QFT limit of TGD: summary about how ideas have evolved

I have been working few months with the QFT limit of TGD. The idea which led to the realization of what QFT limit of TGD could be is simple.

1. Only fermions are fundamental particles in quantum TGD and bosons are fermion-antifermion pairs with fermion and antifermion quantum numbers residing at the opposite 3-D light-like throats of wormhole contacts which are surfaces possessing Euclidian signature of induced metric and are glued to space-time sheets having Minkowskian signature of induced metric. Feynman diagrams can therefore be understood in terms of space-time topology and space-time metric. The interpretation of generalized Feynman diagrams differs dramatically from that for stringy diagrams since vertices are points where light-like 3-surfaces join together just like likes of ordinary Feynman diagram do. Stringy diagrams provide a space-time correlate for the propagation of particle along two different routes followed by fusion and interference.

2. Only fermions are fundamental fields in TGD. This suggests that gauge bosons, which have components of induced spinor connection and projections of CP₂ Killing vector field as classical geometric correlates, should emerge in some sense at QFT limit. In other words, the action for QFT approximating TGD contains nothing but Dirac action coupled to gauge potentials, and the bosonic action containing YM term plus infinite number of vertices defined by closed fermion loops is generated radiatively. This approach leads to a generalization of Feynman rules and in principle predicts all coupling constants and their evolution without any input parameters except CP₂ size and quantum criticality. p-Adic mass calculations demonstrated already 15 years ago that one can understand the mysterious proton mass to Planck mass ratio and elementary particle mass scales and even masses number theoretically.

3. An essential element of the approach is a formulation for UV cutoff. A cutoff in both mass squared and hyperbolic angle is necessary since Wick rotation does not make sense in TGD framework. By assuming a geometrically natural hyperbolic UV cutoff motivated by zero energy ontology one can understand the evolution of the standard model gauge couplings and reproduce correctly the values of fine structure constant at electron and intermediate boson length scales. Also asymptotic freedom follows as a basic prediction. Contrary to the original beliefs propagator generates a mass term unless the hyperbolic cutoffs for time-like and space-like gauge boson momenta are in a definite relation. One could criticize this relation and argue that perhaps super-conformal symmetries might help to get the cancelation with identical cutoffs. It seems that this is not the case.

The UV cutoff for the hyperbolic angle as a function of p-adic length scale is the ad hoc element of the model in its recent form. How to formulate quantitatively the quantum criticality in terms of the behavior of the hyperbolic cutoff as function of p-adic length scale became therefore the basic problem and led what might like a numerics inspired random walk -or perhaps better to say sleep walk - towards what I believe to the solution of the problem. During this kind of heavy numerical calculations one realizes how important it would be to have a colleague replicating the calculations. One can never be quite sure about signs and numerical factors.

1. The process gradually led to an improved understanding of the notion of coupling constant evolution itself. The fermionic loop integral contains a propagator pole contributing imaginary part to the inverse propagator and numerical calculations demonstrated that this contribution is too large to be physically acceptable. Moreover, the sign of coupling strength becomes negative for fermion masses above certain critical mass defining the IR cutoff for the loop momenta. The only manner to avoid difficulties is to assume that loop momenta are always below the p-adic mass scale associated with the momentum of the gauge boson. The assumption eliminates the imaginary part of propagator and keeps coupling constant strength positive. This also gives precise content to the notion of coupling constant evolution since it assigns to the mass scape of p IR cutoff k_max such that for k > k_max coupling constant strength is positive. A nice geometric interpretation is possible in zero energy ontology: loop corrections corresponding to geometric details sufficiently smaller than the length scale assignable to the mass squared.

2. The next idea was that perhaps one could fix the cutoff on hyperbolic angle (hyperbolic cutoff) by some naturally occurring condition. The first guess was that the sign of the coupling constant strength changes at either end of the p-adic half octave for the mass of gauge boson. The motivation to this idea could have come from the calculation of the momentum at which the sign changes for the model reproducing physically reasonable coupling constant evolution: at long length scales the sign indeed changes very near to the end of the half-octave. Unfortunately this did not work.

3. The next guess was that the value of boson momentum at which the sign changes is as near as possible to the end of the mass squared octave. Tedious calculations in a rather arctic numerical environment demonstrated that one obtains a discrete set of coupling constant evolutions but that the hyperbolic cutoff is increasing as a function of k rather than decreasing as required by the coupling constant evolution in standard model. The increase can be understood as a positive feedback effect: the vanishing of the inverse of the coupling constant at given length scale requires a contribution, which increases as a function of the p-adic length scale since the inverse of the coupling constant itself increases. The attempts to modify the model to modify this behavior failed.

4. The next idea was that perhaps p-adic fractality helps to assign the change of the sign at the ends of half octaves or to prime for which p-adic length scale is very near to that defined by the end of the half octave (p <≈ 2^k). p-Adic fractals were one of the first ideas about p-adic physics and I learned quite recently that also mathematicians have discovered them. They are obtained by mapping reals to p-adics by the inverse of the canonical identification I (or a proper variant of it) performing the arithmetics, and map the result back to reals by I. I had not found any direct application except in the case of p-adic mass calculations where p-adic mass squared is mapped to its real counterpart.

   The guess was obvious. Express M-matrix element a function of standard Lorentz invariants with dimensions of mass squared so that a very close connection with mass calculations is obtained. Map the invariants to their p-adic counterparts using the inverse of I, carry out the arithmetics defining the function in the p-adicity under question, and return to the reality using I. Maybe this could allow to achieve the cancelation at the end of the p-adic octave for mass squared. I do not believe this anymore but again a wrong idea led to what looks like a real increase in the understanding of quantum TGD and how p-adic and real physics relate at the level of M-matrix. One nice finding was that p-adic existence forces the loop masses to be above the mass of virtual gauge boson forced by purely physical conditions. It however seems that one must introduce transcendentals like log(2) and π so that an algebraically infinite-dimensional and basically non-algebraic extension of p-adic numbers is unavoidable.

5. The p-adicization program for M-matrix involve a technical difficulty which led to a further progress. It is not possible to perform loop integrals in the p-adic context. All loop integrals must be carried out in the real context and the resulting functions must be p-adicized. For the bosonic vertices defined as purely fermionic loops this is not a problem but the situation changes for the expansion of the M-matrix involving both bosonic and fermionic lines inside loops. The same problem is encountered in the twistorialization and the solution of the problem is based on Cutkosky rules allowing unitarization of the tree amplitudes in terms of TT⁺ contribution involving only light-like momenta seems to be the only working option and requires that TT⁺ makes sense p-adically. This idea is actually very near to the original idea that only light-like momenta appear in loops so that twistorialization is elegant. TT⁺ indeed allows interpretation in terms of loops so that I was not after all totally silly. The p-adic existence of the analytic continuation of TT⁺ by dispersion relations poses strong constraints on otherwise not completely unique continuation.

6. After these steps I was mature to realize how to formulate quantum criticality in such a manner that it could fix the hyperbolic cutoff and hence coupling constant evolution uniquely. The fermionic loops defining bosonic vertices vanish when the incoming momenta are massless. This is it! The condition emerges as a consistency condition: if the vanishing does not occur for on mass shell bosons, one obtains T-matrix expressible in terms of analytic continuation of TT⁺ and one does not have vertex identified as something irreducible anymore. The condition is suggested also by quantum criticality: the vanishing of vertices is very much analogous to the vanishing of higher functional derivatives of the action with respect to gauge fields at criticality (or derivatives of the potential function in Thom's catastrophe theory). Also the fact that only BFF vertex is fundamental vertex if bosonic emergence is accepted suggests the conditions. The vanishing of on mass shell N-vertices gives an infinite number of conditions on the hyperbolic cutoff as a function of the integer k labeling p-adic length scale at the limit when bosons are massless and IR cutoff for the loop mass scale is taken to zero. For a finite cutoff k_max the number of vanishing vertices is finite and correspond to some maximum value N_cr analogous to the order of perturbation theory and identifiable as characterization of the finite measurement resolution. It is not yet clear whether dynamical symmetries, in particular super-conformal symmetries, are involved with the realization of the vanishing conditions or whether hyperbolic cutoff is all that is needed.

A more detailed representation can be found from the last section of the new chapter Quantum Field Theory Limit of TGD from Bosonic Emergence of "Towards M-matrix". I also extracted from the chapter a short piece of text explaining in more detail the ideas discussed here and in the previous posting.

Genes and Water Memory

After long time I had opportunity to read a beautiful experimental article. Not about the latest dramatic experimental breakthroughs in proving F theory to be the only possible theory of everything and even more;-) but about experimental biology. Yolene Thomas, who worked with Benveniste, kindly sent the article to me. The freely loadable article is Electromagnetic Signals Are Produced by Aqueous Nanostructures Derived from Bacterial DNA Sequences by Luc Montagnier, Jamal Aissa, Stéphane Ferris, Jean-Luc Montagnier, and Claude Lavallée published in the journal Interdiscip. Sci. Comput. Life Sci. (2009).

1. Basic findings at cell level

I try to list the essential points of the article. Apologies for biologists: I am not a specialist.

1. Certain pathogenic micro-organisms are objects of the study. The bacteria Mycoplasma Pirum and E. Choli belong to the targets of the study. The motivating observation was that some procedures aimed at sterilizing biological fluids can yield under some conditions the infectious micro-organism which was present before the filtration and absent immediately after it. For instance, one filtrates a culture of human lymphocytes infected by M. Pirum, which has infected human lymphocytes to make it sterile. The filters used have 100 nm and 20 nm porosities. M. Pirum has size of 300 nm so that apparently sterile fluids results. However if this fluid is incubated with a mycoplasma negative culture of human lymphocytes, mycoplasma re-appears within 2 or 3 weeks! This sounds mysterious. Same happens as 20 nm filtration is applied to a a minor infective fraction of HIV, whose viral particles have size in the range 100-120 nm.

2. These findings motivated a study of the filtrates and it was discovered that they have a capacity to produce low frequency electromagnetic waves with frequencies in good approximation coming as the first three harmonics of kHz frequency, which by the way plays also a central role in neural synchrony. What sounds mysterious is that the effect appeared after appropriate dilutions with water: positive dilution fraction varied between 10-7 and 10^-12. The uninfected eukaryotic cells used as controls did not show the emission. These signals appeared for both M. Pirum and E. Choli but for M. Pirum a filtration using 20 nm filter canceled the effect. Hence it seems that the nano-structures in question have size between 20 and 100 nm in this case.

   A resonance phenomenon depending on excitation by the electromagnetic waves is suggested as an underlying mechanism. Stochastic resonance familiar to physicists suggests itself and also I have discussed it while developing ideas about quantum brain (see this). The proposed explanation for the necessity of the dilution could be kind of self-inhibition. Maybe a gel like phase which does not emit radiation is present in sufficiently low dilution but is destroyed in high dilutions after which emission begins. Note that the gel phase would not be present in healthy tissue. Also a destructive interference of radiation emitted by several sources can be imagined.

3. Also a cross talk between dilutions was discovered. The experiment involved two tubes. Donor tube was at a low dilution of E. Choli and "silent" (and carrying gel like phase if the above conjecture is right). Receiver tube was in high dilution (dilution fraction 10^-9) and "loud". Both tubes were placed in mu-metal box for 24 hours at room temperature. Both tubes were silent after his. After a further dilution made for the receiver tube it became loud again. This could be understood in terms of the formation of gel like phase in which the radiation does not take place. The effect disappeared when one interposed a sheath of mu-metal between the tubes. Emission of similar signals was observed for many other bacterial specials, all pathogenic. The transfer occurred only between identical bacterial species which suggests that the signals and possibly also frequencies are characteristic for the species and possibly code for DNA sequences characterizing the species.

4. A further surprising finding was that the signal appeared in dilution which was always the same irrespective of what was the original dilution.

2. Experimentation at gene level

The next step in experimentation was performed at gene level.

1. The killing of bacteria did not cancel the emission in appropriate dilutions unless the genetic material was destroyed. It turned out that the genetic material extracted from the bacteria filtered and diluted with water produced also an emission for sufficiently high dilutions.

2. The filtration step was essential for the emission also now. The filtration for 100 nm did not retain DNA which was indeed present in the filtrate. That effect occurred suggests that filtration destroyed a gel like structure inhibiting the effect. When 20 nm filtration was used the effect disappeared which suggests that the size of the structure was in the range 20-100 nm.

3. After the treatment by DNAse enzyme inducing splitting of DNA to pieces the emission was absent. The treatment of DNA solution by restriction enzyme acting on many sites of DNA did not suppress the emission suggesting that the emission is linked with rather short sequences or with rare sequences.

4. The fact that pathogenic bacteria produce the emission but not "good" bacteria suggests that effect is caused by some specific gene. It was found that single gene - adhesin responsible for the adhesion of mycoplasma to human cells- was responsible for the effect. When the cloned gene was attached to two plasmids and the E. Choli DNA was transformed with the either plasmid, the emission was produced.

3. Some consequences

The findings could have rather interesting consequences.

1. The refinement of the analysis could make possible diagnostics of various diseases and suggests bacterial origin of diseases like Alzheimer disease, Parkinson disease, Multiple Sclerosis and Rheumatoid Arthritis since the emission signal could serve as a signature of the gene causing the disease. The signal can be detected also from RNA viruses such as HIV, influenza virus A, and Hepatitis C virus.

2. Emission could also play key role in the mechanism of adhesion to human cells making possible the infection perhaps acting as a kind of password.

The results are rather impressive. Some strongly conditioned skeptic might have already stopped reading after encountering the word "dilution" and associating it with a word which no skeptic scientist in his right mind should not say aloud: "homeopathy"! By reading carefully what I wrote above, it is easy to discover that the experimenters unashamedly manufactured a homeopathic remedy out of the filtrate! And the motivating finding was that although filtrate should not have contained the bacteria, they (according to authors), or at least the effects caused by them, appeared within weeks to it! This is of course impossible in the word of skeptic.

The next reaction of the skeptic is of course that this is fraud or the experimenters are miserable crackpots. Amusingly, one of the miserable crackpots is Nobelist Luc Montagnier whose research group discovered AIDS virus.

4. How TGD could explain the findings?

Let us leave the raging skeptics for a moment and sketch possible explanations in TGD framework.

1. Skeptic would argue that the filtration allowed a small portion of infected cells to leak through the filter. Many-sheeted space-time suggests a science fictive variant of this explanation. During filtration part of the infected cells is "dropped" to large space-time sheets and diffused back to the original space-time sheets during the next week. This would explain why the micro-organisms were regenerated within few weeks. Same mechanism could work for ordinary molecules and explain homeopathy. This can be tested: look whether the molecules return back to the the diluted solution in the case of a homeopathic remedy.

2. If no cells remain in the filtrate, something really miraculous looking events are required to make possible the regeneration of the effects serving as the presence of cells. This even in the case that DNA fragments remain in the filtrate.
   1. The minimum option is that the presence of these structures contained only the relevant information about the infecting bacteria and this information coded in terms of frequencies was enough to induce the signatures of the infection as a kind of molecular conditioning. Experimentalists can probably immediately answer whether this can be the case.

   2. The most radical option is that the infecting bacteria were actually regenerated as experimenters claim! The information about their DNA was in some form present and was transcribed to DNA and/or RNA, which in turn transformed to proteins. Maybe the small fragment of DNA (adhesin) and this information should have been enough to regenerate the DNA of the bacterium and bacterium itself. A test for this hypothesis is whether the mere nanoparticles left from the DNA preparation to the filtrate can induce the regeneration of infecting molecules.

The notion of magnetic body carrying dark matter quantum controlling living matter forms the basic element of TGD inspired model of quantum biology and suggests a more concrete model. For a possible experimental support for the notion see the earlier posting.

1. If the matter at given layer of the onion-like structure formed by magnetic bodies has large hbar, one can argue that the layer corresponds to a higher evolutionary level than ordinary matter with longer time scale of memory and planned action. Hence it would not be surprising if the magnetic bodies were able to replicate and use ordinary molecules as kind of sensory receptors and motor organs. Perhaps the replication of magnetic bodies preceded the replication at DNA level and genetic code is realized already at this more fundamental level somehow. Perhaps the replication of magnetic bodies induces the replication of DNA as I have suggested.

2. As I have discussed in my earlier postings and and in the books at my homepage, the magnetic body of DNA would make DNA a topological quantum computer (see this). DNA itself would represent the hardware and magnetic bodies would carry the evolving quantum computer programs realized in terms of braidings of magnetic flux tubes. The natural communication and control tool would be cyclotron radiation besides Josephson radiation associated with cell membranes acting as Josephson junctions. Cyclotron frequencies are indeed the only natural frequencies that one can assign to molecules in kHz range. There would be an entire fractal hierarchy of analogs of EEG making possible the communication with and control by magnetic bodies.

3. The values of Planck constant would define a hierarchy of magnetic bodies which corresponds to evolutionary hierarchy and the emergence of a new level would mean jump in evolution. Gel like phases could serve as a correlate for the presence of the magnetic body. The phase transitions changing the value of Planck constant and scale up or down the size of the magnetic flux tubes. They are proposed to serve as a basic control mechanism making possible to understand the properties and the dynamics of the gel phases and how biomolecules can find each other in the thick molecular soup via a phase transition reducing the length of flux tubes connecting the biomolecules in question and thus forcing them to the vicinity of each other.

Consider now how this model could explain the findings.

1. Minimal option is that the the flux tubes correspond to "larger space-time sheets" and the infected cells managed to flow into the filtrate along magnetic flux tubes from the filter. This kind of transfer of DNA might be made possible by the recently discovered nanotubes already mentioned.

2. Maybe the radiation resulted as dark photons invisible for ordinary instruments transformed to ordinary photons as the gel phase assignable with the dark matter at magnetic flux tube network associated with the infected cells and corresponding DNA was destroyed in the filtration.

   This is not the only possible guess. A phase conjugate cyclotron radiation with a large value of Planck constant could also allow for the nanostructures in dilute solute to gain metabolic energy by sending negative energy quanta to a system able to receive them. Indeed the presence of ambient radiation was necessary for the emission. Maybe that for sufficiently dilute solute this mechanism allows to the nanostructures to get metabolic energy from the ambient radiation whereas for the gel phase the metabolic needs are not so demanding. In the similar manner bacteria form colonies when metabolically deprived. This sucking of energy might be also part of the mechanism of disease.

3. What could be the magnetic field inducing the kHz radiation as a synchrotron radiation?
   1. For instance, kHz frequency and its harmonics could correspond to the cyclotron frequencies of proton in magnetic field which field strength slightly above that for Earth's magnetic field (750 Hz frequency corresponds to field strength of B_E, where B_E/=.5 Gauss, the nominal strength of Earth's magnetic field). A possible problem is that the thickness of the flux tubes would be about cell size for Earth's magnetic field from flux quantization and even larger for dark matter with a large value of Planck constant. Of course, the flux tubes could make themselves thinner temporarily and leak through the pores.
   2. If the flux tube is assumed to have thickness of order 20-100 nm, the magnetic field for ordinary value of hbar would be of order .1 Tesla from flux quantization and in the case of DNA the cyclotron frequencies would not depend much on the length of DNA fragment since the it carries a constant charge density. Magnetic field of order .2 Tesla would give cyclotron frequency of order kHZ from the fact that the field strength of .2 Gauss gives frequency of about .1 Hz. This correspond to a magnetic field with flux tube thickness ≈ 125 nm, which happens to be the upper limit for the porosity. Dark magnetic flux tubes with large hbar are however thicker and the leakage might involve a temporary phase transition to a phase with ordinary value of hbar reducing the thickness of the flux tube. Perhaps some genes (adhesin) plus corresponding magnetic bodies representing DNA in terms of cyclotron frequencies depending slightly on precise weight of the DNA sequence and thus coding it correspond to the frequency of cyclotron radiation are the sought for nano-structures.

4. While developing a model for homeopathy based on dark matter I ended up with the idea that dark matter consisting of nuclear strings of neutrons and protons with a large value of hbar and having thus a zoomed up size of nucleon could be involved. The really amazing finding was that nucleons as three quark systems allow to realize vertebrate code in terms of states formed from entangled quarks (see this and this and also the earlier posting)! One cannot decompose codons to letters as in the case of the ordinary genetic code but codons are analogous to symbols representing entire words in Chinese. The counterparts of DNA, RNA, and aminoacids emerge and genetic code has a concrete meaning as a map between quantum states.

   Without any exaggeration this connection between dark hadronic physics and biology has been one of the greatest surprises of my professional life. It suggests that dark matter in macroscopic quantum phase realizes genetic code at the level of nuclear physics and biology only provides one particular (or probably very many as I have proposed) representations of it. If one takes this seriously one can imagine that genetic information is represented by these dark nuclear strings of nanoscopic size and that there exists a mechanism translating the dark nuclei to ordinary DNA and RNA sequences and thus to biological matter. This would explain the claimed regeneration of the infected cells.

5. Genetic code at dark matter level would have far reaching implications. For instance, living matter - or rather, the magnetic bodies controlling it - could purposefully perform genetic engineering. This forces me to spit out another really dirty word, "Lamarckism"! We have of course learned that mutations are random. The basic objection against Lamarckism is that there is no known mechanism which would transfer the mutations to germ cells. In the homeopathic Universe of TGD the mutations could be however performed first for the dark nucleon sequences. After this these sequences would diffuse to germ cells just like homeopathic remedies do, and after this are translated to DNA or RNA and attach to DNA.

We are living exciting times. If someone wants to share this experience with me, she or he can can consult the chapter Homeopathy in Many-Sheeted Space-Time of Bio-Systems as Conscious Holograms where also the nuclear realization of the genetic code is discussed. If the word "hopeopathy" in the title is too much for the stomach of the reader, he can consult also the chapter Nuclear String Model of p-Adic length Scale Hypothesis and Dark Matter Hierarchy.

p-Adicization, twistor program, and quantum criticality

Just a brief note (strongly updated!) about the recent situation concerning bosonic emergence and QFT limit of TGD. There is now a very attractive overall view about how p-adic and real physics are fused together and how p-adic fractality emerges when real Lorentz invariants - typically mass squared for subsystem- are mapped to their p-adic counterparts of a suitable variant of canonical identification which in its simplest form reads as &sum x_npⁿ → &sum x_np^-n. One can say that quantum criticality, bosonic emergence, number theoretic universality, p-adic fractality, and twistor program seem to be very intimately inter-related in TGD Universe.

Loops are the problem of the p-adicization program as also twistor program. In twistorialization the problem can e overcome by using Cutkosky rules which means that one adds to the tree diagram TT⁺ contribution for on mass shell intermediate states allowing unitarization. Since this contribution involves only massless intermediate states twistorialization is possible. This is actually what I suggested earlier (only light-like loop momenta are allowed in twistor context) without properly realizing the connection with Cutkosky rules! If TT⁺ makes sense also p-adically, p-adicization and p-adic fractalization are possible.

Unitarization by Cutkosky rules does not make sense for fermionic loops defining the bosonic vertices as becomes clear by considering B→FFbar→B loop for massless particles. Furthermore, if these vertices were non-vanishing for on mass shell momenta (massless) unitarity would force the introduction of TT⁺ contribution and one could not speak about vertices anymore. Therefore it seems that the fermionic loops defining bosonic vertices vanish when the bosons are on mass shell. These conditions would generalize the quantum criticality condition and hopefully fix completely the vertices. It also means that only BFF vertex is non-vanishing for on mass shell particles as is natural since Dirac action coupled to gauge bosons is the basic action principle. The vanishing of on mass shell N-vertices gives an infinite number of conditions on the hyperbolic cutoff as function of the integer k labeling p-adic length scale at the limit when bosons are massless and IR cutoff for the loop mass scale is taken to zero. For a finite cutoff k_max the number of vanishing vertices is finite and correspond to some maximum value N_cr analogous to the order of perturbation theory and identifiable as characterization of the finite measurement resolution.

Whether the vanishing of the fermionic loops defining vertices is achieved by fixing the hyperbolic cutoff is not clear, and one can wonder whether dynamical on mass shell symmetries -in particular various super-conformal symmetries - could be involved. The first checks suggests that super-symmetry cannot lead to the vanishing of the on mass shell vertices and that hyperbolic cutoff and the non-trivial relation between time-like and space-like hyperbolic cutoffs are necessary.

To me it seems that TGD has forced a rather dramatic simplicification of the very notion of quantum field theory. If so, then the mere assumption about the existence of QFT limit (to say nothing about the assumption that this limit is GUT or a minimally supersymmetric version of standard model (MSSM) for which Lebensraum is shrinking continually) would have led competing unified theorists to a fatal side track.

A more detailed representation can be found from the last section of the new chapter Quantum Field Theory Limit of TGD from Bosonic Emergence of "Towards M-matrix". I also extracted from the chapter a short piece of text explaining in more detail the ideas discussed here and in the previous posting.

Bosonic Emergence, Number Theoretic Universality, p-Adic Fractality, and Twistor Program

Mahndisa made some questions about p-adic fractalization of S-matrix. My reply had too many characters so that I decided to add it as a separate posting.

Dear Manhndisa,

The problems are essentially number theoretical. And "ontological" as philosopher would say. I have a bundle of general ideas developed in various chapters of various books. I try to give impression about just the bare essentials.

1. In TGD framework M-matrix replaces S-matrix: zero energy ontology, etc. The S-matrix accompanying M-matrix is not obtained as a unitary evolution operator as in QFT:s since M-matrix defines time-like entanglement coefficients between positive and negative energy parts of the zero energy state having as counterparts initial and final states of a physical event in positive energy ontology. Heisenberg and Schrodinger pictures relate to positive energy ontology and are therefore not terribly relevant here.

2. Number theoretic universality. M-matrix of same functional form as a function of Lorentz invariants defines complex and p-adic valued variants. The strongest form of this universality is rationality of matrix elements. It should be also possible to algebraically continue this M-matrix to various number fields from the field of rationals (algebraics, reals, and p-adics are completions of rationals and the completion process generalizes to the completion of rational physics in various number fields). p-Adic physics should be visible also at the real side at the level of matrix elements (besides mass spectrum) and here p-adic fractalization enters the stage.

3. Bosonic emergence is essential for QFT limit and emerged from twistor related considerations, started then to look more or less independent of twistors so that I separated it to its own chapter, and finally turned out to be highly relevant for twistorizalition! The counterpart of bosonic YM action emerges from Dirac action via functional integral over fermion fields. In particular, the inverse of the bosonic propagator emerges as fermionic loop. Bosonic emergence in principle predicts all coupling constants and their evolution if one can fix the cutoffs involved with the loop integral over fermion momenta. Quantum criticality should determine the cutoff in hyperbolic angle. What quantum criticality means exactly at the level of MATLAB modules is the problem and with this problem I have worked last months and tested various hypothesis.

4. After tedious and slow calculations it seems that the definition of criticality that I have worked for last month does not work (I managed to calculate yesterday and last night 30 first p-adic length scales using the proposal for criticality based on real physics: the resulting hyperbolic cutoff behaves in non-physical manner if its growth continues to say electron length scale). The proposal is that one should use essentially same definition but adding p-adic fractality. In this picture the real variant of bosonic propagator defines also p-adic propagator: the real Lorentz invariants appearing in matrix element are mapped to p-adic ones by a proper variant of canonical identification. The resulting p-adic sum of various contributions to the propagator from various p-adic mass scales is then mapped back to reals and you get p-adic fractal. This is just the visit from reality to p-adicity and successful return together with brand new p-adically fractal bosonic propagator!;-)

5. What I realized yesterday is that internal consistency is achieved only if the loops involving gauge bosons vanish. This has been one of the one thousand and one formulations of quantum criticality during years. Therefore only tree diagrams with emerging bosonic propagators and free fermionic propagators are needed. One obtains non-trivial coupling constant evolution and tree diagrams! Both real and p-adic versions of perturbation theory exist since the mathematically existence bosonic loop integrals are absent. Number theoretical universality and p-adic fractality are both obtained. That p-adicity is visible also at the level of real scattering amplitudes is highly satisfactory. This picture generalizes also to the quantum TGD proper.

6. Whether the general definition of quantum criticality as vanishing of the bosonic loops is equivalent with definition of quantum criticality allowing to deduce hyperbolic cutoff and which I have studied during last month (whose technical definition I will not discuss here) after p-adic fractalization remains an open question.

7. What is interesting that twistorialization, which was the starting point of the work one half year ago, works for tree diagrams. Physics requires non-trivial coupling constant evolution and thus loops in standard framework but loops are the basic problem of twistor approach since particles in loops are massive and twistorialization for them is not elegant. All the fantastic results of twistorialization program (say this) are for tree diagrams. Bosonic loops would not be present p-adically (would vanish in real sense) in quantum critical TGD Universe.

This is the general picture now. I have the feeling that no big changes are needed anymore.

The relevant text can be found from the last section of the new chapter Quantum Field Theory Limit of TGD from Bosonic Emergence of "Towards M-matrix".

Silence

I have not had time for blog postings. My response to the Mahndisa in earlier posting gives the reason why and also some ideas about the recent situation in coupling constant evolution. I hope that I can write as summary within few days.

Dear Mahndisa,

numerical work makes me rather non-communicative;-). I have been working with the numerical realization for the model of coupling constant evolution based on quantum criticality. Numerical work is is not easy at this age and would require a hard wired brain at any age. To make challenge even more difficult, I am forced to use MATLAB without compiler and there are a lot of loops. The basic challenge is how to make things fast by making as much as possible analytically. This kind of problems sound of course childish and crackpottish in the ears of anyone allowed to use the computational resources of any physics department of any University. Perhaps I should be ashamed;-)!

There are also other reasons for being not so communicative. Last week MATLAB went completely mad: probably I became a victim of a clever virus attack, nothing standard against which I am well-shielded. This was not the first time.

Add to this jeremiad the problems of everyday life (for a week I have had nothing at my bank account and getting support social office is slow nowadays since there are long ques) and you begin to understand why the working conditions are not very inspiring. We are however in Finland and in the academic circles of this country thinkers are regarded next to criminals and the best manner to treat them has been found to be the academic equivalent of Siberia.

In any case I have made a lot of progress in understanding coupling constant evolution. The question whether the proposed realization of quantum criticality works is still open. In any case, at ultrahigh energies the behavior of em couplings strength would be like that for asymptotic free theory if criticality is accepted. For low energies the criticality is consistent with standard model behavior for fine structure constant (its value at electron and intermediate boson scale are the constraints). I do not yet know whether the low energy and high energy behaviors are consistent with each other or not. The calculations are desperately slow.

This problem led to the ask whether p-adicization of the theory is necessary to realize criticality. Within two days this led to a rather precise recipe for how to p-adicize the theory in terms of p-adic fractals- creatures which I discovered within first year of p-adic TGD but for which I have not found direct application in TGD hitherto.

The recipe was very simple: consider real Lorentz invariant amplitudes, map Lorentz invariant kinematic quantities to their p-adic counterparts by some variant of canonical identification to get p-adic calued functions with same functional form, carry out arithmetic operations such as the summation of perturbative contributions using p-adic arithmetics, and map the result back to reals to get a p-adic fractal.

You just go to p-adicity, perform arithmetics there and return to reality to see what you got! In this manner the difficulties related to p-adicization such as the non-existence of p-adic definite integral, and the problems with minus sign and imaginary unit can be circumvented and the outcome cannot differ too much from real physics prediction.

I hope that I can write about this within few days. The recent situation concerning bosonic emergence in quantum TGD framework given in the new chapter Quantum Field Theory Limit of TGD from Bosonic Emergence of "Towards M-matrix".

Best Regards,

Matti