Color confinement and its dual

In yesterday's posting "Precise form of ew-color duality" an important step in the understanding of HO-H duality was summarized. The short and long p-adic length scales L_k and L_p, p=about 2^k correspond to short and long distance limits in HO-H duality. In particular, the duality was formulated in terms of symmetries of the world of the classical worlds as a kind of super-symmetry permuting configuration space degrees of freedom with corresponding spin degrees of freedom. The description of duality at the configuration space level can be applied to gain a view about color confinement and its dual for electro-weak interactions at short distance limit as a process in which configuration space degrees of freedom begin to dominate, field theory picture becomes obsolete, and it is better to go the dual description. The correct prediction is that SO(4) should appear as dynamical symmetry group of low energy hadron physics. There are two basic types of vacuum extremals: CP_2 type extremals representing elementary particles, and vacuum extremals having CP_2 projection which is at most 2-dimensional Lagrange manifold representing for instance hadrons. It is not surprising that HO-H duality can be interpreted in terms of these vacuum extremals and they provide a more precise view about what happens at the limits when either CH or CHO degrees of freedom begin to dominate over space-time degrees of freedom describable ordinary quantum field theory.

1. Short distance limit

Consider first the short distance limit at which electro-weak confinement is expected and HO picture becomes more appropriate. a) Ew-color duality would suggests that at the limit of short distances something analogous to color confinement occurs for electro-weak interactions. Also the large value of U(1) coupling supports this expectation. The vacuum property of CP_2 type extremals means that induced spinor fields become vacuum spinor fields with an identically vanishing Dirac action. Therefore these spinor fields effectively disappear at space-time level for the maxima of Kähler function, and contribute only via quantum fluctuations, which correspond to configuration space dynamics. Color partial waves are left as a genuine configuration space degree of freedom and the expectation is that only the lowest color partial waves corresponding to singlet and triplet remain and become spin like degrees of freedom analogous to QCD color in HO picture. b) Duality suggests SO(4) confinement in E^4 degrees of freedom at this limit. The nearly vacuum property should allow very large fluctuations of the ordinary fermion and anti-fermion numbers at the limit when the fermions become pure vacuons for which creation and annihilation operators reduce to anti-commuting Grassmann numbers. In HO picture this would mean that high SO(4) partial waves in E^4 are possible for composites although net ew quantum numbers vanish. Hence electro-weak spins become analogous to classical angular momentum at this limit.

2. Long distance limit

Consider next color confinement at the long length scale limit as a dual of this picture. a) In the case of color interactions very high color partial waves for quarks and gluons appear at the confinement limit. For instance, vacuum extremals representable as maps M^4--> CP_2 identifiable as hadronic space-time sheets correspond to color confinement limit. Strong fluctuations due to high color partial waves in CH appear, and corresponds in CHO$picture to the presence of high colored hyper-octonionic fermion and anti-fermion numbers. Since configuration space degrees of freedom begin to dominate, color confinement limit transcends the descriptive power of QCD just as high energy limit transcends the descriptive power of standard model of electro-weak interactions. b) The success of SO(4) sigma model in the description of low lying hadrons could directly relate to the fact that this group labels also the E^4 Hamiltonians in HO picture. SO(4) quantum numbers can be identified as right and left handed strong isospin for which no natural interpretation exists in H picture. Family replication phenomenon is described in the same manner in both cases so that quantum numbers like strangeness and charm are not fundamental. Indeed, p-adic mass calculations allowing fractally scaled up versions of various quarks allow to replace Gell-Mann mass formula with highly successful predictions for hadron masses. c) Ordinary fermion numbers do not fluctuate at the color confinement limit. That this does not occur must relate to the facts that modified Dirac action relates by super-symmetry to Kähler action and the variations of Kähler action vanish up to third order around canonically embedded M^4 whereas for the CP_2 type extremals the situation is completely different. The absence of fluctuations corresponds to the possibility to describe low energy hadrons using simple valence quark model without the inclusion of the quark and gluon sea. At the asymptotic freedom limit sea becomes increasingly important. d) Baryons should be analogous to color partial waves of quarks, and just as CP_2 spinors allow at CH level color triplet partial waves also hyper-octonionic fermions should allow at CHO level SO(4) partial waves transforming as doublets under SU(2)_L or SU(2)_R. e) Ordinary fermion numbers do not fluctuate at the color confinement limit. That this does not occur must relate to the facts that modified Dirac action relates by super-symmetry to Kähler action and the variations of Kähler action vanish up to third order around canonically embedded M^4 whereas for the CP_2 type extremals the situation is completely different. The absence of fluctuations in ew spin degrees of freedom suggests the possibility of describing low energy hadrons using simple valence quark model without quark color with quark and gluon sea modelling the presence hyper-octonionic quark pairs. The problems due to statistics might be resolved by anyonic statistic possible for 2-D partonic surfaces. At the asymptotic freedom limit hyper-octonionic sea becomes less and less important.

3. Proton spin crisis as a signature of hyper-octonionic quarks

Hyper-octonionic quarks carry neither ordinary nor electro-weak spin since these quantum numbers correspond to orbital quantum numbers in HO. Hence in the ideal colored quark description the contribution of quarks to both spin and electro-weak spin of proton should vanish whereas in H quark description quarks should give proton spin. Obviously, these descriptions correspond to colored current quark description and to a static color singlet quark descriptions possible for anyonic statistics. This prediction would sound crazy unless the essence of proton spin crisis were just the finding that the contribution of quarks to proton spin is small. Spin-statistics paradox is avoided if configuration space degrees of freedom are taken into account. Quantum-classical correspondence, if taken at extreme, would suggest that configuration space degrees of freedom might have some kind of space-time correlate. The 2-dimensionality of stringy and partonic surfaces suggests that anyons might provide this correlate. In HO picture spin-statistics paradox at space-time level would be avoided by the 2-dimensionality of partonic surfaces allowing to have braid representations of the rotation group and colored quarks can have half-odd integer valued anyonic spin and electro-weak spin. A possible physical mechanism transforming H quarks without color spin but with ew- and ordinary spin to HO quarks having only color spin is following. Anyonic and ordinary contributions to ew- and ordinary spin of H quark cancel each other and color spin is generated anyonically. In TGD framework anyons are associated with punctures assignable to the thin flux threads connecting partonic 2-surfaces and these punctures appear always as pairs with the ends of thread carrying opposite anyonic quantum numbers. OH fermions would correspond to fermion plus the second end of the anyon thread. In H picture the approach to confinement means large fluctuations also in SO(3) degrees of freedom and the emergence of Regge trajectories. In HO picture the angular momentum of hadron would be due the angular momentum of a large number of colored quark pairs. For more details see the chapter TGD as a Generalized Number Theory: Quaternions, Octonions, and their Hyper Counterparts. Matti Pitkänen

Primes and Quantum Computation

The morning visit to hep-th is nowadays an irritating experience since most of the stuff is wastes produced by the dying M-theory. In this morning cross-listings however contained a highly interesting looking article quant-ph/0503013 with title "Entanglement Distillation Protocols and Number Theory" by H. Bombin, M.A. Martin-Delgado. I attach here the abstract.

We show that the analysis of entanglement distillation protocols for qudits of arbitrary dimension D benefits from applying basic concepts from number theory, since the set \zdn associated to Bell diagonal states is a module rather than a vector space. We find that a partition of \zdn into divisor classes characterizes the invariant properties of mixed Bell diagonal states under local permutations. We construct a very general class of recursion protocols by means of unitary operations implementing these local permutations. We study these distillation protocols depending on whether we use twirling operations in the intermediate steps or not, and we study them both analytically and numerically with Monte Carlo methods. In the absence of twirling operations, we construct extensions of the quantum privacy algorithms valid for secure communications with qudits of any dimension D. When D is a prime number, we show that distillation protocols are optimal both qualitatively and quantitatively.

The result means that qupits represented by quantum systems with p states, p prime, are in a very special position as far quantum computation is considered. I have been talking about the importance of qupits for quantum computation and information processing in TGD Universe for about half decade now, see for instance the chapters Negentropy Maximization Principle, Intentionality, Cognition, and Physics as Number Theory or Space-Time Point as Platonia , Topological Quantum Computation in TGD Universe . As one might guess, colleagues who read only respected journals have not shown a slightest interest. There are however notable exceptions. In particular, Roger Penrose who mentions my work with p-adic numbers in his newest book "Road to Reality". My sincere hope is that this could help these revolutionary ideas to find young brains as templates for growth.

1. Number theory allows definition of entanglement negentropy

p-Adic approach leads to the generalization of the notion of Shannon entropy and thus of entanglement entropy defined by the same formula. The idea is to replace logarithms of probabilities appearing in the formula with the logarithms of their p-adic norms. This certainly makes sense if probabilities are rational numbers. It makes also sense for algebraic extensions of rationals provided an appropriate extension of p-adic numbers is introduced and norm generalized appropriately. What is fascinating is that the resulting entanglement entropy can be negative so that entropy becomes negentropy! One can always find the prime for which it is maximally negative and identify it as a prime characterizing the system in question. This assignment can be generalized so that it applies at space-time level too. In this case maximal non-deterministic regions of space-time surface define an ensemble, and the prime p assignable to space-time sheet is some prime factor of the number N of these regions. One can say that rational and even algebraic entanglement carries genuine information. When the number of entangled state is prime or power of prime, entanglement negentropy increases dramatically. This would predict that physical systems for which state space is p^n-dimensional, p prime, should be of special importance. In p-adic physics this kind of systems appear naturally. Also this idea generalizes to space-time level. In TGD inspired theory of consciousness rational entanglement can be identified as a correlate for an experience of understanding. Quite generally, rational, and more generally algebraic, entanglement suggests strongly the identification as bound state entanglement. Thus in quantum jump this kind of entanglement would be stable whereas transcendental entanglement would be unstable. For ideal quantum computer entanglement probabilities are indeed simple rational numbers. That bound state entanglement is in question might imply stability and allow to circumvent the fundamental problem caused by the fact that entanglement is highly unstable in the world governed by the standard quantum theory. In quantum computing p^n-state systems should be of special importance since qupits are expected to be exceptionally stable. A general mechanism of macroscopic and macrotemporal quantum coherence based on spin glass property of TGD Universe, and possible applications to quantum biology are discussed in Macro-Temporal Quantum Coherence and Spin Glass Degeneracy .

2. Rationals and algebraic numbers are special also physically

The exceptional role of rationals has a nice correlate in what I take a liberty to call arithmetic dynamics. Rationals can be regarded as islands of order in the chaotic sea of transcendentals in the sense that the expansion of rational in terms of any integer n> 1 is periodic and thus analog to a periodic orbit of a dynamical system. For transcendentals this is not the case. Algebraic numbers are islands of order in weaker sense since they become rational by a finite number of algebraic operations. Rationals have a special role also in classical mechanics. Resonances occur in many particle systems (such as planetary systems) when the frequencies of periodic motions are rational multiples of basic frequency so that the entire system can behave strictly periodically and oscillate coherently. This is in accordance with the interpretation as formation of bound states. One of the basic characteristics of brain is entrainment with external frequencies. There is evidence that radiation with cyclotron frequencies of important biological ions in Earth's magnetic field has special effects on brain. Schumann frequencies appear in EEG also. TGD interpretation would be that the magnetic body associated with the biological body is of Earth size at least (entire scale hierarchy of them is predicted), and that there is also a strong resonant coupling with the magnetic body of Earth, even to the extent that bound states are formed. Libet's findings about strange time delays of conscious experience find a nice explanation if EEG mediates sensory input from brain to magnetic body and intentional motor control from magnetic body to brain.

3. Primes for tensor product are prime-dimensional vector spaces

For physicist it might become as a surprise how general the notion of prime is. When you have some algebraic product you can immediate ask whether there exist elements of algebra serving as "elementary particles" of the algebra in the sense that you can express any algebra element as product of these elements. In case of number fields these building bricks are highly unique. For instance, all classical division fields allow the notion of primeness and the notions of ordinary, complex, hyper-quaternionic and hyper-octonionic primes are central for TGD. "Hyper" means that one considers primes in the complexification of number field and having the property that imaginary part of integer is multiplied by commuting sqrt(-1): "hyper" is needed in order to obtain number theoretic norm with Minkowskian signature. Also the tensor product of vector spaces defines an algebraic multiplication and now primes are vector spaces with prime-valued dimension. Hence p-adic hierarchy emerges already at the level of basic mathematics of quantum theory. Obviously this observation should have deep relevance if Universe is topological quantum computer as TGD indeed suggests. Matti Pitkänen

Precise Form of Electroweak-Color Duality

During the last months dramatic progress in the understanding of TGD in terms of what I call HO-H duality, has occurred. HO denotes 8-D space of hyper-octonions obtained from octonions by multiplying the imaginary units with a commuting square root of -1. H denotes the imbedding space H=M^4xCP_2 obtained by replacing each point of Minkowski space with CP_2. The strongest form of HO-H duality is that TGD can be described either in H picture or in HO picture and one can speak about number theoretical compactification (no actual compactification of course occurs). To get a glimpse about what is involved for the earlier postings and the chapter Physics as a Generalized Number Theory: Quaternions, Octonions, and their Hyper Counterparts. HO-H duality seems to physically correspond to electroweak-color duality and I have tried to understand what is really involved. After reading emails and visiting at blog groups to find that nothing interesting had appeared (I do not find the usual battles between alpha males interesting) I suddenly got the feeling that I might be able to write something new about this duality. My gut feeling was correct. The e(lectro)w(eak)-color duality associated with H-HO duality reflects the fact that in both pictures the dynamics of single space-time surface can provide only a partial description of quantum dynamics, and that configuration space level(configuration space is the world of classical worlds, space-time surfaces) is needed in order to code all quantum numbers and all interactions. The situation cries for a more precise formulation for the ew--color duality. The sought for formulation can be expressed as a single concise statement.

In H (HO) picture ew-spin (color) degrees freedom correspond to spin like quantum numbers and color (ew) degrees of freedom to classical conserved quantum numbers.

Consider now what this statement really means.

1. Spin-like quantum numbers and conserved charges in H-picture

In H picture ew quantum numbers and spin are manifestly present whereas color quantum numbers and interactions emerge as spin like quantum numbers only at configuration space level as does also four-momentum via Kac-Moody representations. Classical(!) color and Poincare charges are well defined also in H picture. There is also a non-trivial interaction between color and ew degrees of freedom since color transformations are accompanied by ew rotations in accordance with the fact that U(2)_{ew} can be mapped to a subgroup of SU(3) via the coset construction.

2. Spin-like quantum numbers and conserved charges in HO-picture

Hyper-octonion HO spinors decompose to representations of color group whereas H spinors decompose to the representations of ew and Lorentz group. Hence for HO picture color is manifestly present as spin degrees of freedom but ew spin and spin are absent. By ew-color duality at space-time level ew and spin charges should somehow emerge also in HO picture as classical conserved quantities. The first observation is that the automorphism group G_2 corresponds to 2 conserved commuting charges. Translating H picture directly to HO level this would mean that the classical conserved charges associated with WZW + Dirac action have identification as ew charges. Also now a non-trivial relation between electro-weak and color quantum numbers is involved. There are also symmetries not respecting hyper-octonion real-analyticity and analogous to those affecting the moduli characterizing complex structure. SO(7) leaves the spatial part of the hyper-octonionic norm invariant and this extends the number of conserved charges to 3 bringing in spin. The full isometry group of the hyper-octonionic norm is SO(7,1) so that also Lorentz boost would be included to the Cartan algebra. Also translations are symmetries of HO picture since the shift of the origin gives rise to a new solution family which is however not hyper-octonion analytic in the original coordinate system. Four-momentum should emerge at quantum level via Kac-Moody type realization also now. The correspondences M^4xSO(7,1)<--> PxSU(3) and SU(3)<--> U(2)_{ew} code for the ew--color duality. Interestingly, the four M^4 coordinates depending on X^4 define a local Kac-Moody algebra identifiable in terms of the Cartan algebra of SO(7,1) and extendable by k=1 vertex operator construction to a representation of SO(7,1) Kac-Moody algebra. On the other hand, the Euclidian stringy degrees of freedom in M^4 give rise to SU(3) Kac Moody algebra and to SU(3)/U(2) WZW model serving as a candidate for a model of ew interactions. A very tight web of correspondences between various symmetries is involved.

3. Ew-color duality and and parton-string duality

G_2/SU(3) coset WZW theory would correspond naturally to QCD and geometrically to the identification of a preferred hyper-octonion imaginary unit defining a foliation of HO by string like 2-surfaces. SU(3)/U(2) corresponds naturally to ew gauge theory. These WZW models are obviously mutually exclusive if defined at time-like stringy surfaces Y^2. One can however consider defining SU(3)/U(2) WZW type model at the partonic 2-surfaces X^2 in H picture. These two dual space-time pictures might allow a rather satisfactory quantum-classical correspondence. That information from entire light-like causal determinants is needed, conforms with the quantum gravitational holography, and the objection that the theory is equivalent to a string model can be avoided. The duality corresponds two kinds of conformal symmetries: hyper-octonionic conformal invariance and the conformal invariance associated with the light-like causal determinants. 4. Ew-color duality and duality of long and short p-adic length scales The first formulation for ew-color duality was in terms of p-adic length scale hypothesis selecting the primes p\simeq 2^k, k positive integer, preferably prime or power of prime, as preferred p-adic length scales. L_p\propto \sqrt{p} corresponds to the p-adic length scale defining the size of the space-time sheet at which elementary particle represented as CP_2 type extremal is topologically condensed and is of order Compton length. L_k\propto \sqrt{k} represents the p-adic length scale of the wormhole contacts associated with the CP_2 type extremal and CP_2 size is the natural length unit now. The proposal was that QCD type description based on quarks and gluons corresponds to a description in the ultra-short length scale L_k and the description in terms of hadrons possessing only electro-weak quantum numbers and spin corresponds to the hadronic length scale L_p. The order of magnitude for alpha_s is predicted correctly directly from the fact that it is proportional to alpha_K and as U(1) coupling increases towards short p-adic length scales in a manner predicted by heuristic arguments assuming that gravitational constant does not run appreciably as a function of p-adic length scale. Indeed, HO picture describing color as a spin like quantum number is more appropriate near CP_2 length scale whereas H picture describing color classically (as in color flux tube models) is more appropriate in hadronic length scales. Perturbative--non-perturbative QCD duality would thus also correspond to HO-H duality. Strictly speaking, non-perturbative QCD would be a meaningless notion.

5. H-HO duality in the world of classical worlds

An interesting challenge is to translate H-HO duality to the level of configuration space geometry and spinor structure. a) In H picture CH Hamiltonians correspond to Hamiltonians of \delta M^4_+xCP_2 in representations of SO(3)xSU(3) whereas spin and electro-weak spin correspond to spin degrees of freedom associated with complexified gamma matrices acting as super-generators. b) In HO picture CH is replaced with what might be called CHO. The guess is that also CHO allows Kähler and symplectic structures. CHO Hamiltonians cannot correspond to Hamiltonians of E^7 (imaginary hyper-octonions) since E^7 has wrong dimension. 7-D light-cone is in turn metrically a 6-sphere. If S^6 does not allow complex structure as Chern's last theorem claims, it does not allow Kähler structure neither. Situation changes if onen considers \delta M^4_+xE^4 metrically equivalent to S^2xE^4, which certainly allows Kähler and symplectic structures. This choice is of course perfectly natural and consistent with the view that number theoretical compactification takes effectively E^4 to CP_2 by attaching to it a 2-sphere at infinity. SO(3)xSO(4) would assign to Hamiltonians spin and ew quantum numbers. Color quantum numbers would correspond to spin degrees of freedom associated with CHO gamma matrices acting also as super generators. H-HO duality could be also interpreted as a super-symmetry permuting bosonic and fermionic degrees of freedom at the level of configuration space. With a technical skills of a super-string dualist it would be probably easy to derive strong testable predictions from this duality. For more details see Physics as a Generalized Number Theory: Quaternions, Octonions, and their Hyper Counterparts. Matti Pitkanen

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.

What about zero modes?

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.

What about renormalization?

The elimination of infinities relates closely to the renormalization of coupling constants, and one could argue that the proposed beautiful scenario is in conflict with basic experimental facts. This not the case if Kähler coupling strength has an entire spectrum of values labelled by primes or subset of primes labelling p-adic length scales. In this picture p-adic primes label p-adic effective topologies of non-deterministic extremals of Kähler action. p-Adic field equations possess an inherent non-determinism and the hypothesis is that this non-determinism gives in an appropriate length scale rise to fractality characterized by an effective p-adic topology such that the prime p is fixed from the constraint that the non-determinism of Kähler action correspond to the inherent p-adic non-determinism in this length scale range. The highly non-trivial prediction is that quantum non-determinism is not just randomness since p-adic non-determinism involves long range correlations due to the fact that in p-adic topology evolution is continuous. The proposal is that the long range correlations of locally random looking intentional/purposeful behavior could correspond to p-adic non-determinism with p characterizing the "intelligence quotient" of the system. This is a testable prediction. The coupling constant evolution is replaced by the p-adic length scale dependence of Kähler coupling strength and of other coupling constants determined by it. The emergence of the analogs of loop corrections might be understood if there is a natural but non-orthogonal state basis labelled by quantum numbers which allow a natural grading. The orthogonalized state basis would be obtained by a Gram-Schmidt orthonormalization procedure respecting the grading. Orthonormalization introduces a cloud of virtual particles and the dependence of this Gram-Schmidt cloud on prime p would induces the TGD counterpart of renormalization group evolution. It is clear that the classical non-determinism of Kähler action is the red thread of the proposed vision. By quantum classical correspondence space-time surface is not only a representation for a quantum state but even for the final states of quantum jump sequence. Classical non-determinism would thus correspond to the space-time correlate of quantum non-determinism. Matti Pitkänen

Melancholic Moods and Road to Reality

These periods of stagnation are difficult to tolerate. You have been experiencing a flow of mathematical ideas continuing more than year with only brief periods of recovery from physical exhaustion. Then you lose the contact. No ideas pop up. You have been asked to write articles but you cannot do it: a passive organizing of existing material into articles simply cannot motivate you, and you cannot develop enough self discipline to do this by brute force. Chess problems are your latest invention in the war against depression: your brain gets depressed unless it receives its daily portion of problems. You also write to your blog page these melancholic musing to stay sane. You begin to feel more and more strongly your role in the society: an unemployed academic village fool, trying to survive with a minimum possible unemployment money. Without any future and thus without any hope. The academic world of Finland will never forgive you that you have carried out without their permission a lifework summarized in these four 1000-page books which have become your identity. The punishment is the cruellest possible: they have stolen your future. You feel that you have lost completely your faith in humanity but at the same time know that you should be able to cheat yourself to believe on the ultimate justice. Fear fills you: how could you recover some of this naive and healthy belief that all power holders of science are not gangsters. In this mood you open monday's email box and ... experience something completely unexpected. An email tells that Sir Roger Penrose mentions in his newest book "Road to Reality" your work relating to p-adic physics. Although you have talked a lot about name magic, you cannot hide your happiness. You feel it really consolating that there are also honest and intelligent people in this cruel, and you notice notice it now, beautiful world. You know that this kind of gesture has miraculous effect: some collegue might lower himself to touch your work. Even better, this book will inspire many young people who still have an open mind. While looking out you see that Sun is almost shining. You hope to have some day the opportunity to read Road to Reality yourself and remember that there was a discussion in Not-Even-Wrong about Road to Reality. Matti Pitkanen

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

Parody as means to see what went wrong

The world of science is a strange world. You readily see that its inhabitants are strangely twisted creatures having somehow surreal shapes. You also recognize that all of them have in some difficult-to-define sense lost their way. You gradually begin to realize what the problem might be. It seems that these creatures have lost themselves into a forest. This forest is not an ordinary forest. No. The trees of this forests are concepts and expressions of language. It is obvious that these meanderes in the forest have forgot where they came from and what they were. They do not have the familiar five senses anymore, they have replaced reality with symbols. These creatures without senses look depressed and unhappy. What is left from real life are occasional bursts of aggression. Clearly, they can still hate and be bitter and do so intensively but what has happened to love and trust and joy? Something has gone badly wrong and you would be happy if you could help them somehow but you do not know what to do. Certainly it seems that it is not a good idea to go and tap on the shoulder and suggest a beer or two and a friendly chat. You ask how it is possible that a happy gifted child came an angry and bitter academic identifying himself with his curriculum vitae. You gradually learn that this tragic somehow relates to meanings. Clearly, these people have started to assign meanings to symbols of reality instead of reality. Life goes by as they have their endless bloody debates about priorities. You find it really heart breaking to see how these tiny beings fight passionately about something which has absolutely no meaning for you or any healthy human being who has learned what are those few really signficant things in life. How could even a best Zen guru teach these people the joy of carrying water and chopping firewood. Superstring community certainly represents an extreme example about this anomalous association of meanings. It is its own micro-culture talking strange stringy language expressing strange stringy concepts. This closed society of stringists is like a religious sect with its own difficult-to-understand rules and habits. In order to understand what I mean, try to imagine a community which has replaced spoken language with written scientific jargon with every sentence ornamented with minimum of one reference or foonnote and discussing via published articles rather than face-to-face. W. Siegel, a string theorist himself but in a strangely ambivalent manner also an outsider, has produced hilarious parodies about typical super string articles. Here is one example. Do not miss the other ones. The following piece of text made me laughing to science and myself and I realized how insane this world (I do not try to pretend that it includes also me) is. My hope is that this text fragment might give even a reader without any background in science a healthy burst of laugh. "The moonlight danced on her face as it reflected off the warm August waves. He watched her graceful body gradually emerge as she stepped closer to the shore. As she lay down in the sand beside him he leaned close to whisper in her ear: "String theory is presently the most successful model to describe quantum gravity [1]. Unfortunately, like QCD before it, the proof of its relevance to the real world was shortly followed by the realization that it was impossible to calculate anything with it..." 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...".

...and how Lubos Motl responded...

Here is Lubos Motl's response which reflects his characteristic deep respect of intellectual integrity: "Matti - well, let me admit that I don't believe any of the things about "TGD" you wrote." The response was not surprising in view of what I have learned about Lubos Motl's typical behavioral patterns. I wonder whether the open censoring of ideas and opinions which do not conform with those of hegemony relates to change that has occurred during the regime of Bush. For instance, in a recent poll it was found that about one half of student agreed when it was asked whether the full freedom of speech guaranteed in constitution law should be restricted. Food for a worried thought! My response, which I unfortunately did not store to my computer, was related to the basic trauma suffered by theoretical physicists at the end of the era of hard-boiled reductionism. It could be called Planck length syndrome and manifests itself as desperate attempts to explain all elementary particle masses in terms of Planck mass scale although the first glance at elementary particle mass spectrum demonstrates that there are obviously several mass mass scales involved: electron, mu, and tau or quarks simply cannot belong to same multiplet of symmetry group in any conceivable scenario based on a unifying gauge group. p-Adic length scale hypothesis resolves the paradox and allows to understand among other things the basic mystery number 10^38=about 2^127-1 (largest Mersenne prime of physical signficance in human scales) of physics. Also a far reaching generalization of the scope of what physics can describe emerges: p-adic physics is physics of cognition which must be also a part of theory of everything. The response of Lubos is a school example about how difficult it is to communicate new, radical, and working idea. Other typical and equally brilliant response would have been "You cannot be right because Witten would have discovered it before you!". Matti Pitkanen

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

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

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.

End of an epoch as an era of moral degeneration

One might try to interpret this sad state of theoretical physics (and science in general of course), which affects deeply also experimental physics, in particular particle physics, using notions like sociology of science, and talking about the fierceful competion about positions and research money implying that people with keen brain but without intelligent elbows are doomed to be destroyed. I think however that there is something deeper involved. My conclusion from what has happened during these decades, and from a personal work in wide range of topics ranging from particle physics to cosmology to biology to consciousness, is that we are living an end of an epoch, not only in science but also in the evolution of western civilization. Epochs are always characterized by some sacred philosophical ideas, and when they lose their explanatory and guiding power, the civilization breaks down and experiences a phase transition to a "New Age". The irony is that the end of epoch is always seen as the final victory of the old dogmas (see the last quotations in the beginning of these musings!). In physics only this and comparison with what happened for a century ago, helps to understand how M-theories, regarded now by most professionals (privately of course!) as a pathetical failure, are still successfully sold for media as the final word about physics. Of course, things have begun to change rapidly. For instance, for some time ago the physics department of Boston University made the decision that string models cannot be regarded as physics so that string theorists must find their positions from Mathematics department. Do not take me wrongly: many of the mathematical ideas of string models are fantastics and will be pieces of future theories. The problems is that the utterly wrong philosophy makes it impossible build a physical theory based on these ideas. In physics several philosophical dogmas have lost their power during the last century. For century ago before advent of quantum theory the idea about deterministic clockwork universe was the final truth. The attribute "Newtonian" is often assigned with this clockwork. This does injustice to Newton, who was five centuries ahead of his time and regarded even planetary systems as living systems. This was described in some popular science document as "the dark side of Newton"! Although quantum physics in microscopic length scales turned out to be in conflict with the clockwork idea and although the evidence for macroscopic quantum effects is increasing, the average physic is still, at the temporal distance of century from Bohr, stubbornly refuses to consider the possibility that quantum effects might be important in our everyday life, and that we might not be deterministic machines. This collective silliness has meant after the days of von Neumann nothing interesting has been said about quantum measurement theory so that its paradoxes are still there, virtually untouched. What a gem for a brilliant theoretician these paradoxes could be but how few are those who are ready to spend life in science as crackpots! The sad consequence is that theoretical physicists have practically nothing to add to what they could had said about biology and consciousness before Bohr. Second sacred and very dead idea is reductionism. String theories and M-theory amplify this idea to sheer madness. The dogma is that the whole physics reduces to Planck scale: a very short distance about 10^(-33) meters. After a huge amount of theoretical activity during last 20 years we have a a theory which cannot predict anything even in principle, and the proponents of this theory are now seriously suggesting that we must accept this state of affairs since M-theory is the final theory! I know...! I know that I have said this many times before but this is so scandalous that it must be said aloud again and again. Of course, one need not go to Planck length to see the failure of reductionism: again the phenomenon of life tells that reductionism is wrong. In fact, all transitions in the sequence quark physics--> hadron physics --> nuclear physics ---> atomic physics --> molecular physics --> ... are all ill-defined, and the belief that lower levels predict the higher ones is just a belief without any justification. Despite this most colleagues enjoying monthly salary are still repeating the liturgy which I heard for the first time in my student days: life as nothing but Schrödinger equation applied to a very complex system. It is understandable that a person at the age of 20 years takes this platitude seriously but it is unforgivable that professors of physics still repeat this kind of trash. The seeds of the new age are already here: fractality is now a relatively well established notion even in physics, and means an entire hierarchy of scales giving good hopes of expressing quantitatively how reductionism fails. A lot of new, but already existing mathematics is however needed. Unfortunately, and totally unexpectedly, for consistency of interpretation one cannot avoid talking about physical correlates of cognition once this mathematics is introduced. The transition to the New Age will thus be very painful since cognition and consciousness do not exist for a physicist. Next big idea. Average physicist familiar with special relativity identifies without hesitation the time of conscious experience as the fourth space-time coordinate. This despite the fact that brief analysis shows that these notions are quite different. The sole justification for their identification is the materialistic dogma: in the materialistic world order consciousness is a mere epiphenomenon and has no active role: everything about contents of consciousness is known once the state of the clockwork is known. Despite all the paradoxes that this view implies, it dominates still even neuroscience as I painfully learned during the decade that I devoted to TGD inspired theory of consciousness and learned basics of neuroscience in discussions and by reading. Energy is dual to time and subjectivetime=geometric time problematics repeats itself as the poorly understood relationship between inertial energy and gravitational energy. Einstein's proposal was that they are one and the same thing but even academic person could privately wonder why it is possible that inertial energy is conserved exactly whereas gravitational energy is not. Of course, friendly Einstein has so long a shadow that the academic person does not allow this kind of thoughts to pop up into his conscious mind.

The sociological side

There are also social factors involved and these factors are amplified during a period of degeneration that we are living. To make this more concrete I tell about my experiences in certain discussion groups related to string models and M-theory during last half year. My motivation for spending (or rather wasting as it turned out) my time in these groups was besides satisfying my basic social needs, to learn what the general situation is, and perhaps even communicate something: it is extremely frustrating to see that physics is in a state of deep stagnation and, this I can say without any hybris, to realize that one reason for this is simply that I have not used all my energy to communicate the bottleneck ideas which are the only way out of the dead end. To be specific, I have read the comments at the Not-Even-Wrong blog page administered by Peter Woit. For half a year ago I was enthusiastic: this would be a page where the problems of M-theory and string models could be discussed openly without repeating the usual liturgy starting with "M-theory is the only known quantum theory of gravitation...". It however soon became clear that the discussion group is not intended to serve as a forum for new ideas. The dream about return back to the good old days of quantum field theories seems to be the basic motivation for its existence, and any new idea is therefore regarded as an enemy. For a month or so ago an open censorship was established meaning that everything which is not about the topic of discussion is censored away. To stay in the "topic of discussion" is to repeat the already-too-familiar mantra like arguments against M-theory. M-theory is dead but it does not help to say this again and again: something more constructive is needed. This discussion group has taught me a lot about the difference between western and eastern communication culture manifests. Typically the discussions are battles: people express their verbal skills by insulting each other personally in all imaginable manners. Open verbal sadism is completely acceptable and often regarded as a measure of intelligence. Paradoxically, I myself find that I take seriously Lubos Motl, young extremely aggressive M-theory fanatic who has been continually ranting and raving in Not-Even-Wrong. It is easy to disagree in a well-argumented manner with most of what he says, and most of his arguments are cheap rhetoric, but still! Why do I pay attention to this empty verbalism. Sad to confess, I am part of this culture which confuses aggression with intellect. In Eastern cultures, which we tend to assign labels like "New-Age", both physical and intellectual integrity are the basic values, and my dream is that the New Age would see these people continually insulting each other simply as what they are: uncivilized barbarians. NAMEs are important ghostly participants in any academic discussion group. Again and again I find, that people are asking was it this or was it that what Feynman/Gross/Witten..... said and what is the URL where it can be found. Somehow it seems completely impossible to be taken seriously unless one continually drops names and citations. With all respect to these brilliant NAMEs, one must admit that their often casual comments are just casual comments and that 65 year old NAME probably has not very interesting things to say about forefront science. A further amusing feature of the sociology of academic discussion groups is the attitude towards "crackpots". "Crackpots" are by definition those who have something original to say and strong motivations to do this. The strategy for dealing with crackpots is following. In the first stage "serious researchers" do not notice in any manner the presence of "crackpot", he feels that he is just air. If this does not help, ironic and sadistic comments communicating the conviction that the "crackpot" is a complete idiot begin to appear. If even this does not help, the "crackpot" is told to leave the discussion group since people want to have "serious discussions". If even this does not beat the visitor, his messages are censored away. The irony is that also in this group the "crackpots" have been the most interesting participants. They have something interesting to say, they have a passion of understanding, they develop real arguments instead of cascades of insults and citations from authorities, and they have a strong urge to express clearly what they want to say. This cannot be said about many "respected" contributions, which often contain about ten typos and grammatic errors per line, and from which it is often completely impossible to figure out whether the attempt is to say something or just to teach by example that the writer is an an-alphabet. One gloomy conclusion from these experiences about the sociology of science might be that organized good is impossible. Perhaps this true to some degree. I can however imagine the feelings of those enlightened people who discovered the idea of gathering together intelligent people devoting their lives to science. Certainly they dreamed of spirited debates, endless discussions, openness of minds, evolution of ideas. It is not the fault of these visionaries that in a typical university people sitting in the same room for years know nothing about each other's work and could not be less interested, that hatred and envy are the dominant emotions of an average career builder, and that science has transformed from passion to a fierceful competition to a war in which the best colleague is a dead colleague. Perhaps the New Age inspired by new Great Ideas and Ideals will create a new kind of university in which people could could live like ordinary simple people like to live: listening, helping and loving each other. Matti Pitkanen

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