Is dark matter warped?

The formula for the gravitational Planck constant contains the parameter v₀/c=2^-11. This velocity defines the rotation velocities of distant stars around galaxies. The presence of a parameter with dimensions of velocity should carry some important information about the geometry of dark matter space-time sheets.

Velocity like parameters appear also in other contexts. There is evidence for the Tifft's quantization of cosmic red-shifts in multiples of v₀/c=2.68× 10^-5/3: also other units of quantization have been proposed but they are multiples of v₀ (see this).

The strange behavior of graphene includes high conductivity with conduction electrons behaving like massless particles with light velocity replaced with v₀/c=1/300. The TGD inspired model explains the high conductivity as being due to the Planck constant h(M⁴)= 6h₀ increasing the delocalization length scale of electron pairs associated with hexagonal rings of mono-atomic graphene layer by a factor 6 and thus making possible overlap of electron orbitals. This explains also the anomalous conductivity of DNA containing 5- and 6-cycles (same reference).

1. Is dark matter warped?

The reduced light velocity could be due to the warping of the space-time sheet associated with dark electrons. TGD predicts besides gravitational red-shift a non-gravitational red-shift due to the warping of space-time sheets possible because space-time is 4-surface rather than abstract 4-manifold. A simple example of everyday life is the warping of a paper sheet: it bends but is not stretched, which means that the induced metric remains flat although one of its components scales (distance becomes longer around direction of bending). For instance, empty Minkowski space represented canonically as a surface of M⁴× CP₂ with constant CP₂ coordinates can become periodically warped in time direction because of the bending in CP₂ direction. As a consequence, the distance in time direction shortens and effective light-velocity decreases when determined from the comparison of the time taken for signal to propagate from A to B along warped space-time sheet with propagation time along a non-warped space-time sheet.

The simplest warped imbedding defined by the map M⁴→ S¹, S¹ a geodesic circle of CP₂. Let the angle coordinate of S¹ depend linearly on time: Φ= ω t. g_tt} component of metric becomes 1-R²ω² so that the light velocity is reduced to v₀/c=(1-R²ω²)^1/2. No gravitational field is present.

The fact that M⁴ Planck constant n_ah₀ defines the scaling factor n_a² of CP₂ metric could explain why dark matter resides around strongly warped imbeddings of M⁴. The quantization of the scaling factor of CP₂ by R²→ n_a²R² implies that the initial small warping in the time direction given by g_tt=1-ε, ε=R²ω², will be amplified to g_tt= 1-n_a²ε if ω is not affected in the transition to dark matter phase. n_a=6 in the case of graphene would give 1-x≈ 1- 1/36 so that only a one per cent reduction of light velocity is enough to explain the strong reduction of light velocity for dark matter.

2. Is c/v₀ quantized in terms of ruler and compass rationals?

The known cases suggests that c/v₀ is always a rational number expressible as a ratio of integers associated with n-polygons constructible using only ruler and compass.

1. c/v₀=300 would explain graphene. The nearest rational satisfying the ruler and compass constraint would be q= 5× 2¹⁰/17≈ 301.18.

2. If dark matter space-time sheets are warped with c₀/v=¹¹ one can understand Nottale's quantization for the radii inner planets. For dark matter space-time sheets associated with outer planets one would have c/v₀= 5× 2¹¹.

3. If Tifft's red-shifts relate to the warping of dark matter space-time sheets, warping would correspond to v₀/c=2.68× 10^-5/3. c/v₀= 2⁵× 17× 257/5 holds true with an error smaller than .1 per cent.

3. Tifft's quantization and cosmic quantum coherence

An explanation for Tifft's quantization in terms of Jones inclusions could be that the subgroup G of Lorentz group defining the inclusion consists of boosts defined by multiples η= nη₀ of the hyperbolic angle η₀≈ v₀/c. This would give v/c= sinh(nη₀)≈ nv₀/c. Thus the dark matter systems around which visible matter is condensed would be exact copies of each other in cosmic length scales since G would be an exact symmetry. The property of being an exact copy applies of course only in single level in the dark matter hierarchy. This would mean a delocalization of elementary particles in cosmological length scales made possible by the huge values of Planck constant. A precise cosmic analog for the delocalization of electron pairs in benzene ring would be in question.

Why then η₀ should be quantized as ruler and compass rationals? In the case of Planck constants the quantum phases q=exp(imπ/n_F) are number theoretically simple for n_F a ruler and compass integer. If the boost exp(η) is represented as a unitary phase exp(imη) at the level of discretely delocalized dark matter wave functions, the quantization η₀= n/n_F would give rise to number theoretically simple phases. Note that this quantization is more general than η₀= n_F,1/n_F,2.

For more details see the chapter TGD and Astro-Physics.

Orbital radii of exoplanets and Bohr quantization of planetary orbits

Orbital radii of exoplanets save as a test for the Bohr quantization of planetary orbits. Hundreds of them are already known and in tables (Masses and Orbital Characteristics of Extrasolar Planets using stellar masses derived from Hipparcos, metalicity, and stellar evolution) basic data for for 136 exoplanets are listed. The tables also provide references and links to sources giving data about the stars, in particular star mass M using solar mass M_S as a unit. Hence one can test the formula for the orbital radii given by the expression

r/r_E= (n²/5²) ×(M/M_S)× X ,

X= (n₁/n₂)², n_i=2^k_i× ∏_siF_si ,

F_si in the set {3,5,17,257, 2¹⁶+1} . Here a given Fermat prime F_si can appear only once.

It turns out that the simplest option assuming X=1 fails badly for some planets: the resulting deviations of of order 20 per cent typically but in the worst cases the predicted radius is by factor of ≈ .5 too small. The values of X used in the fit correspond to X having values in {(2/3)², (3/4)², (4/5)², (5/6)², (15/17)², (15/16)², (16/17)²} ≈ {.44, .56,.64,.69,.78, .88,.89} and their inverses. The tables summarizing the resulting fit using both X=1 and X giving optimal fit are here. The deviations are typically few per cent and one must also take into account the fact that the masses of stars are deduced theoretically using the spectral data from star models. I am not able to form an opinion about the real error bars related to the masses.

The appendix of the chapter TGD and Astrophysics of "Classical Physics in Many-Sheeted Space-Time" contains more details.

Magnetic bodies and hierarchy of Planck constants

The proposed hierarchy of Planck constants is given as h(M⁴_+/-)=n_ah₀ and h(CP₂)=n_bh₀, where the integers n_a and n_b correspond to orders of the maximal cyclic subgroups of groups G_a subset SU(2) subset SL(2,C) (Lorentz group) and G_b subset SU(2) subset SU(3) (color group). G_a defines covering of CP₂ by G_a related M⁴_+/- points and G_b covering of M⁴_+/- by G_b related CP₂ points. Fixed points correspond to orbifold points. The covariant metric of M⁴_+/- (CP₂) is proportional to n_b² (n_a²). Different copies of imbedding space are glued together along common M⁴_+/- or CP₂ factors to form an infinite tree with each noding containing infinitely many branches. The effective Planck constant appearing in Schrödinger equation is given by h_eff= (n_a/n_b)h₀.

The question concerns the interpretation of G_a. The first observation that apart from two exceptions, which correspond to symmetries of tedrahedron and dodecahedron, the group G_a acts in plane. G_a =Z_n consists of rotations by multiples of 2π/n and for G_a =D_2n rotations by 2π/2n combined with reflection. The orbit of D_2nwhich could be genuinely 3-dimensional decomposing to discrete orbits of Z_2n above and below plane. For small values of n say, n=5 and 6 orbits correspond to cycles and a highly attractive idea is that 5- and 6-cycles appearing in the fundamental bio-chemistry correspond to these orbits. Electron pairs are associated with 5- and 6-rings and the hypothesis would be that these pairs are in dark phase with n_a=5 or 6. Graphene which is a one-atom thick hexagonal lattice could be also an example of (conduction) electronic dark matter with n_a=6.

For some of the proposed physical applications the order n_a/n_b is very large. In particular, the Bohr quantization of planetary orbits, which stimulated the idea about dark matter as a large Planck constant phase, requires n_a/n_b= GMm/v₀, v₀=2^-11 so that the values are gigantic. A possible interpretation is in terms of a dark (gravi)magnetic body assignable to the system playing a key role in TGD inspired quantum biology. Topological quantization of magnetic field means a decomposition of the (gravi)magnetic field to flux tubes represented as space-time sheets. In the case of (say) dipole field the rotational and mirror symmetry with respect to the dipole axis would break down to Z_n or D_2n. This would also correspond to a transition to a phase with quantum group symmetry. Of course, the field body could contain also electric part consisting of radial flux tubes and obeying same symmetry. If star and each planet interact via a field body connecting only them, one could understand why the gravitational Planck constant depends on both M and m rather than being something universal.

Particles of dark matter would reside at the flux tubes but would be delocalized (exist simultaneously at several flux tubes) and belonging to irreducible representations of G_a. What looks weird is that one would have an exact macroscopic or even astroscopic symmetry at the level of generalized imbedding space. Visible matter would reflect this symmetry approximately. This representation would make sense also at the level of biochemistry and predict that magnetic properties of 5- and 6-cycles are of special significance for biochemistry. Same should hold true for graphene.

The chapter Does TGD Predict the Spectrum of Planck Constants? of "Towards S-matrix" contains more details.

Mersenne primes: connection between particle physics and quantum computation?

I found in the blog of Scott Aaronson an interesting piece about Mersenne primes and quantum computation: the title of posting is Why complexity is better than cannabis. I glue the text here.

Whether there exist subexponential-size locally decodable codes, and sub-n^ε-communication private information retrieval (PIR) protocols, have been major open problems for a decade. A new preprint by Sergey Yekhanin reveals that both of these questions hinge on -- wait for this -- whether or not there are infinitely many Mersenne primes. By using the fact (discovered a month ago) that 2^32,582,657-1 is prime, Yekhanin can already give a 3-server PIR protocol with communication complexity O(n^1/32,582,658), improving the previous bound of O(n^1/5.25). Duuuuuude. If you've ever wondered what it is that motivates complexity theorists, roll this one up and smoke it.

I do not pretend of understanding much about this statement but I have the dim gut feeling that the existence of infinite number of Mersenne primes would be wonderful for communications. In any case I can optimistically click the link and try to read the abstract.

1. At the first line of abstract I make a head on collision with "Q-query Locally Decodable Code (LDC)". LDC codes n-bit message to an N-bit code word C(x) such that one can probabilistically recover any bit of the message by querying only q bits of the code words, even after some constant fraction of bits have been corrupted. Error correction seems to be in question. Knowledge of these q bits allow to save all bits. "Locally decodable" has only heuristic meaning for me.

2. Authors assign to Mersenne primes M_t=2^t-1 (t is also prime) 3-query codes satisfying N=exp(n^-1/t) for all values of the bit number n. As Mersenne prime increases N approaches to unity. I guess that the equality is a shorthand for N=P_n(t)exp(n^-1/t), P_n(t) a polynomial.

The reason why I saw the trouble of trying to understand the abstract is that Mersenne primes are in a preferred position in the p-adic world order and p-adic physics is physics of cognition in TGD Universe. I have also discussed generalizations of genetic code at my own primitive level of understanding, in particular codes associated with Mersenne primes and Combinatorial hierarchy M(n)= M_M(n-1) starting with 3,2³-1=7,2⁷-1=127,2¹²⁷-1,?.. defining a kind of abstraction hierarchy of Boolean statements about statements about.....with one statement thrown away at each step. Only the listed candidates are known to be primes. Hilbert has conjectured that the hierarchy continues indefinitely.

The original discovery was that the mass scale ratios for elementary particles seem to be near to ratios of Mersenne primes. Eventually this led to p-adic mass calculations where the prime p characterizing p-adic number field characterizes elementary particle mass scale. It turns out that quite many elementary particles correspond to Mersennes or Gaussian Mersennes (also primes p near 2^k, k prime or even integer, are possible). In particular, Mersenne primes label bosons mediating fundamental interactions: electro-weak gauge bosons correspond to M₈₉, M₁₀₇ to gluons, and gravitons can be assigned to M₁₂₇ defining the largest non-super-astronomical p-adic length scale of this kind.

Also Gaussian Mersennes are interesting. The length scale range between 10 nm (cell membrane thickness) and 2.5 micrometers (size of small cell) contains four(!) Gaussian Mersennes G_k = (1+i)^k-1, k=151,157,163,167. All primes k in this range define Gaussian Mersennes. This number theoretic miracle suggests that also Gaussian Mersennes are crucial for communications and especially so in living matter!

The heuristics is that Mersenne primes and their Gaussian cousins are winners in the number theoretical fight for survival. Maybe the ability to communicate reliably with minimum number of "parity bits" is the key to the success. Mersennes and possibly Fermat primes (very few of latter are known or might even exist) would be preferred since they are so near to powers of two. If the number of Mersenne primes is infinite, TGD would predict infinite hierarchy of exotic bosons and maximally interesting Universe. TGD inspired theory of consciousness suggests that real fermionic partons and their p-adic counterparts form particle and its cognitive representation so that cognition, information processing, and communications would be present already at elementary particle level. Exciting!

Some thoughts inspired by certain blog posting

Lubos represented quite an interesting series of arguments (Googolplex of e-foldings relating to the arrow of time, second law, initial values in cosmology, anthropic principle, etc... Even God was mentioned and I thought to bring in also Angels;-).

1. Why we need a proper theory of consciousness?

I agree full-heartedly with Lubos that our arguments (we, the physicists;-)) should be independent of our personal idiosyncracies, the details of our particular species, our culture, etc... It seems however that to achieve this freedom, "we" should have some kind of abstract overview about all possible species and all possible cultures. The only hope to achieve this kind of rough overview is by extending physics to a theory of consciousness allowing to say something very universal about life and intelligence. I am a little bit surprised how few theoreticians are taking this challenge seriously although the quantum consciousness movement began already at eighties (Esalem, Finkelstein). Perhaps super string models caught the attention of the most imaginative brains at that time.

2. Geometric time and experienced time

A proper theory of consciousness could remove a lot of mist floating around the concept of time. Experienced time and geometric time are very different things but for some odd reason most physicists stubbornly continue to identify them. The paradoxes are unavoidable. For instance, Lubos mentions the paradox implied by "Now as the maximum of entropy" assumption made by some cosmologists.

I think that the lack of understanding about the relation between experienced time (or "de-coherence arrow of time") and geometric time is the main source of confusion and a treasure trove of misleading arguments. If one accepts that these two times are different notions, one must among other things ask whether both of these times have an arrow. I would guess "Yes": by quantum classical correspondence space-time geometry should provide a representation for the sequence of quantum jumps and therefore for the contents of consciousness (in my own theory world). Note that this requires the failure of strict classical determinism for the dynamics of space-time surfaces and can serve as a valuable guideline. The directions of these arrows of time could in principle be independent. Indeed, for phase conjugate laser beams the arrow of geometric time and de-coherence time seem to be opposite. Self-assembly in living systems would be a decay in a reverse direction of geometric time and consistent with the second law a deeper level.

3. The issue of initial conditions

Lubos talked also about the issue of initial conditions and the new view about time allows a fresh approach to the problem. First of all, I would talk about boundary conditions in the geometric past since geometric time is in the same role as spatial dimensions in any relativistic theory. Secondly, the idea about the initial/boundary conditions as something fixed by God at the moment of creation carries a flavor of medieval theology. If theorist must fix initial conditions by some educated guess of what medieval God willed, most of the predictive power of theory is lost, and in principle the theory becomes intestable since one cannot compare the time evolutions of different classical worlds. It is somewhat surprising that theoreticians still use this medieval conceptualization. Perhaps an introductory course to philosophical problems of physics without professional jargon and in science fictive spirit would help.

The notion of boundary conditions at big bang has different interpretation if one accepts some amount of consciousness theory. Suppose that you believe that conscious existence is a sequence of quantum jumps replacing the quantum superposition of classical four-worlds (analogous to Bohr orbits) with a new one (what these quantum jumps are, is of course a non-trivial question). The key observation is that the average boundary/initial conditions defined by the quantum superposition change in every quantum jump. Quantum jump re-creates also the geometric past. Quantum jump sequences are expected to lead to asymptotic self organization patterns and they could correspond to rather specific boundary conditions in the geometric past (somewhat misleadingly, "moment of big bang"). The basin of attractor leading asymptotically to a particular effective cosmological history becomes a more natural notion.

In this framework the victories of anthropic principle could be interpreted differently. Asymptotic self organization patterns resulting in quantum jump sequences correspond to very specific values of coupling constant parameters of the theory. If one also accepts that quantum jump sequence gives rise to as genuine evolution then these values would favor intelligent life in some form. Genuine evolution would however require a new kind of variational principle. There is room for this kind of principle since there must be some law telling what it is possible to tell about dynamics of quantum jumps defining the dynamics of conscious existence. The first guess for the principle would be that the amount of conscious information in quantum jump is maximized (Negentropy Maximization Principle). The task would be to show that this principle is not conflict with the second law.

4. Universe or Universes?

Lubos talks about Universe as the only one of its kind and having age of about 13.7 billion years. I would talk in plural. This particular universe of ours would be a self-organization pattern extending 13.7 billion years to the geometric past. My motivation is that if one accepts space-time as a 4-D surface, one cannot avoid a profound generalization of space-time concept to what I call many-sheeted space-time. As the first step you end up with what you might call Russian doll cosmology. This idea can be probably expressed also in branish although I personally would prefer plain english.

5. What is space-time correlate for mind stuff?

Many-sheeted space-time is not all that is needed. Number theoretical vision about physics as something which is number theoretically universal leads to the introduction of infinite number p-adic variants of real number based physics requiring the fusion of real and p-adic variants of imbedding space together along algebraic points. This allows the identification of space-time correlates of cognition and intentions, the mind stuff of Descartes. But even this is not enough.

6. Dark matter and quantized Planck constant

Lubos did not mention dark matter at all. Perhaps he believes that dark matter is just some exotic particle. I do not believe this. Here I must introduce some more TGD related stuff. The basic dogma of quantum physics is still that Planck constant is a universal constant, just a conversion factor which can be taken to be hbar=1 in suitable units and nothing else. History has not respected the constancy of fundamental constants so that one has right to ask whether Planck constant is really a universal constant, could it be quantized, and if so, how.

Associating Planck constant with dark matter (there are also empirical motivations to do so) one could ask where there could exist a hierarchy of dark matters with levels labelled by different values of Planck constant. I believe that this is possible. At more technical level I propose that the hierarchy of dark matters corresponds to the hierarchy of Jones inclusions for hyper-finite factors of type II₁ labelled by ADE groups appearing in McKay correspondence. The motivation comes from the observation that the infinite-dimensional Clifford algebra of the world of classical worlds is hyper-finite factor of type II₁, and from the vision that entire TGD emerges in some sense from this kind of structure.

Topologically/geometrically the hierarchy of Planck constants requires a considerable generalization of the notion of 8-D imbedding space (counterpart of 10-D target space in super string models) since one cannot allow quantized Planck constant in quantum theory based on the standard view about space-time. Very loosely, different Planck constants correspond to different branches of imbedding space replaced with a tree like structure obtained by gluing various copies of imbedding space together along common M⁴ or CP₂ factor. In this framework dark matter at a given level of hierarchy can quantum control the lower levels. This control hierarchy would make the matter living. Even phase transition changing Planck constants and identified as leakage between different branches becomes a well defined notion.

7. Angels and Gods and physicist

The predicted infinite hierarchy of conscious entities identified as levels of dark matter would be the physicists counterpart for the angels and spirits of religious world views. The core of religious world view is that there exists something better than this cruel everyday world and that we can directly experience this something now and then. Dark matter hierarchy could give a justification for this belief. The Universe itself would take the role of a dynamical God re-creating itself again and again ("moment of mercy" instead "updating" is perhaps a more proper word here;-)). Our recent physics would be only a humble beginning. A new period of voyages of discovery to the higher, more spriritual, levels of hierarchy using the methods of empirical science would be waiting for us whereas anthropic principle would define a foolproof recipe for the end of physics.

The truth about quantization of Planck constants

The development of ideas about quantization of Planck constants has been a trial-and-error process which only a person like me possessing an exceptionally fuzzy and pathologically associative brain function could go through. I however try to be honest and I can convinve the reader that there is no danger that reading the scandalous truth could infect the brain of reader with the same intolerable fuzziness.

The moment of truth

1. The big idea was that the claims for the Bohr quantization of planetary radii with a gigantic value of Planck constant (called cautiously gravitational Planck constant at that stage) could be understood as a genuine quantization of Planck constant for dark matter serving as a template for ordinary matter. Recall that Nottale had already proposed effective quantization based on hydrodynamics. As a warm up exercise I went on to propose a rather complex and wrong formula for the gravitational Planck constant in terms of Beraha numbers B_n= 4cos²(π/n) assignable to quantum phases q=exp(iπ/n). The inspiration came from Jones inclusions which I believed to relate closely to the quantization.

2. Soon I realized that the formula for Planck constant does not really work. Anyonic arguments based on the idea that Riemann surface like coverings of M⁴ are involved, led to an extremely simple formula for h as h=n×h₀.

3. It became also clear that as far Lie algebras of symmetries are considered, there are actually two Planck constants corresponding to Lie-algebra commutators associated with M⁴_+/- and CP₂ degrees of freedom and that these Planck constants would correspond to Jones inclusions of Clifford subalgebras identifiable in terms of gamma matrix algebras for the world of classical worlds consisting of 3-surfaces in H. These inclusions are characterized by subgroups G_a of SU(2) subset SL(2,C) and G_b of SU(2) subset SU(3). These discrete groups define orbifold coverings of M⁴_+/- by G_b related CP₂ points and vice versa so that a generalization of the notion of imbedding space emerges.

   It also became clear that one must glue various copies of imbedding space along M⁴_+/- in the case that G_b is same for two copies and vice versa. This generalization was easy to discover since also the p-adic variants of the imbedding space are glued together in same manner along common rational and algebraic points making it possible to fuse real and various p-adic physics to single coherent whole. Now this fusion has reached a rather concrete form and involves a lot of fascinating number theory, in particular Riemann Zeta.

   To be honest, I tried to cheat here;-). It took some time to realize that strictly speaking it is M⁴_+/- rather than M⁴ whose coverings are involved but I could argue that this is just an inaccurate use of language. Sincerely, in the middle of a flood of ideas rusing through your head you do not notice this kind of details.

4. It became also clear that the covariant metric of M⁴_+/- must be proportional to n_b² and that of CP₂ proportional to n_a². Here n_a is the order of maximal cyclic subgroup of G_a and n_b the order of maximal cyclic subgroup of G_b.

5. By an anyonic argument Planck constants are given by h(M⁴_+/-)=n_a h₀. For some time I however believed that h(M⁴_+/-)=n_b h₀ so that Schrödinger equation would be invariant under phase transition changing the Planck constants in an obvious contradiction with the the quantization of planetary orbits requiring gigantic Planck constant(!).

6. There was also the characteristic fuzziness in the identification of the observed Planck constant. As one might guess, I made first the wrong identification as h(M⁴_+/-) although I had realized from the beginning that only the ratio n_a/n_b appears in the Kähler action and that the natural interpretation for the nonlinearity is a universal geometric coding of radiative corrections to the induced geometry of the space-time sheet via the nonlinear dependence on the induced metric. Therefore the only sensible conclusion would have been that the observed Planck constant is given by h_eff/h₀= n_a/n_b! This predicts that Planck constant can in principle have all rational values: both larger and smaller than the value for the ordinary matter.

7. In the middle of this flux of good and bad ideas emerged also the hypothesis that the preferred values of n_a and n_b correspond to integers defining n-polygons constructible using ruler and compass alone. The motivation was that quantum phases are expressible in this case using only iterated square rooting of rationals so that number theoretically (and thus cognitively) simple levels of hierarchy of algebraic extensions of p-adic number fields expected to be abundant in cosmos would be in question. One would have n_F= 2^k ∏_s;F_s, where F_s =2^2s +1 are Fermat primes. The known Fermat primes are 3,5,17,257, and 2¹⁶+1. This would mean that the preferred values of h_eff are given as ratios of these integers. In living matter the fractal hierarchy n_F=2^11k seems to be favored and the number 2¹¹ corresponds to a fundamental dimensionless constant in TGD.

And now a desperate attempt to defend myself

After having revealed this scandalous multiple blundering process which has lasted two years I still dare to hope that reader has not made her final conclusions and is willing to listen my excuses. I sincerely hope that some examples about how the quantization of Planck constants could manifest itself in physics anomalies might induce merciful feelings also in the readers who identify themselves as serious scientists.

1. The evidence for Bohr quantization of planetary orbits can be interpreted in terms of a huge value of h_eff= GMm/v₀, where M and m are the masses of, say, Sun and planet (for Nottale's orginal paper explaining quantization hydrodynamically see astro-ph/0310036). The ruler-compass hypothesis means very strong constraints on the ratios of planetary masses satisfied with an accuracy of 3 per cent and also on solar mass satisfied if the fraction of non-dark matter is around 4 per cent. A dramatic prediction is that in this phase the value of n_a is gigantic meaning that dark matter obeys spatial symmetry corresponding to either the cyclic group Z_na or group obtained by adding planar reflection to it. Ring like structures of dark matter analogous to ring like structures analogous to benzene ring in chemistry perhaps assignable to planetary orbits suggest themselves (see this).

   An even more dramatic prediction is that the scaling of CP₂ metric by n_a means that it has astrophical size so that in dark sector imbedding space looks like uncompactified M⁸ in human length scales! But this is dark sector and since the mass spectra of elementary particles do not depend at all on the values of Planck constants no obvious contradictions with observations are predicted! Contrary to what super string model suggest, big hyper-space dimensions would not be seen in particle accelerators but in astrophysics.

2. Hierarchy of scaled variants of atomic physics

The binding energy scale of hydrogen atom is proportional to 1/h_eff²= (n_b/n_a)² so that a fractionization of occurs.

1. The findings of Mills about fractionization of energy spectrum of hydrogen atom (hydrino atom) with scaling factor k=2,3,4,5,6,7,9,10 can be understood for k= n_b/n_a. For some time ago I constructed a model for hydrino using q-Laquerre equation and predicting k=2 as a new state. Also k> 2 result approximately. It seems that these states could serve as intermediate states in the transition to a phase with modified Planck constants. In this case effective Planck constant is smaller than its standard value and the sizes of hydrino atoms are smaller. Also zoomed up versions of ordinary atoms with identical chemical properties but sizes scaled up by n_a=n_b are predicted.

2. Exotic atoms with increased sizes and reduced binding energy scale are predicted and the integers n_F are especially interesting. Atomic nucleus can be or ordinary and one obtains N-atoms by putting electron on several sheets of n_a-fold covering of CP₂. This leads to a model for hydrogen bonds and active catalyst sites and for how symbolic level emerges in bio-chemistry. It is also possible that only the valence electrons of ordinary atom are in dark phase (live at different branch of generalized imbedding space) so that only these electronic orbitals are scaled up. This could make possible anomalous conductivity and even super-conductivity.

   For instance, the 5- and 6-rings characteristic for the fundamental bio-molecules (sugars, DNA, important neurotransmitters including those containing four aminoacids having 5- and 6-cycles, hallucinogens) could correspond to n_a= 5 or 6 for free electron pairs characterizing these rings. The mysterious conductivity of DNA (Science (1997), vol. 275, 7. March 1997) could be understood in terms of the delocalization of the aromatic electron pairs associated with the 5- and 6-rings due to n_a² fold scaling of the orbitals making possible overlap between the rings in the DNA ladder (see this).

3. The weird looking properties of graphene (in particular its high conductivity) forming hexagonal carbon atom lattice of thickness of single atom could be understood if one has n_a=n_b=6 for the free electron pairs assignable to Carbon rings. TGD based model for particle massivation implying that conformal weight and thus mass squared is additive for hadron type bound states of partons can also explain neatly why conduction electrons behave as massless particles. The mass of bound state parton is m²- p_T² and transverse momentum squared p_T² can compensate the mass of quark/electron completely. This could also explain why massive quarks seem to behave as massless particles inside hadrons. TGD also predicts that warped vacuum imbedding without gravitational fields can induce anomalous time dilation and large reduction of effective light velocity: this could explain why light velocity for these electrons is c/300 (see this).

3. Anomalous properties of water

   The anomalous properties of water, in particular the chemical formula H_1.5O suggesting itself in attosecond scale, provided first challenge for the proposed model of dark matter (see this). Tedrahedral and icosahedral clusters are characteristic for water and the corresponding subgroups of SU(2) correspond to the exceptional Lie groups E₆ and E₈ via ADE correspondence: these are the only genuinely 3-dimensional discrete subgroups of SU(2). The value of Planck constant would be scaled up by a factor n_a=3 or 5 for these sub-groups and n_a=5 is the minimal value making possible topological quantum computation using braid S-matrix. Icosahedral and dual odecahedral structures are abundant in living matter: viruses being only one example.

4. Mono-atomic elements

   Mono-atomic elements or ORMEs (see this) are transition elements claimed by Hudson to have strange properties. They are claimed to be non-visible in ordinary emission spectroscopy but become visible after a time which is 90 s instead of 15 seconds for ordinary elements in typical case. The ratio of times is 6 and Golden rules suggests that one has n_a=6 at least for valence electrons. Chemistry would not be affected if one has n_b=6 too. The atomic clusters are predicted to have hexagonal symmetry. The other strange properties of these compounds such as claimed super-conductivity suggest that also the phase with n_b=1 is present as indeed required by the general scenario for phase transitions changing the values of Planck constants as a leakage between different sectors of the imbedding space.

5. Dark EEG hierarchy

   In dark phase the energy associated with a photon of given frequency is scaled up by a factor n_a. If n_a is large enough, even EEG photons can have energies above thermal energy. The finding that ELF photons with frequences which are harmonics of cyclotron frequencies for biologically important ions have effects on vertebrate brain supports this idea. This observation leads to a model for a fractal hierarchy of EEGs based on the assumption that the integers n_a=2^11k are especially favored: the motivation is that 2¹² corresponds to a fundamental dimensionless constant in TGD. Even individual narrow peaks in beta and theta bands are predicted correctly (see this).

The chapter Does TGD Predict the Spectrum of Planck Constants? of "Towards S-matrix" explains in detail the recent view about the quantization of Planck constants.

Comments about p-adic mass calculations

I have been reformulating basic quantum TGD using partonic formulation based on light-like 3-surfaces identifiable as parton orbits. This provides a precise and rigorous identification of various conformal symmetries which have been previously identified as mathematical necessities. Also concrete geometric picture emerges by using quantum classical correspondence. This kind of reformulation of course means that some stuff appears to be obsolete or simply wrong.

1. About the construction of physical states

The previous construction of physical states was still far from complete and involved erraneous elements. The partonic picture confirms however the basic vision. Super-canonical Virasoro algebra involves only generators L_n, n<0, and creates tachyonic ground states required by p-adic mass calculations. These states correspond to null states with conformal weight h<0 and annihilated by L_n, n<0. The null state property saves from an infinite degeneracy of ground states and thus also of exotic massless states. Super-canonical generators and Kac-Moody generators applied to this state give massless ground state and p-adic thermodynamics for SKM algebra gives mass squared ientified as the thermal expectation of conformal weight. The non-determinism of almost topological parton dynamics partially justifies the use of p-adic thermodynamics.

The hypothesis that the commutator of super-canonical and SKM algebras annihilates physical states seems attractive and would define the analog of Dirac equation in the world of classical worlds and eliminate large number of exotic states.

2. Consistency with p-adic thermodynamics

The consistency with p-adic thermodynamics provides a strong reality test and has been already used as a constraint in attempts to understand the super-conformal symmetries at the partonic level. In the proposed geometric interpretation inspired by quantum classical correspondence p-adic thermal excitations could be assigned with the curves ζ(n+1/2+iy) at S²subset CP₂ for CP₂ degrees of freedom and S² subset δ M⁴_+/- for M⁴ degrees of freedom so that a rather concrete picture in terms of orbits of harmonic oscillator would result.

There are some questions which pop up in mind immediately.

1. The most crucial consistency test is the requirement that the number of SKM sectors is N=5 to yield realistic mass spectrum. The SKM sectors correspond to SU(3)× SO(3)× E² isometries and to SU(2)_L× U(1) electro-weak holonomy algebra having only spinor realization. SO(3) holonomy is identifiable as the spinor counterpart of SO(3) rotation. If E² can be counted as a single sector rather than two (SO(2)subset SO(3) acts as rotations in E² sector) the number of sectors is indeed 5.

2. Why mass squared corresponds to the thermal expectation value of the net conformal weight? As already explained this option is forced among other things by Lorentz invariance but it is not possible to provide a really satisfactory answer to this question yet. The coefficient of proportionality can be however deduced from the observation that the mass squared values for CP₂ Dirac operator correspond to definite values of conformal weight in p-adic mass calculations. It is indeed possible to assign to the center of mass of partonic 2-surface X² CP₂ partial waves correlating strongly with the net electro-weak quantum numbers of the parton so that the assignment of ground state conformal weight to CP₂ partial waves makes sense. In the case of M⁴ degrees of freedom it is not possible to talk about momentum eigen states since translations take parton out of δ H_+/- so that momentum must be assigned with the tip of the light-cone containing the particle and serving the role of argument of N-point function at the level of particle S-matrix.

3. The additivity of conformal weight means additivity of mass squared at parton level and this has been indeed used in p-adic mass calculations. This implies the conditions

   (∑_i p_i)²= ∑_i m_i²

   The assumption p_i²= m_i² makes sense only for massless partons moving collinearly. In the QCD based model of hadrons only longitudinal momenta and transverse momentum squared are used as labels of parton states, which would suggest that one has

   p_i,II² = m_i² , -∑_i p_i,perp² +2∑_i,j p_i· p_j=0 .

   The masses would be reduced in bound states: m_i²→ m_i²-(p_T²)_i. This could explain why massive quarks can behave as nearly massless quarks inside hadrons. Conduction electrons in graphene behave as massless particles and dark electrons forming hadron like bound states (say Cooper pairs) could be in question.

4. Single particle conformal weights can have also imaginary part and if only sums y=∑_kn_ky_k, n_k≥ 0, are allowed, y is always rather sizable in the scale for conformal weights. The question is what complex mass squared means physically. Complex conformal weights have been assigned with an inherent time orientation distinguishing positive energy particle from negative energy antiparticle (in particular, phase conjugate photons from ordinary photons). This suggests an interpretation of y in terms of a decay width. p-Adic thermodynamics suggest that the measured value of y is a p-adic thermal average. This makes sense if the values of y_k are algebraic (or perhaps even rational) numbers as the sharpening of Riemann Hypothesis states and the number theoretically universal definition of Dirac determinant requires. The simplest possibility is that y does not depend on the thermal excitation so that the decay width would be characterized by the massless state alone. Perhaps a more reasonable option is that y characterizes the decay rates for massive excitations and is in principle calculable.

   For instance, if a massless state characterized by p-adic prime p has y=p× s y_k, where s is the denominator of rational valued y_k=r/s, the lowest order contribution to the decay width is proportional to 1/p by the basic rules of p-adic mass calculations and the decay rate is of same order of magnitude as mass. If y is of form pⁿy_k for massless state then a decay width of order Γ≈ p^(n-1)/2m results. For electron n should be rather large. This argument generalizes trivially to the case in which massless state has vanishing value of y.

The chapter Massless states and Particle Massivation of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy" contains a more detailed about the topic.

About the identification of Kac Moody algebra and corresponding Virasoro algebra

It is relatively straightforward to deduce the detailed form of the TGD countetpart of Kac-Moody algebra identified as X³-local infinitesimal transformations of H__+/-=M⁴_+/-× CP₂ respecting the lightlikeness of partonic 3-surface X³. This involves the identification of Kac-Moody transformations and corresponding super-generators carrying now fermion numbers and anticommuting to a multiple of unit matrix. Also the generalization the notions of Ramond and N-S algebra is needed. Especially interesting is the relationship with the super-canonical algebra consisting of canonical transformations of δ M⁴_+/-× CP₂.

1. Bosonic part of the algebra

The bosonic part of Kac-Moody algebra can be identified as symmetries respecting the light-likeness of the partonic 3-surface X³ in H=M⁴_+/-× CP₂. The educated guess is that a subset of X³-local diffeomorphisms of is in question. The allowed infinitesimeal transformation of this kind must reduce to a conformal transformation of the induced metric plus diffeomorphism of X³. The explicit study of the conditions allows to conclude that conformal transformations of M⁴_+/- and isometries of CP₂ made local with respect to X³ satisfy the defining conditions. Choosing special coordinates for X³ one finds that the vector fields defining the transformations must be orthogonal of the light-like direction of X³. The resulting partial differential equations fix the infinitesimal diffeomorphism of X³ once the functions appearing in Kac-Moody generator are fixed. The functions appearing in generators can be chosen to proportional to powers of the radial coordinate multiplied by functions of transversal coordinates whose dynamics is dictated by consistency conditions

The resulting algebra is essentially 3-dimensional and therefore much larger than ordinary Kac-Moody algebra. One can identify the counterpart of ordinary Kac-Moody algebra as a sub-algebra for which generators are in one-one correspondence with the powers of the light-like coordinate assignable to X³. This algebra corresponds to the stringy sub-algebra E²× SO(2)×SU(3) if one selects the preferred coordinate of M⁴ as a lightlike coordinate assignable to the lightlike ray of δ M⁴+/- defining orbifold structure in M⁴_+/- ("massless" case) and E³× SO(3)×SU(3) if the preferred coordinate is M⁴ time coordinate (massive case).

The local transformation in the preferred direction is not free but fixed by the condition that Kac-Moody transformation does not affect the value of the light-like coordinate of X³. This is completely analogous to the non-dynamical character of longitudinal degrees of freedom of Kac-Moody algebra in string models.

The algebra decomposes into a direct sum of sub-spaces left invariant by Kac-Moody algebra and one has a structure analogous to that defining coset space structure (say SU(3)/U(2)). This feature means that the space of physical states is much larger than in string models and Kac Moody algebra of string models takes the role of the little algebra in the representations of Poincare group. Mackey's construction of induced representations should generalized to this situation.

Just as in the case of super-canonical algebra, the Noether charges assignable to the Kac-Moody transformations define Hamiltonians in the world of classical worlds as integrals over the partonic two surface and reducible to one-dimensional integrals if the SO(2)× SU(3) quantum numbers of the generator vanish. The intepretation is that this algebra leaves invariant various quantization axes and acts as symmetries of quantum measurement situation.

2. Fermionic sector

The zero modes and generalized eigen spinors of the modified Dirac equation define the counterparts of Ramond and N-S type super generators.

The hypothesis inspired by number theoretical conjectures related to Riemann Zeta is that the eigenvalues of the generalized eigen modes associated with ground states correspond to non-trivial zeros of zeta. Also non-trivial eigenvalues must be considered.

1. Neveu-Schwartz type eigenvalues which are expressible as λ=1/2+i∑_kn_ky_k, where s_k=1/2+iy_k is zero of Riemann zeta. Higher Virasoro excitations would correspond to conformal weights λ=n+1/2+i∑_kn_ky_k.

2. Zero modes correspond naturally Ramond type representations for which the ground state conformal weight vanishes so that a zero mode (solution of the modified Dirac equation is in question) and higher conformal weights would be integer valued.

3. If one accepts non-trivial zeros as generalized eigenvalues one would have additional Ramond type representations with a tachyonic ground state conformal weight lambda= -2n, n>0.

Thus N-S type ground state conformal weights would involve also imaginary part and this has an interpretation in terms of an inherent arrow of time associated with particles distinguishing positive energy particle propagating to the geometric future from negative energy particle propagating to geometric past. p-Adic mass calculations suggest that y could characterize the decay width of the particle. The action of Kac-Moody generator to the state can be defined and affects the conformal weight in the expected manner.

3. Super Virasoro algebras

The identification of SKM Virasoro algebra as that associated with radial diffeomorphisms is obvious and this algebra replaces the usual Virasoro algebra associated with the complex coordinate of partonic 2-surface X² in the construction of states and mass calculations. This algebra does not not annihilate physical states and this gives justification for p-adic thermodynamics. The commutators of super canonical and super Kac-Moody (and corresponding super Virasoro) algebras would however annihilate naturally the physical states. Four-momentum does not appear in the expressions for the Virasoro generators and mass squared is identified as p-adic thermal expectation value of conformal weight. There are no problems with Lorentz invariance.

One can wonder about the role of ordinary conformal transformations assignable to the partonic 2-surface X². The stringy quantization implies the reduction of this part of algebra to algebraically 1-D form and the corresponding conformal weight labels different radial SKM representations. Conformal weights are not constants of motion along X³ unlike radial conformal weights. TGD analog of super-conformal symmetries of condensed matter physics rather than stringy super-conformal symmetry would be in question.

The last sections of the chapter Construction of Quantum Theory: Symmetries of "Towards S-matrix" and the chapter The Evolution of Quantum TGD of "TGD: an Overall View" give detailed summary of the recent picture.

Comment to Not-Even-Wrong

Below a comment inspired by one of the comments of Bert Schroer in Not-Even-Wrong Bert Schroer's comment in Not-Even-Wrong relating to a discussion about Maldacena conjecture.

Bert Schroer Says:

October 8th, 2006 at 4:15 pm

I repeat, there is the maldacena conjecture about a relation between a duality relation between a N–> infinity gauge theory and some form of 5-dim. gravity and there is a mathematical theorem about a AdS–CFT holography (or better correspondenc) and both have, according to our best knowledge nothing to do with each other. From a conceptual point of view the correspondence is rather trivial because the same substrate of matter is only changing its spacetime encoding (for more details see may essay) and olthough the physical interpretation changes it does not change miraculously (e.g. a spin 2 particle must have been there already on the CFT side). If the more than 400 people who worked on this problem would in addition to their computations have leaned back a while and looked at the published theorem may be we would have known by now what the Maldacena conjecture really mean conceptually. But I would predict that nobody at this late time will do this, the right time has passed. Certainly I would not loose time on such a physically fruitless project, but on the other hand as mathematical physicist I find this change of spacetime encoding in holographic projections very interesting; most interesting if the smaller spacetime is not a brane but rather a null-surface (see my last section and the cited literature).

I added the bold face at the end. My comment is here.

Some comments to Bert Schroer about null surfaces, or lightlike surfaces, as I have used to call them.

3-D lightlike surfaces in 4-D space-time, which itself is a surface in 8-D space-time H=M⁴×CP₂, can be seen as fundamental quantum dynamical objects in Topological Geometrdynamics. They are identified as parton orbits. The effective metric 2-dimensionality with ensuing super-conformal symmetries makes D=4 as a space-time dimension unique.

The infinitesimal transformations respecting null surface property form a Kac-Moody type algebra of conformal transformations of H localized with respect to X³ decomposing to representations of 1-D Kac Moody algebra.

The cones H_+/-= δ M⁴_+/-×CP₂ are also crucial for the formulation of theory and the 3-D lightlikeness of δ M⁴_+/- makes possible super-conformal symmetries of new kind based on canonical algebra of H_+/- and its super-counterpart. General Coordinate Invariance predicts quantum holography at level of H_+/-apart from effects implied by the failure of the complete classical determinism of the classical theory.

The resulting theory at fundamental parton is almost (absolutely important physically!) topological CFT defined by Chern-Simons action for Kähler gauge potential of CP₂ projected to X³. The second quantized fermionic counterpart of C-S action is fixed by the requirement of super-conformal symmetry. The theory allows N=4 super-conformal symmetries of various kinds broken for lightlike 3-surfaces which are not extremals of C-S action (have CP₂ projection with dimension D>3). No space-time (Poincare) super-symmetries and thus no sparticles are predicted.

Super-symmetrization of super-canonical algebra is possible for a sub-algebra of superconformal symmetries for which Noether charges defined as 2-D integrals over partonic 2-surface reduces to 1-D integrals as duals of closed 2-forms. The (super-)Hamiltonians of this sub-algebra have vanishing spin and color quantum numbers and thus leave invariant the choice of various quantization axis.

The vertices of the theory are described by almost topological having stringy character. Correlations between partons (propagators) involve interior dynamics determined by a vacuum functional defined as a determinant of the Dirac operator and assumed to reduce to an exponent of Kähler action for absolute extrema playing the role of Bohr orbits for particles identified as 3-surfaces: this would be quantum holography at the level of space-time surface. Interior dynamics of space-time surface codes for non-quantum fluctuating classical observables allowing to realize quantum measurement theory at fundamental level. There would be thus a direct connection between quantum holography and quantum measurement theory.

The mathematical methods of string theory can be applied to TGD and one can see the target space of string theories as a fictive concept associated with the vertex operator construction assigning to the Cartan algebra of Kac-Moody algebra a target space. In TGD framework spontaneous compactification can be seen only as an ad hoc attempt to give physical content to the theory.

For more details see my blog and homepage, in particular What's New sections to get view about the recent rapidly evolving situation in TGD.

With Best Regards,

Matti Pitkanen

Quantization of the modified Dirac action

The modified Dirac action for the light-like partonic 3-surfaces is determined uniquely by the Chern-Simons action for the induced Kähler form (or equivalently classical induced color gauge field possessing Abelian holonomy) and by the requirement of super-conformal symmetry. This action determines quantum physics of TGD Universe at the fundamental level. The classical dynamics for the interior of space-time surface is determined by the corresponding Dirac determinant. This classical dynamics is responsible for propagators whereas stringy conformal field theory provides the vertices. The theory is almost topological string theory with N=4 super-conformal symmetry.

The requirement that the super-Hamiltonians associated with the modified Dirac action define the gamma matrices of the configuraion space in principle fixes the anticommutation relations for the second quantized induced spinor field when one notices that the matrix elements of the metric in the complexified basis for super-canonical Killing vector fields of the configuration space ("world of classical worlds") are simply Poisson brackets for complexified Hamiltonians and thus themselves bosonic Hamiltonians. The challenge is to deduce the explicit form of these anticommutation relations and also the explicit form of the super-charges/gamma matrices. This challenge is not easy since canonical quantization cannot be used now. The progress in the understanding of the general structure of the theory however allows to achieve this goal.

1. Two options for fermionic anticommutators

The first question is following. Are anticommutators proportional

1. to 2-dimensional delta function as the expression for the bosonic Noether charges identified as configuration space Hamiltonians would suggest, or
2. to 1-dimensional delta function along 1-D curve of partonic 2-surfaces conformal field theory picture would suggest.

For the full super-canonical algebra the 1-D form is certainly impossible and the question is under which restriction on isometry Hamiltonians they reduce to duals of closed but in general non-exact 2-forms expressible in terms of 1-form analogous to a vector potential of a magnetic field.

It turns out that stringy option is possible if the Poisson bracket of Hamiltonian with the Kähler form of δ M⁴×CP₂ vanishes. The vanishing states that the super-canonical algebra must commute with the Hamiltonians corresponding to rotations around spin quantization axis and quantization axes of color isospin and hypercharge. Therefore hese quantum numbers must vanish for allowed Hamiltonians and super-Hamiltonians acting as symmetries. This brings strongly in mind weak form of color confinement suggested also by the classical theory (the holonomy group of classical color gauge field is Abelian).

The result has also interpretation in terms of quantum measurement theory: the isometries of a given sector of configuration space corresponding to a fixed selection of quantization axis commute with the basic measured observables (commuting isometry charges) and configuration space is union over sub-configuration spaces corresponding to these choices.

It is possible to find the explicit form of super-charges and their anticommutation relations which must be also consistent with the huge vacuum degeneracy of the bosonic Chern-Simons action and Kähler action.

2. Why stringy option is so nice?

An especially nice outcome is that string has purely number theoretic interpretation. It corresponds to the one-dimensional set of points of partonic 2-surface for which CP₂ projection belongs to the image of the critical line s=1/2+iy containing the non-trivial zeros of ζ at the geodesic sphere S₂ of CP₂ under the map s→ ζ(s).

The stimulus that led to the idea that braids must be essential for TGD was the observation that a wide class of Yang-Baxter matrices can be parametrized by CP₂, that geodesic sphere of S² of CP₂ gives rise to mutually commuting Y-B matrices, and that geodesic circle of S² gives rise to unitary Y-B matrices. Together with braid picture also unitarity supports the stringy option, as does also the unitarity of the inner product for the radial modes r^Δ, Δ=1/2+iy, with respect to inner product defined by scaling invariant integration measure dr/r. Furthermore, the reduction of Hamiltonians to duals of closed 2-forms conforms with the almost topological QFT character.

3. Number theoretic hierarchy of discretized theories

Also the hierarchy of discretized versions of the theory which does not mean any approximation but a hierarchy of physics characterizing increasing resolution of cognition can be formulated precisely. Both

• the hierarchy for the zeros of Riemann zeta assumed to define a hierarchy of algebraic extensions of rationals,

• the discretization of the partonic 2-surface by replacing it with a subset of the discrete intersection of the real partonic 2-surface and its p-adic counterpart obtained by algebraic continuation of algebraic equations defining the 2-surface, and

• the hierarchy of quantum phases associated with the hierarchy of Jones inclusions related to the generalization of the notion of imbedding space

are essential for the construction.

The mode expansion of the second quantized spinor field has a natural cutoff for angular momentum l and isospin I corresponding to the integers n_a and n_b characterizing the orders of maximal cyclic subgroups of groups G_a and G_b defining the Jones inclusion in M⁴ and CP₂ degrees of freedom and characterizing the Planck constants. More precisely: one has l≤ n_a and I≤ n_b. This means that the the number modes in the oscillator operator expansion of the spinor field is finite and the delta function singularity for the anticommutations for spinor field becomes smoothed out so that theory makes sense also in the p-adic context where definite integral and therefore also delta function is ill-defined notion.

The almost topological QFT character of theory allows to choose the eigenvalues of the modified Dirac operator to be of form s= 1/2+i∑_kn_ky_k, where s_k=1/2+iy_kare zeros of ζ. This means also a cutoff in the Dirac determinant which becomes thus a finite algebraic number if the number of zeros belonging to a given algebraic extension is finite. This makes sense if the theory is integrable in the sense that everything reduces to a sum over maxima of Kähler function defined by the Dirac determinant as quantum criticality suggests (Duistermaat-Heckman theorem in infinite-dimensional context).

What is especially nice that the hierarchy of these cutoffs replaces also the infinite-dimensional space determined by the configuration space Hamiltonians with a finite-dimensional space so that the world of classical worlds is approximated with a finite-dimensional space.

The allowed intersection points of real and p-adic partonic 2-surface define number theoretical braids and these braids could be identified as counterparts of the braid hierarchy assignable to the hyperfinite factors of type II₁ and their Jones inclusions and representing them as inclusions of finite-dimensional Temperley-Lieb algebras. Thus it would seem that the hierarchy of extensions of p-adic numbers corresponds to the hierarchy of Temperley-Lieb algebras.

For more details see the chapter Construction of Quantum Theory: Symmetries of "Towards S-matrix" .