several equivalent diagrams would conform with the non-determinism of Kähler action implying several equivalent space-time surfaces having given 3-surfaces as boundaries. This of course correlates directly with the fact that the functional integral and canonical quantization fail completely. The generalized Feynman diagrams could be also interpreted as space-time counterparts for different analytic continuations of configuration space spinor fields (classical spinor fields in the world of classical worlds) from a sector of configuration space with a given 3-topology to another sector with different topology (initial and final states of particle reaction in the language of elementary particle physicist). This continuation can be performed in very many manners but the final result is same always, just as in case of equivalent computations.

Getting rid of standard divergences

It is possible to get rid of path integrals in TGD framework but not from the functional integral over the infinite-dimensional world of classical worlds. This integration means performing an average over these well-defined generalized Feynman diagrams, one might say over predictions of finite quantum field theories. This functional integral in question could bring back the basic difficulties but it does not. a) The vacuum functional over quantum fluctuating degrees of freedom defining the functional integral is completely analogous to a thermal partition function defined as an exponent of Hamiltonian in thermodynamics at a critical temperature. Kähler coupling strength is analogous to critical temperature, which means that the values of the only free parameter of the theory are predicted as they should in any respectable TOE. The good news is that Kähler function is a non-local functional of 3-surface X^3. Hence the local divergences unavoidable in any local QFT are absent. If one would try to integrate over all X^4, one would have Kähler action and locality and all the problems of standard approach would be magnified since the action is extremely non-linear. b) Vacuum functional is the exponent of Kähler function and in the perturbation theory configuration space contravariant metric becomes propagator. The Gaussian determinant is the inverse of the metric determinant and these two ill-defined determinants neatly cancel each other so that also aesthetic is perfect! Note that the coefficient of the exponent of Kähler function is also fixed. A further good news is that there are hopes that the functional integral might be carried out exactly by performing perturbation theory around the maxima of Kähler function. These hopes are stimulated by the fact that the world of classical worlds is a union of symmetric spaces and for a symmetric space all points are metrically equivalent. In the finite-dimensional case there are a lot of examples about the occurrence of this phenomenon. The conclusion is that the standard divergences are not present and that this result is basically due to a new philosophy rather than some delicate cancellation mechanism.

Is there something that could still go wrong? Yes. The existence of configuration space metric requires that it is a union over infinite dimensional symmetric spaces labelled by zero modes whose contribution to CH line element vanishes. An infinite union is indeed in question: if CH would reduce to single symmetric space, a 3-surface with size of galaxy would be equivalent with a 3-surface associated with electron. The zero modes characterize classical degrees of freedom: shape, size, and the induced Kähler form defining a classical Maxwell field on X^4(X^3). In zero modes there is no proper definition of the functional integral. Here comes however quantum measurement theory in rescue. Zero modes are non-quantum fluctuating degrees of freedom and thus behave like genuine classical macroscopic degrees of freedom. Therefore a localization in these degrees of freedom is expected to occur in each quantum jump as a counterpart of quantum measurement. These degrees of freedom should be also correlated in one-one manner with quantum fluctuating degrees of freedom like the pointer of measurement apparatus with the direction of electron spin. A kind of duality between quantum fluctuating degrees of freedom and zero modes is required. We would experience the macroworld as completely classical because each moment of consciousness identifiable as quantum jump makes it classical. It is made again non-classical during the unitary U process stage of the next quantum jump. Dispersion in zero modes, localization in zero modes, dispersion in zero modes,.... Like Djinn getting out of the bottle and representing a very long list of classical wishes of which just one is realized. With this complete localization or localization to a discrete union of points in zero mode degrees of freedom, S-matrix elements become well defined. Note however that the most general option would be a localization into finite-dimensional symplectic subspaces of zero modes in each quantum jump. The reason is that zero modes allow a symplectic structure and thus all possible finite-dimensional integrals are well defined using the exterior powers of symplectic form as integration measure.

Saturday, February 26, 2005

Numb3rs

In to-day's Not-Even-Wrong I learned that M-theory God Ed Witten has followed his brother's Matt Witten's example and is now a TV writer(;-)! The TV series Numb3rs is very ambitious project. The basic goal is to fight against the anti-intellectualism, which has got wings during Bush's era and attack the stereotype about mathematician as a kind of super book-keeper and super-calculator with zero real life intelligence. A TV series in which mathematician's solve crimes is an ingenious choice since detectives must be real-life-intelligent even if they are mathematicians. The team coworks with real mathematicians in Caltech (as you see they look very hippie like) since the goal is to be as autenthic as possible. Even the formulas on blackboard must be sensible so that even mathematician can enjoy the series without fear of sudden strong visceral reactions. Ed Witten got a manuscript to read and proposed an episode in which a rogue mathematician proves Riemann Hypothesis to destroy internet security. My interest is keen since I have proposed a proof, or more cautiously A Strategy for a Proof of Riemann Hypothesis, which has been published in Acta Math. Univ. Comeniae, vol. 72. . I have proposed also a TGD inspired conjecture about the zeros of Zeta. The postulate is that real number based physics of matter and various p-adic physics (one for each prime p) describing correlates of cognition are obtained by algebraic continuation from rational number based physics. This translates to the mathematics the idea that cognitive representations are mimicries of reality and cognitive representation and reality meet each other in a finite number of rational points. This is just what happens in the numerical modelling of the real world since we can represent only rationals using even the best computers. This vision leads to concrete conjectures about the number theoretical anatomy of the zeros of Riemann Zeta which appear in a fundamental role in quantum TGD. The conformal weights of the so called super-canonical algebra creating physical states are suitable combinations of zeros of Zeta. The conjecture is following: for any zero z=1/2+iy of Zeta at critical line the numbers p^(iy) are algebraic numbers for every prime p. Therefore any number q^(iy) is an algebraic number for any rational number q. This assumption guarantees that the expansion of Zeta makes sense also in various p-adic senses for z=n+1/2+iy. A related conjecture is that ratios of logarithms of rationals are rationals: this hypothesis could in principle be tested numerically by looking whether ratios of this kind have periodic expansions in powers of any chosen integer n>1. I would be happy if I had even a slight gut feeling about how the "Strategy for a Proof of Riemann Hypothesis" might relate to Internet safety. Here I meet the boundaries of my narrow mathematical education. So, at this moment it seems that I will not be a notable risk for Internet safety. A word warning is however in order: TGD will certainly become a safety risk for M-theory: sooner or later;-)! Matti Pitkanen

Mersenne Primes and Censorship

Lubos Motl commented at his blog site about the largest known Mersenne prime , which is 2^24,036,583 -1. This inspired me to write a comment copied below (I have added a couple of links and added some detail).

....Not only Mersennes ....

Mersenne primes are in Topological Geometrodynamics framework the most interesting primes since they correspond to most important p-adic length scales. Only Mersennes up to M_127 =2^127-1 are interesting physically since next Mersenne corresponds to a completely super astrophysical length scale. M_127 corresponds to electron whereas M_107 corresponds to the hadronic length scale (QCD length scale). M_89 corresponds to intermediate boson length scale. There is an interesting number theoretic conjecture due to Hilbert that iterated Mersennes M_{n+1}= M_{M_n} form an infinite sequence of primes: 2,3,7,127,;M-_{127},.... etc. Quantum computers would be needed to kill the conjecture. Physically the higher levels of this hierarchy could be also very interesting.

...but also Gaussian Mersennes are important in TGD Universe

Also Gaussian primes associated with complex integers are important in TGD framework. Gaussian Mersennes defined by the formula (1\pm i)^n-1 exist also and correspond to powers p=about 2^k, k prime. k=113 corresponds to the p-adic length scale of muon and atomic nucleus in TGD framework. Neutrinos could correspond to several Gaussian Mersennes populating the biologically important length scales in the range 10 nanometers 5 micrometers. k=151,k=157,k=163, k=167 all correspond to Gaussian Mersennes. There is evidence that neutrinos can appear with masses corresponding to several mass scales. These mass scales do not however correspond to these mass scales but to scale k=13^2=169 about 5 micrometers and k=173. The interpretation is that condensed matter neutrinos are confined by long range classical Z^0 force predicted by TGD inside condensed matter structures at space-time sheets k=151,...,167 and those coming from say Sun are at larger space-time sheets such as k=169 and k=173. p-Adic mass calculations are briefly explained here and here, where also links to the relevant chapters of p-Adic numbers and TGD can be found. That Gaussian Mersennes populate the biologically most interesting length scale range is probably not an accident. The hierarchical multiple coiling structure of DNA could directly correspond to these Gaussian Mersennes. The ideas about the role of Gaussian Mersennes in biology are discussed briefly here can be found. For more details see the chapter Biological realization of self hierarchy of "TGD Inspired Theory of Consciousness...".

Friday, February 25, 2005

What UFOs could teach about fundamental physics?

Here is my comment about UFOs to Not-Even-Wrong discussion group, where it was mentioned that Michio Kaku, string theorists and author of "Hyper-Space", has expressed publicly as his opinion that UFOs might be a real. Leaving aside ontological considerations and the question what one should think about people taking seriously UFOs, one could take UFOs as an inspiration for a thought experiment. Suppose for a moment that UFOs represent a real technology. According to the reports, UFOs seem to have a very small inertial mass (butterfly like motions involving sudden accelerations and changes of direction of motion without producing any shock waves). A technology able to reduce dramatically inertial mass of a material object would thus exist. What could this tell about fundamental physics? A possible answer would be a modification of Equivalence Principle. Gravitational mass would be absolute value of inertial mass, which can have both signs. One of the most obvious implications is an explanation for why gravitational energy is definitely not conserved in cosmological scales whereas there is no evidence for the non-conservation of inertial energy. The simplest cosmology would be that created from inertial vacuum by energetic vacuum polarizations creating regions of positive and negative density of inertial mass. The 4-D universe replacing itself by a new one quantum jump by quantum jump would become possible and the difficult philosophical problems formulated as questions like "What was the initial state of the Universe and what were the initial values/densities of conserved quantities at the moment of big bang" would disappear. The observations motivating the anthropic principle would find a natural explanation: the universe has gradually quantum engineered itself so that the values of these constants are what they are. Technological implications would be also interesting. Forming an tightly bound state of systems with positive and negative inertial mass a large feather light system could be created. Could UFOs utilize this kind of technology? Accepting negative energies, one cannot avoid the questions whether negative energy signals propagate backwards in (geometric) time and whether phase conjugate light discovered at seventies could be identified as signals of this kind. Positive answer would have quite interesting technological implications. Negative energy signals time reflected as positive energy signals from time mirrors (lasers with population reversal for instance) would allow communications with geometric past. Our memory might be based on this mechanism: to recall memories would be to scan the brain of geometric past by using reflected in time direction (rather than in spatial direction as seeing in the ordinary sense). Communications with the civilizations of the geometric future and past might become possible by a similar mechanism. Matti Pitkanen

Saturday, February 19, 2005

Color confinement and conformal field theory

The discovery of number theoretical compactification has meant a dramatic increase in the understanding of quantum TGD. There are two manners to undestand the theory.
• Number theoretic view: Space-time surfaces can be regarded as hyper-quaternionic 4-surfaces in 8-dimensional hyper-octonionic space HO.
• Physics view: Space-time surfaces can be seen as 4-surfaces in 8-D space M^4xCP_2 minimizing the so called Kähler action which essentially means that they minimize their non-commutativity measured by Lagrangian density of Kähler action.
These views seem to be complementary, and at this moment the very existence of this duality (conjecture of course) is what has the strongest implications. A lot remains to do in order to see whether the conjecture is indeed correct and what it really implies. At this moment I am trying to find whether this duality, very reminiscent of various M-theory dualities, is internally consistent. One of the possible implications is the possibility to interpret TGD also as a kind of string theory, not in the usual sense of the world, but as a generalization of so called topological quantum field theories, where the notion of braids is central. Whether this duality is completely general or holds true only for selected space-time surfaces, such as space-time sheets corresponding to maxima of Kähler function (most probable space-times) or space-time sheets representing asymptotic behavior, is an open question. I have explained the duality in earlier posts and do not go to the details here. Suffice it so say that so called Wess-Zumino-Witten action for group G_2, a group which as a Lie group is completely exceptional, and acts as the automorphism group of octonions, suggests itself as a characterizer of the dynamics of these strings. G_2 has group SU(3) as maximal subgroup and can be said to leave these strings invariant. The interpretation is as the color group and G_2/SU(3) coset theory is the natural guess for the dynamics. SU(3) takes indeed the role of color gauge group. The so called primary fields of the theory correspond to two color singlets, triplet and antitriplet and the natural guess is that they relate to leptons and quarks. Indeed in the H picture the basic fields are lepton and quark fields and all other particles are constructed from leptonic and quark like excitations. The beauty of this approach is that QCD might be replaced with an exactly solvable conformal field theory allowing also to deduce how correlation functions change in hyper-octonion analytic transformations affecting space-time surface. There are however also objections against this picture. a) The basic objection is that G_2 Kac-Moody algebra contains triplet and anti-triplet generators and triplet generators commute to anti-triplet. It is hard to imagine any sensible physical interpretation for these lepto-quark generators, whose commutation relations break the conservation of lepton and quark number. The point is however that triplet generators affect e_1, and thus S^6 coordinates and also the SU(3) subgroup acting as isotropy group changes. Thus correlation functions involving these currents are not physically meaningful. Indeed, in G/H coset theory only the H Kac-Moody currents appear naturally in correlation functions since the construction involves functional integral only over H connections. b) If 14-dimensional adjoint representation of G_2 would appear as primary field, also 3 and \overline{3} lepto-quark like states for which baryon and lepton number are not conserved would appear in the spectrum. This is in conflict with H picture. The choice k=1 for Kac-Moody central charge provides however a unique manner to circumvent this difficulty. Integrability condition for the highest weight representations allows for a given value of k only the highest weights \lambda_R satisfying Tr(\phi \lambda_R)\leq k, where \phi is the highest root for Lie-algebra. Since the highest root has length squared 2, adjoint representation is not possible as a highest weight representation for k=1 WZW model, and the primary fields of G_2 model are singlet and 7-plet corresponding to the hyper-octonionic spinor field and defining in an obvious manner the primary fields 1+3+\overline{3} of G_2/SU(3) coset model. Fusion rules for 1\oplus 7 correspond to octonionic multiplication. The absence of G_2 gluons saves from lepto-quark like bosons, and the absence of SU(3) gluons can be interpreted as HO counterpart for the fact that all particles, in particular gluons, can be regarded bound states of fermions and anti-fermions in TGD Universe. This picture conforms also with the claims that 3+\overline{3} part of G_2 algebra does not allow vertex operator construction whereas SU(3) allows the construction in terms of two free bosonic fields. These fields would naturally correspond to the two X^4 directions transversal to the string orbit defined by 1 and e_1. One could say that strings in X^4 are able to represent color Kac-Moody algebra and that SU(3)is inherent to 4-dimensional space-time. c) The third objection is that conformal field theory correlation functions obeying simple scaling laws are not consistent with the exponentially decreasing correlation functions suggested by color confinement. A resolution of the paradox could be based on the role of classical gravitation. At light-like causal determinants the time-like component g_{tt} of the induced metric vanishes meaning that classical gravitational field is very strong. Hence also the normal component g_{nn} of the induced metric is expected to become very large so that hadron would look like the interior of black hole. A finite X^4 proper time for reaching the outer boundary of the hadronic surface can correspond to a very long M^4 time and the finite M^4 distance from the boundary can mean very long distance along hadronic space-time surface. Hence quarks and gluons can behave as almost free particles when viewed from hadronic space-time sheet but look confined when seen from imbedding space. If the hyper-quaternionic coordinates appearing in the correlation functions correspond to internal coordinate of the space-time surface, the correlation functions when expressed in terms of M^4 coordinates can look confining. For more details see the chapter TGD as a Generalized Number Theory: Quaternions, Octonions, and their Hyper Counterparts. Matti Pitkanen

Friday, February 18, 2005

Approaching the end of an epoch

The following quotes from the page Suppression, Censorship and Dogmatism in Science" serve as a good introduction to the recent sad situation in science. Do not miss the chronological ordering. "Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense." Buddha (563BC-483BC). "There must be no barriers to freedom of inquiry. There is no place for dogma in science. The scientist is free, and must be free to ask any question, to doubt any assertion, to seek for any evidence, to correct any errors." J. Robert Oppenheimer, quoted in Life, October 10, 1949. "The more important fundamental laws and facts of physical science have all been discovered, and these are now so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote.... Our future discoveries must be looked for in the sixth place of decimals." Albert Abraham Michelson, speaking at the University of Chicago, 1894. "The great era of scientific discovery is over.... Further research may yield no more great revelations of revolutions, but only incremental, diminishing returns." Science journalist John Horgan, in The End of Science (1997).

Suppression in science

Anyone who has devoted life to some new idea has experienced the arrogance and cruelty of those who are in power. This applies also to me. I have used 26 years of my life to a revolutionary idea and I have summarized the resulting world view in 4 books making about 5000 pages full of original ideas developed in highly detailed manner. Both in quality and quantity this output is exponentially higher than that of an average professor. One might think that with this background it would not be difficult to find a financiation for my research which is still continuing. The reality however loves paradoxes. There is absolutely no hope of getting any support in my home country for my work, and I do not believe that situation might be better elsewhere. Even worse. Average theoretical physicist colleague refuses to read anything that I have written, and there is absolutely no manner to communicate these ideas to a mainstream career builder living in a typical academic environment. I am doomed to be a crackpot. Although American Mathematical Society lists Topological Geometrodynamics in Mathematical Subject Classification Tables, something which can be regarded as a rare honor for a physicist, I am a crackpot. It is easier to change water into wine than change the conviction of an average research person receiving a monthly salary in University about my crackpot-ness. I am not of course alone. The suppression in science has become rule rather than a rare exception, and even Nobelists like Brian Josephson, are punished with censorship for the courage of talking aloud about phenomena not understood within the confines of existing dogmas recent. The victims of this suppression cannot publish anything and even e-print archives such as arXiv.org are closed for those who think in a new manner. These people have started to organize. For instance, at the web page Archivefreedom.org scientific dissidents tell their personal horror stories.

Infinite primes and physics as a generalized number theory

Layman might think that something which is infinite is also something utterly non-physical. The notion of infinity is however much more delicate and it depends on topology whether things look infinite or not. Indeed, infinite primes have become besides p-adicization and the representation of space-time surface as a hyper-quaternionic sub-manifold of hyper-octonionic space, basic pillars of the vision about TGD as a generalized number theory.

1. Two views about the role of infinite primes and physics in TGD Universe

Two different views about how infinite primes, integers, and rationals might be relevant in TGD Universe have emerged. a) The first view is based on the idea that infinite primes characterize quantum states of the entire Universe. 8-D hyper-octonions make this correspondence very concrete since 8-D hyper-octonions have interpretation as 8-momenta. By quantum-classical correspondence also the decomposition of space-time surfaces to p-adic space-time sheets should be coded by infinite hyper-octonionic primes. Infinite primes could even have a representation as hyper-quaternionic 4-surfaces of 8-D hyper-octonionic imbedding space. b) The second view is based on the idea that infinitely structured space-time points define space-time correlates of mathematical cognition. The mathematical analog of Brahman=Atman identity would however suggest that both views deserve to be taken seriously.

2. Infinite primes and infinite hierarchy of second quantizations

The discovery of infinite primes suggested strongly by the possibility to reduce physics to number theory. The construction of infinite primes can be regarded as a repeated second quantization of a super-symmetric arithmetic quantum field theory. Later it became clear that the process generalizes so that it applies in the case of quaternionic and octonionic primes and their hyper counterparts. This hierarchy of second quantizations means an enormous generalization of physics to what might be regarded a physical counterpart for a hierarchy of abstractions about abstractions about.... The ordinary second quantized quantum physics corresponds only to the lowest level infinite primes. This hierarchy can be identified with the corresponding hierarchy of space-time sheets of the many-sheeted space-time. One can even try to understand the quantum numbers of physical particles in terms of infinite primes. In particular, the hyper-quaternionic primes correspond four-momenta and mass squared is prime valued for them. The properties of 8-D hyper-octonionic primes motivate the attempt to identify the quantum numbers associated with CP_2 degrees of freedom in terms of these primes. The representations of color group SU(3) are indeed labelled by two integers and the states inside given representation by color hyper-charge and iso-spin.

3. Infinite primes as a bridge between quantum and classical

An important stimulus came from the observation stimulated by algebraic number theory. Infinite primes can be mapped to polynomial primes and this observation allows to identify completely generally the spectrum of infinite primes whereas hitherto it was possible to construct explicitly only what might be called generating infinite primes. This in turn led to the idea that it might be possible represent infinite primes (integers) geometrically as surfaces defined by the polynomials associated with infinite primes (integers). Obviously, infinite primes would serve as a bridge between Fock-space descriptions and geometric descriptions of physics: quantum and classical. Geometric objects could be seen as concrete representations of infinite numbers providing amplification of infinitesimals to macroscopic deformations of space-time surface. We see the infinitesimals as concrete geometric shapes!

4. Various equivalent characterizations of space-times as surfaces

One can imagine several number-theoretic characterizations of the space-time surface.
• The approach based on octonions and quaternions suggests that space-time surfaces might correspond to associative or hyper-quaternionic surfaces of hyper-octonionic imbedding space.
• Space-time surfaces could be seen as an absolute minima of the Kähler action. The great challenge is to rigorously prove that this characterization is equivalent with the others.

5. The representation of infinite primes as 4-surfaces

The difficulties caused by the Euclidian metric signature of the number theoretical norm forced to give up the idea that space-time surfaces could be regarded as quaternionic sub-manifolds of octonionic space, and to introduce complexified octonions and quaternions resulting by extending quaternionic and octonionic algebra by adding imaginary units multiplied with \sqrt{-1}. This spoils the number field property but the notion of prime is not lost. The sub-space of hyper-quaternions resp.-octonions is obtained from the algebra of ordinary quaternions and octonions by multiplying the imaginary part with \sqrt{-1}. The transition is the number theoretical counterpart for the transition from Riemannian to pseudo-Riemannian geometry performed already in Special Relativity. The notions of hyper-quaternionic and octonionic manifold make sense but it is implausible that H=M^4xCP_2 could be endowed with a hyper-octonionic manifold structure. Indeed, space-time surfaces are assumed to be hyper-quaternionic or co-hyper-quaternionic 4-surfaces of 8-dimensional Minkowski space M^8 identifiable as the hyper-octonionic space HO. Since the hyper-quaternionic sub-spaces of HO with a fixed complex structure are labelled by CP_2, each (co)-hyper-quaternionic four-surface of HO defines a 4-surface of M^4xCP_2. One can say that the number-theoretic analog of spontaneous compactification occurs. Any hyper-octonion analytic function HO--> HO defines a function g: HO--> SU(3) acting as the group of octonion automorphisms leaving a selected imaginary unit invariant, and g in turn defines a foliation of OH and H=M^4xCP_2 by space-time surfaces. The selection can be local which means that G_2 appears as a local gauge group. Since the notion of prime makes sense for the complexified octonions, it makes sense also for the hyper-octonions. It is possible to assign to infinite prime of this kind a hyper-octonion analytic polynomial P: HO--> HO and hence also a foliation of HO and H=M^4xCP_2 by 4-surfaces. Therefore space-time surface could be seen as a geometric counterpart of a Fock state. The assignment is not unique but determined only up to an element of the local octonionic automorphism group G_2 acting in HO and fixing the local choices of the preferred imaginary unit of the hyper-octonionic tangent plane. In fact, a map HO--> S^6 characterizes the choice since SO(6) acts effectively as a local gauge group. The construction generalizes to all levels of the hierarchy of infinite primes and produces also representations for integers and rationals associated with hyper-octonionic numbers as space-time surfaces. A close relationship with algebraic geometry results and the polynomials define a natural hierarchical structure in the space of 3-surfaces. By the effective 2-dimensionality naturally associated with infinite primes represented by real polynomials 4-surfaces are determined by data given at partonic 2-surfaces defined by the intersections of 3-D and 7-D light-like causal determinants. In particular, the notions of genus and degree serve as classifiers of the algebraic geometry of the 4-surfaces. The great dream is of course to prove that this construction yields the solutions to the absolute minimization of Kähler action.

6. Generalization of ordinary number fields: infinite primes and cognition

The introduction of infinite primes, integers, and rationals leads also to a generalization of real numbers since an infinite algebra of real units defined by finite ratios of infinite rationals multiplied by ordinary rationals which are their inverses becomes possible. These units are not units in the p-adic sense and have a finite p-adic norm which can be differ from one. This construction generalizes also to the case of hyper-quaternions and -octonions although non-commutativity, and in the case of octonions also non-associativity, pose technical problems. Obviously this approach differs from the standard introduction of infinitesimals in the sense that sum is replaced by multiplication meaning that the set of real units becomes infinitely degenerate. Infinite primes form an infinite hierarchy so that the points of space-time and imbedding space can be seen as infinitely structured and able to represent all imaginable algebraic structures. Certainly counter-intuitively, single space-time point is even capable of representing the quantum state of the entire physical Universe in its structure. For instance, in the real sense surfaces in the space of units correspond to the same real number 1, and single point, which is structure-less in the real sense could represent arbitrarily high-dimensional spaces as unions of real units. For real physics this structure is completely invisible and is relevant only for the physics of cognition. One can say that Universe is an algebraic hologram, and there is an obvious connection both with Brahman=Atman identity of Eastern philosophies and Leibniz's notion of monad. For more details see the chapter TGD as a Generalized Number Theory III: Infinite Primes. Matti Pitkanen

Tuesday, February 15, 2005

Comment to Not-Even-Wrong

The discovery that strings in a fixed flat background could describe gravitation without any need to make the background dynamical was really momentous. The discovery should have raised an obvious question: How to generalize the theory to the physical 4-dimensional case by replacing string orbits with 4-surfaces? Instead, the extremely silly idea of making also imbedding space dynamical emerged and brought back and magnified all the problems of general relativity, which one had hoped to get rid of. I have tried for more than two decades to communicate simple core ideas about an alternative approach but have found that theoretical physicists are too arrogant to listen to those without name or position. a) The fusion of special relativity with general relativity is achieved by assuming that space-times are 4-surfaces in M^4xCP_2. The known quantum numbers pop out elegantly from this framework. The topological complexity of space-time surfacse allows to circumvent objection that the induced metrics are too restricted. Light-like 3-D causal determinants allow generalization of super-conformal invaraince by their metric 2-dimensionality and dimension 4 for space-time is the only possibility. b) The maximal symmetries of H=M^4xCP_2 have an excellent justification when quantum theory is geometrized by identifying physical states of the Universe as classical configuration space spinor fields, configuration space being defined as the space of 3-surfaces in H. The only hope of geometrizing this infinite-dimensional space is as union of infinite-dimensional symmetric spaces labelled by zero modes having interpretation as non-quantum fluctuating classical degrees of freedom. Infinite-dimensional variant of Cartan's problem of classifying symmetric spaces emerges as the challenge of finding TOE. Mathematical existence fixes physical existence. Just as in the case of loop space, and with even better reasons, one expects that there are very few choices of H allowing internally consistent Kaehler geometry. Fermion numbers and super-conformal symmetries find an elegant geometrization and generalization in terms of complexified gamma matrices representing super-symmetry generators. c) M^4xCP_2 follows also from purely number theoretical considerations as has now become clear. The theory can be formulated in two equivalent manners. *4-surfaces can be regarded as hyper-quaternionic 4-surfaces in M^8 possessing what I call hyper-octonionic tangent space structure (octonionic imaginary units are multiplied by commutative sqrt(-1) to make number theoretical norm Minkowskian). *Space-times can be regarded also as 4-surfaces in M^4xCP_2 identified as extrema of so called Kaehler action in M^4xCP_2. Spontaneous compactification has thus purely number theoretical analog but has nothing to do with dynamics. The surprise was that under some additional conditions (essentially hyper-octonion real-analyticity for the dynamical variables in M^8 picture) the theory can be coded by WZW action for two-dimensional string like 2-surfaces in M^8. These strings not super-strings but generalizations of braid/ribbon diagrams allowing n-vertices in which string orbits are glued together at their ends like pages of book. Vertices can be formulated in terms of octonionic multiplication. Both classical and quantum dynamics reduce to number theory and the dimensions of classical division algebras reflect the dimensions of string, string orbit, space-time surface, and imbddding space. The conclusion is that both particle data table, the vision about physics as free, classical dynamics of spinor fields in the infinite-dimensional configuration space of 3-surfaces, and physics as a generalized number theory, lead to the same identification: space-time can be regarded as 4-surfaces in M^4xCP_2. In the case that someone is more interested of learning about real progress instead of wasting time to heated arguments at the ruins M theory, he/she can read the chapter http://www.helsinki.fi/~matpitka/tgd.html#visionb summarizing part of the number theoretical vision, and also visit my blog at http://matpitka.blogspot.com/ where I have summarized the most recent progress and great ideas of TGD. With Best Regards, Matti Pitkanen

Monday, February 14, 2005

Kähler calibrations, number theoretical compactification, and general solution to the absolute minimization of Kähler action

The title probably does not say much to anyone who is not a theoretical physicist working with theories of everything. I thought however it appropriate to glue this piece of text from my homepage in hope that the information about these beautiful discoveries might find some readers. So what follows is rather heavy mathematical jargon. Calibrations represent a good example of those merciful "accidents", which happen just the right time. Just for curiosity I decided to look what the word means, and it soon became clear that the notion of calibration allows to formulate my proposal for how to construct general solution of field equations defined by Kähler action in terms of a number theoretic spontaneous compactification in a rigorous and..., perhaps I dare say it aloud, even in convincing manner. For an excellent popular representation about calibrations, spinors and super-symmetries see the homepage of Jose Figueroa-O'Farrill . 1. The notion of calibration Calibrations allow a very elegant and powerful formulation of minimal surface property and have been applied also in brane-worldish considerations. Calibration is a closed p-form, whose value for a given p-plane is not larger than its volume in induced metric. What is important that if it is maximum for tangent planes of p-sub-manifold, minimal surface with smallest volume in its homology equivalence class results. Could absolute minima of Kähler action found using Kähler calibration?! For instance, all surfaces X^2xY^2 subset M^4xCP_2, X^2 and Y^2 minimal surfaces, are solutions of field equations. Calibration theory allows to concluded that Y^2 is any complex manifold of CP_2! A very general solution of TGD in stringy sector results and there exists a deformation theory of calibrations to produce moduli spaces for the perturbations of these solutions! In fact, all known solutions of field equations are either minimal surfaces or have a vanishing Kähler action density. This probably tells more about my simple mind-set than reality, and there are excellent reasons to believe that, since Lorentz-Kähler force vanishes, the known solutions are space-time correlates for asymptotic self-organization patterns. The question is how to find more general solutions. Or how to generalize the notion of calibration for minimal surfaces to what might be called Kähler calibration? It is here, where the handsome and young idea of number theoretical spontaneous compactification enters the stage and the outcome is a happy marriage of two ideas. 2. The notion of Kähler calibration It is intuitively clear that the closed calibration form omega which is saturated for minimal surfaces must be replaced by Kähler calibration 4-form omega_K= L_K omega . L_K is Kähler action density (Maxwell action for induced CP_2 Kähler form).

Important point: omega_K is closed but not omega as in the case of minimal surfaces. L_K acts as an integrating factor. This difference is absolutely essential. When L_K is constant you should get minimal surfaces. L_K is indeed constant for the known minimal surface solutions. The basic objection against this conjecture is following: L_K is a four dimensional action density. How it is possible to assign it to a 4-form in 8 dimensional space-time? Here number theoretical spontaneous compactification shows its power. 3. Number theoretic compactification allows to define Kähler calibration The calibrations are closely related to spinors and the number theoretic compactification based on 2-component octonionic spinors satisfying Weyl condition, and therefore equivalent with octonions themselves, tells how to construct omega.

• The hyper-octonion real-analytic maps of HO=M^8 to itself define octonionic 2 spinors satisfying Weyl condition. Octonionic massless Dirac equation reduces to d'Alembert equation in M^8 by the generalization of Cauchy-Riemann conditions.
• Octonions and thus also the spinors have 1+1+3+3bar decomposition with respect to (color) SU(3) sub-group of octonion automorphism G_2. SU(3) leaves a preferred hyper-octonionic imaginary unit invariant. The unit can be chosen in local manner and the choices are parameterized by local S^6.
• 3x3bar tensor product defines color octet identifiable as SU(3) Lie algebra generator and its exponentiation gives SU(3) group element.
• The canonical bundle projection SU(3)-->CP_2 assigns a CP_2 point to each point of M^8, when a preferred octonion unit is fixed at each point of M^8.
• Canonical projection M^8-->M^4 assigns M^4 point to point of M^8.
Conclusion: M^8 is mapped to M^4xCP_2 and metric of H and CP_2 Kähler form can be induced. M^8 having originally only the number theoretical norm as metric inherits the metric of H.
• Here comes the key point of the construction. CP_2 parameterizes hyper-quaternionic planes of hyper-octonions and therefore it is possible to assign to a given point of M^8 a unique hyper-quaternion 4-plane. Thus also the projection J of Kähler form to this plane and also the dual *J of this projection. Therefore also L_K=J\wedge*J as the value of Kähler action density!
• The Kähler calibration omega_K= L_K*omega

is defined in an obvious manner. As found, L_K is associated with the local hyper-quaternionic plane is assigned to each point of M^8. The form omega is obtained from the wedge product of unit tangent vectors for hyper-quaternionic plane at a given point by lowering the indices using the induced metric in M^8. omega is not a closed form in general. For a given 4-plane it is essentially the cosine of the angle between plane and hyper-quaternionic plane and saturated for hyper-quaternionic plane so that calibration results.

• Kähler calibration is the only calibration that one can seriously imagine. Furthermore, the spinorial expression for omega is well defined only if the form omega saturates for hyper-quaternionic planes or their duals. The reason is that non-associativity makes the spinorial expression involving an octonionic product of four tangent vectors for the calibration ill defined for non-associative 4-planes. Hence number theory allows only hyper-quaternionic saturation. Note that also co-hyper-quaternionicity is allowed and required by the known extremals of Kähler action. A 4-parameter foliation of M^8, and perhaps even that of M^4xCP_2 (discrete set of intersections probably occurs) by 4-surfaces results and the parameters at given point of X^4 define the dual space-time surface.
• A surprise, which does not flatter theoretician's vanity, emerges. Closed-ness of omega_K implies that if absolute value of Kähler action density replaces K\"ahler action, minimization indeed occurs for hyper-quaternionic surfaces in a given homology class assuming that the hyper-quaternionic plane at given point minimizes L_K (is this equivalent this closed-ness of omega_K?). Thus L_K should be replaced with L_K so that vacuum extremals become absolute minima, and universe would do its best to save energy by staying as near as possible to vacuum. The 3 surfaces for which CP_2 projection is at least 2-dimensional and not Lagrange manifolds would correspond to non-vacua since since conservation laws do not leave any other option. The attractiveness of this option from the point of calculability of TGD would be that the initial values for the time derivatives of the imbedding space coordinates at X^3 at light-like 7-D causal determinant could be computed by requiring that the energy of the solution is minimized. This could mean a computerizable solution to the absolute minimization.
• There is a very beautiful connection with super-symmetries allowing to express absolute minimum property as a condition involving only the hyper-octonionic spinor field defining the Kähler calibration (discovered for calibrations by Strominger and Becker).
4. Could TGD reduce to string model like theory in HO picture? Conservation laws suggests that in the case of non-vacuum extremals the dynamics of the local automorphism associated with the hyper-octonionic spinor field is dictated by field equations of some kind. The experience with WZW model suggests that in case of non-vacuum extremals G_2 element could be written as a product g=g_L(h)g^{-1}_R(h*) of hyper-octonion analytic and anti-analytic complexified G_2 elements. g would be determined by the data at hyper-complex 2-surface for which the tangent space at a given point is spanned by real unit and preferred hyper-octonionic unit. Also Dirac action would be naturally restricted to this surface. The amazing possibility is that TGD could reduce in HO picture to 8-D WZW string model both classically and quantally since vertices would reduce to integrals over 1-D curves. The interpretation of generalized Feynman diagrams in terms of generalized braid/ribbon diagrams and the unique properties of G_2 provide further support for this picture. In particular, G_2 is the lowest-dimensional Lie group allowing to realize full-powered topological quantum computation based on generalized braid diagrams and using the lowest level k=1 Kac Moody representation. Even if this reduction would occur only in special cases, such as asymptotic solutions for which Lorentz Kähler force vanishes or maxima of Kähler function, it would mean enormous simplification of the theory. 5. Why extermals of Kähler action would correspond to hyper-quaternionic 4-surfaces? The resulting over all picture leads also to a considerable understanding concerning the basic questions why (co)-hyper-quaternionic 4-surfaces define extrema of Kähler action and why WZW strings would provide a dual for the description using Kähler action. The answer boils down to the realization that the extrema of Kähler action minimize complexity, also algebraic complexity, in particular non-commutativity. A measure for non-commutativity with a fixed preferred hyper-octonionic imaginary unit is provided by the commutator of 3 and 3bar parts of the hyper-octonion spinor field defining an antisymmetric tensor in color octet representation: very much like color gauge field. Color action is a natural measure for the non-commutativity minimized when the tangent space algebra closes to complexified quaternionic, instead of complexified octonionic, algebra. On the other hand, Kähler action is nothing but color action for classical color gauge field defined by projections of color Killing vector fields. Here it is!That WZW + Dirac action for hyper-octonionic strings would correspond to Kähler action would in turn be the TGD counterpart for the proposed string-YM dualities. 6. Summary To sum up, the following conjectures are direct generalizations of those for minimal surfaces.
• L_K acts as an integrating factor and omega_K= L_K*omega is a closed form.
• Generalizing from the case of minimal surfaces, closed-ness guarantees that hyper-quaternionic 4-surfaces saturating this form are absolute minima of Kähler action.
• The hyper-octonion analytic solutions of hyper-octonionic Dirac equation defines those maps M^8-->M^4xCP_2 for which L_K acts as an integrating factor. Classical TGD reduces to a free Dirac equation for hyper-octonionic spinors!
For more details see the chapter TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts.

Friday, February 11, 2005

Great Ideas

It occurred to me that I could try to summarize the great ideas behind TGD. There are many of them and I cannot summarize them in single page. My dream is to communicate some holistic vision about what I have become conscious of and therefore I start just by listing the great ideas that come to my mind just now and continue later by giving details. These ideas are also summarized in the chapter Overview about the Evolution of Quantum TGD of TGD.
• Classical physics as the geometry of space-times regarded as 4-surfaces in certain 8-dimensional space-time. This generalizes and modifies Einstein's vision. Note that in quantum context the plural "space-times" indeed makes sense.
• Quantum physics as infinite-dimensional geometry of the world of classical worlds=space-time surfaces. This vision generalizes further the vision of Einstein. One of the paradoxes of the sociology of post-modern physics is that M-theorists refuse to realize the enormous unifying power of infinite-dimensional geometry.
• Physics as number theory vision involves several ideas bigger than life.