Constraints on the fermionic realization of genetic code from the model for color qualia

The original model for DNA as topological quantum computer assigns to DNA nucleotides quarks at ends of flux tubes or quark pairs at the ends of wormhole flux tubes. This is only the realization that came first to my mind in TGD Universe where dark variants of quarks can define QCD like physics even in cellular length scales. One can actually imagine several realizations of the genetic code and the first realization is far from being the simplest one. It is enough to have four different particles or many-particle quantum states to build at least formally a map from A,T,C, G to four states. It is obvious that the number of possible formal realizations is limited only by the imagination of the theoretician. Additional conditions are required to fix the model.

Fermionic representation

Consider first the fermionic representations in the general case without specifying what fermions are.

1. The original proposal was that DNA nucleotides correspond to flux tubes with quark q and antiquark qbar at the ends of the parallel flux sheets extremely near to each other. Second options relies on wormhole magnetic flux tubes in which case quark pair qqbar is at both ends. Quarks u, d and their antiquarks would code for A,T,C,G. The spin of quarks is not taken into account at all in this coding: why not restrict the consideration to single quark. The total quark charge at given end of flux tube pair vanishes and flux tube ends carry opposite quark charges.
   

   The nice feature of this option is that one could understand the generation of color qualia in the model of sensory receptor in simple manner to be discussed below. Even if one accepts the arguments supporting the view that dark quarks in cell scale are natural outcome of the hierarchy of Planck constants, one could argue that the presence of both quarks and antiquarks does not conform with matter antimatter asymmetry (not that one can however identify the analog of matter antimatter asymmetry at DNA level).
   
   

2. Spin states for fermion pairs assigned with two parallel magnetic flux tubes with the magnetic field generated by spin provide much simpler representation for nucleotides. Similar fermion pair would reside at the second end of flux tube pair.
   
   1. It is is essential that rotational symmetry is broken and reduces to rotational symmetry around the direction of flux tubes so that spin singlet and spin 0 state of triplet mix to form states for which each fermion is in spin eigenstate. The states must be antisymmetric under exchange of the protons and spin 1/0 states are antisymmetric/symmetric in spatial degrees of freedom (wave functions located to the ends of flux tubes). The states with definite spin for given flux tube are mixtures of s=1 states with vanishing spin projection and s=0 state.
      
      
   2. It is not quite clear whether one should treat fermion pairs as identical bosons with 3+1 spin states since in TGD framework one considers disjoint partonic 2-surfaces and the situation is not that of QFT in M⁴. This interpretation would require totally symmetry of the states under permutations of bosonic states defined by the 3+1 spin states. Coding by spin requires that each nucleotide corresponds to a state with a well defined spin. In field theory language the state would be obtained by applying bosonic oscillator operators generating states of given spin localized to a given nucleotide position.
      
      
   3. The classical correlate for the permutations of coordinates of fermions has interpretation as braiding for the flux tubes of the flux tube pair. In the similar manner the permutation of the flux tube pairs associated with nucleotides has interpretation as braiding of the 3-braids formed form from flux tube pairs. Braiding therefore gives a representation of spin analogous to the well-known orientation entanglement relation invented by Dirac and providing geometric representation of spin 1/2 property.
      

Various options for the fermionic representation of A,T,C,G

Fermionic representations allows several options since fermion can be electron, u or d quark, or proton. Wormhole magnetic fields would not be needed in this case.

1. The problem of electron and proton options is that it does not allow realization of color qualia. There is also the well-known problem related to the stability of DNA caused by the phosphate charge of -2 units per nucleotide. Somehow this charge should be screened. In any case, the charge -2 should correspond to the electron pair at the DNA end of the flux tube for electron option. For proton option the charge would be screened completely. One could of course consider also the large hbar color excitations of ordinary protons instead of quark at its nucleotide ends. This option would however require the modification of quark wave functions inside proton and this option will not be discussed here.
   
   
2. Quark option would give rise to both color and allow also to reduce the electronic charge of -2 units by 4/3 units to -2/3 units in the case of u quark pair. This would help to stabilize DNA. In the case of d quarks the charge would increase to -10/3 units and is not favored by stability argument. Flux tube pairs assigned to single nucleotide define diquarks with spin 1 or spin 0.
   
   
   1. Diquarks behave ass identical bosons with 3+1 spin states and 3× 3 color states. The states with well defined symmetry properties in spin degrees of freedom have such properties in spatial degrees of freedon. This means that one obtains a superposition of flux tube pairs with are either braided or unbraided. Triplet/singlet state is symmetric/antisymmetric and total asymmetry could be guaranteed by
      assuming symmetry/antisymmetry in spatial degrees of freedom and antisymmetry/symmetry in color degrees of freedom. This would give anti-triplet/6-plet in color degrees of freedom. Spatial symmetry would favor antitriplet and diquark would behave like antiquark with respect to color. Let us assume antitriplet state for definiteness.
      
      
   2. DNA codon corresponds to three-di-quark state. This state must be totally symmetric under the exchange of bosons. One can have total symmetry in both spatial and color degrees of freedom or total antisymmetry/symmetry in spatial and total antisymmetry/symmetry in color degrees of freedom.The first option gives 10-dimensional color multiplet and the second one color singlet. Braiding is maximal and symmetric/antisymmetric in these case. One can consider also mixed symmetries. In this case one has color octet which is antisymmetric with respect to the first nucleotide pair and symmetric with respect to first nucleotide pair and third nucleotide. The braiding of the first two nucleotides must be antisymmetric and the braiding of this pair with third nucleotide. The conclusion would be that color multiplets correspond to well defined braidings and one would therefore have directed connection with topological quantum computation. Color octet is especially interesting concerning the representation of color qualia.
      

The challenge of all these options (note that the representability of color selects quark option) is to find a good justification for why the assignment of A,T,C,G to quark states or spin states is unique dynamically. Stability argument is expected to help here.

Realization of color qualia for quark option

Consider now how one could understand the generation of qualia for quark option.

1. The generation of qualia involves interaction with external world giving rise to a sensory percept. In the case of visual colors it should correspond to a measurement of quark color and should give rise to eigenstages of color at the ends of flux tubes at DNA nucleotides for a nucleus or cell of photoreceptor. A modification of capacitor model is needed. Color polarization is still essential but now polarization in nucleus or cell scale is transformed in the generation of color quale to a polarization in longer length scale by the reconnection of flux tubes so that their ends attach to "external world". The nucleus/cell becomes color and state function reduction selects well defined quantum numbers. It is natural to assume that the entanglement in other degrees of freedom after color measurement is negentropic.
   
   
2. Does the "external world" corresponds to another cell or to the inner lipid layers of the cell membrane containing the nucleus. In the first case flux tubes would end to another cell. If the nuclei of receptor cells are integrate to a larger structure by magnetic flux sheets traversing through them one can also consider the possibility that the polarization in the scale of cell nucleus (recall that the nucleus has also double lipid layer) is transformed to a polarization in cell scale so that similar process in cell scale gives rise to qualia.
   
   
3. The entire receptor unit must have net color charge before the state function reduction. This requires that there are flux tubes connecting the receptor unit to a unit representing "external world and having vanishing color charge. If second cell is the "external world" these flux tubes must go through the pair of lipid layers of both cell membrane and end up to the nucleus of cell in the environment. If external world correspond to the complement of nucleus inside cell the inner layers of cell membrane represents external world. Cell membrane indeed serves as sensory receptor in cell length scale. One can of course have sensory qualia in various length scales so that both options are probably correct and a kind of fractal hierarchy is very natural giving rise also to our qualia at some higher level. Living matter as conscious hologram metaphor suggests a fractal hierarchy of qualia.
   

   After state function reduction reducing the entanglement the flux tubes split and the receptor becomes un-entangled with external world and has vanishing color charges. At the level of conscious experience this means that there can be only memory about the quale experience. The sensation of quale lasts with respect to subjective time as long as the negentropic entanglement prevails. There is an obvious analogy with Orch OR proposal of Hamerofr and Penrose in which also period of conscious experience ends with state function reduction.
   
   

4. Consider now how the color qualia are generated.
   
   
   1. There must be two flux tube states. In the first state there are two flux tube beginning from cell nucleus A and ending to the inner lipid layer a₁ and flux tube beginning from the outer lipid layer a₂ and ending cell nucleus B. Both flux tubes have vanishing net color so that cells have vanishing net colors. This could be regarded as the resting state of the receptor. The lipids in layers a₁ and a₂ are connected by another short flux tube. Same for b₁ and b₂.
      
      
   2. The second flux tube state corresponds to long flux tubes connecting the nuclei of cells A and B. The ends carry opposite color charges. In this case the net color of both A and B is non-vanishing. This state would be an outcome of a reconnection process in which the flux tubes from A to a₁ and B to a₂ re-connect with the short flux tube connecting lipid layers a₁ and a₂.
      
      
   3. When these flux tubes carry opposite colors numbers at their ends, the cell possess net color charge and can represent color quale. Or rather, creation of this kind of flux tube connections would give rise to the color charging of the receptor cell with external world carrying opposite color charge.
      
      

One can argue that this mechanism is not quite in spirit with color capacitor model. Polarization is still essential but now polarization in receptor scale is transformed to polarization in longer length scale by the reconnection of flux tubes. The analog of di-electric breakdown however still applies in the sense that its analog induces large polarization. Several mechanisms generating larger polarization are of course possible. One can ask how essential the electromagnetic polarization of cell membrane is for the generation of qualia at cell level. Note also that biomolecules are quite generally polar molecules.

The unexpected prediction of the model is that braiding would correlate directly with qualia. This would mean also a connection between quantum computation and qualia. This condition emerges from Fermi/Bose-Einstein statistics correlating braiding with symmetric properties of color states and spin states. Quite generally, the correlation of braiding with the symmetries of wave functions as functions of points of braid end points would allow to have direct geometric correlate between induced entanglement and braiding as naive intuitive expectations have suggested.

This model is not consistent with the naive expectation that the quale is generated after state function reduction. Rather, the beginning of sensation of quale means beginning of negentropic entanglement and fusion with external world and state function usually associated with the quantum measurement would mean the end of the sensation and separation from the external world! Maybe one can say that state function reduction means that experience is replaced with a memory "I had the sensation of quale"! Krishnamurti would certainly agree!;-)

Indications for M89 hadron physics from cosmic rays

Sabine Hossenfelder has written two excellent blog postings about cosmic rays. The first one is about the GKZ cutoff for cosmic ray energies and second one about possible indications for new physics above 100 TeV. This inspired me to read what I have said about cosmic rays and Mersenne primes- this was around 1996 - immediately after performing for the first time p-adic mass calculations. It was unpleasant to find that some pieces of the text contained a stupid mistake related to the notion of cosmic ray energy. I had forgotten to take into account the fact that the cosmic ray energies are in the rest system of Earth- what a shame! Therefore I did some corrections to the text contained as last section to New Particle Physics Predicted by TGD: Part I of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy". I glue here the text in the beginning of this section.The recent version should be free of worst kind of blunders.

TGD suggests the existence of a scaled up copy of hadron physics associated with each Mersenne prime M_n=2ⁿ-1, n prime: M₁₀₇ corresponds to ordinary hadron physics. Also leptohadrons are predicted. Also Gaussian Mersennes (1+i)^k-1, could correspond to hadron physics. Four of them (k=151,157,163,167) are in the biologically interesting length scale range between cell membrane thickness and the size of cell nucleus. Also leptonic counterparts of hadron physics assignable to certain Mersennes are predicted and there is evidence for them.

The scaled up variants of hadron physics corresponding to k<107 are of special interest. k=89 defines the interesting Mersenne prime at LHC, and the near future will probably tell whether the 125 GeV signal corresponds to Higgs or a pion of M₈₉ physics - or perhaps something else. Also cosmic ray spectrum could provide support for M₈₉ hadrons and quite recent cosmic ray observations are claimed to provide support for new physics around 100 TeV. M₈₉ proton would correspond to .5 TeV mass considerably below 100 TeV but this mass scale could correspond to a mass scale of a scaled up copy of a heavy quark of M₁₀₇ hadron physics: a naive scaling of top quark mass by factor 512 would give mass about 87 TeV. Also the lighter hadrons of M₈₉ hadron physics should contribute to cosmic ray spectrum and there are indeed indications for this.

The mechanisms giving rise to ultra high energy cosmic rays are poorly understood. The standard explanation would be acceleration of charged particles in huge magnetic fields. TGD suggests a new mechanism based on the decay cascade of cosmic strings. The basic idea is that cosmic string decays cosmic string → M₂ hadrons k → M₃ hadrons ....→ M₆₁ → M₈₉ → M₁₀₇ hadrons could be a new source of cosmic rays. Also variants of this scenario with decay cascade beginning from larger Mersenne prime can be considered. One expects that the decay cascade leads rapidly to extremely energetic ordinary hadrons, which can collide with ordinary hadrons in atmosphere and create hadrons of scaled variants of ordinary hadron physics. These cosmic ray events could serve as a signature for the existence of these scale up variants of hadron physics.

1. Centauro events and the peculiar events associated with E>10⁵ GeV radiation from Cygnus X-3. E refers to energy in Earth's rest frame and for a collision with proton the cm energy would be E_cm=(2EM)^1/2>10 TeV in good approximation whereas M₈₉ variant of proton would have mass of .5 TeV. These events be understood as being due to the collisions of energetic M₈₉ hadrons with ordinary hadrons (nucleons) in the atmosphere.
   
   
2. The decay π_n→ γγ produces a peak in the spectrum of the cosmic gamma rays at energy m(π_n)/2. These produce peaks in cosmic gamma ray spectrum at energies which depend on the energy of π_n in the rest system of Earth. If the pion is at rest in the cm system of incoming proton and atmospheric proton one can estimate the energy of the peak if the total energy of the shower can be estimated reliably.
   
   
3. The slope in the hadronic cosmic ray spectrum changes at E=3 ×10⁶ GeV. This corresponds to the energy E_cm=2.5 TeV in the cm system of cosmic ray hadron and atmospheric proton. This is not very far from M₈₉ proton mass .5 TeV. The creation of M₈₉ hadrons in atmospheric collisions could explain the change of the slope.
   
   
4. The ultra-higher energy cosmic ray radiation having energies of order 10⁹ GeV in Earth's rest system apparently consisting of protons and nuclei not lighter than Fe might be actually dominated by gamma rays: at these energies γ and p induced showers have same muon content. E=10⁹ GeV corresponds to E_cm=(2Em_p(1/2= 4× 10⁴ GeV. M₈₉ nucleon would correspond to mass scale 512 GeV.
   
   
5. So called GKZ cutoff should take place for cosmic gamma ray spectrum due to the collisions with the cosmic microwave background. This should occur around E=6× 10¹⁰ GeV, which corresponds to E_cm=3.5× 10⁵ GeV. Cosmic ray events above this cutoff are however claimed. There should be some mechanism allowing for ultra high energy cosmic rays to propagate over much longer distances as allowed by the limits. Cosmic rays should be able to propagate without collisions. Many-sheeted space-time suggests manners for how gamma rays could avoid collisions with microwave background. For instance, gamma rays could be dark in TGD sense and therefore have large value of Planck constant. One can even imagine exotic variants of hadrons, which differ from ordinary hadrons in that they do not have quarks and therefore no interactions with the microwave background.
   
   
6. The highest energies of cosmic rays are around E=10¹¹ GeV, which corresponds to E_cm=4× 10⁵ GeV. M₆₁ nucleon and pion correspond to the mass scale of 6× 10⁶ GeV and 8.4× 10⁵ GeV. These events might correspond to the creation of M₆₁ hadrons in atmosphere.
   

The identification of the hadronic space-time sheet as a super-symplectic mini black-hole suggests the science fictive possibility that part of ultra-high energy ∈dexcosmic rays cosmic rays could be also protons which have lost their valence quarks. These particles would have essentially same mass as proton and would behave like mini black-holes consisting of dark matter. They could even give a large contribution to the dark matter. Since electro-weak interactions are absent, the scattering from microwave background is absent, and they could propagate over much longer distances than ordinary particles. An interesting question is whether the ultrahigh energy cosmic rays having energies larger than the GZK cut-off of 5× 10¹⁰ GeV in the rest system of Earth are super-symplectic mini black-holes associated with M₁₀₇ hadron physics or some other copy of hadron physics.

Higgs and finnish folklore

The belief of the average physics blogger has been that Higgs has been discovered and that experimentalists are just too conservative to say it aloud. Gradually this belief has begun to shatter as I have told in previous postings.

Lubos Motl was one of the most fanatic believers in Higgs and labeled Higgs skeptics as madmen and idiots whereas I had to take the non-rewarding role of a wise guy. This wise guy appears also in finnish folklore telling about the people of "Hölmölä" and - believe or not - has name "Matti";-). I did not find any direct translation for "Hölmölä" in English. In any case, it is a village populated by friendly but rather simple fellows getting into trouble all the time and Matti has to solve their problems. I cannot avoid the feeling that physics blogs might define the counterpart for "Hölmölä" in weblore.

Name obliges and I have tried to fulfill my duty by repeatedly telling to the web villagers that this signal at 125 GeV does not look quite like Higgs and that Higgs mechanism is mathematically an extremely ad hoc and ugly manner to reproduce the particle masses. It seems that in vain.

In his last posting related to growing suspicions about the real nature of Higgs signal Lubos makes clear that those who have dared to suspect the correctness of the interpretation of 125 GeV state as Higgs, are mentally unstable. Lubos himself however considers now seriously the possibility that it is not Higgs after all, and reveals to the reader that even him - Lubos! - can be wrong!

Lubos promotes the interpretation as a "radion", a prediction of an extremely ad hoc theory known as Randall-Sundrum model deriving its respectability from the claim that its inspiration comes from string theory. Surprisingly, Lubos does not regard himself as mentally unstable whereas those who were more far sighted are such according to Lubos. Could the reason be that the radion model has been developed by Names! Or is this just a new example about the strange non-Boolean logic of Lubos? Reader perhaps remembers from the postings of Lubos that standard SUSY should have been found long time ago. Lubos also revealed the rumor about discovery of stop and hinted about possession of super-secret information about the discovery of SUSY to be revealed within few months.

In light of this reader perhaps understands why I find difficult to avoid the association between physics blog community and "Hölmölä".

Addition: Colleagues seem to be realizing that something might be wrong with the interpretation of 125 GeV signal as Higgs. For instance, J. W. Moffat suggests interpretation of 125 GeV state as pseudo-scalar analogous to pion. Also TGD based interpretation corresponds to pseudoscalar: the fact that one works 8-D context however forces to modify pseudoscalar to the analog of pseudo-vector in CP₂ degrees of freedom having both neutral and charged components. Moffat does not propose charged companions.

Standard views about cosmic rays and galactic dark matter in difficulties

The morning walk in web resulted in two interesting news. Both told about experimental findings which challenge existing theories. The first theory assumes that highly energetic cosmic rays get their energy in ultra strong magnetic fields. Second theory states what galactic dark matter consists of wimps or some other particles with exotic name surrounding galaxy as a spherical or nearly spherical halo.

Standard view about very high energy cosmic rays is in difficulties

The latest news is that cosmic ray models which assume that cosmic rays receive their enormous energies (up to 10⁹ GeV = 10⁶ TeV at least) by acceleration in gigantic magnetic fields are in difficulties. The high energy protons resulting in this manner should be accompanied by neutrinos. Too few neutrinos are detected.

TGD based explanation for ultra-high cosmic ray energies introduced already around 1995 (see "Cosmic rays and Mersenne primes" here) would be that the energies are due to the fact that they are decay products of scaled up copies of hadron physics assignable to Mersenne primes M_k= 2^k-1 with k=89 with mass scale of .5 TeV (scaled up proton mass) that LHC is painfully trying to avoid to discover;-), k=61 with mass scale of 8×10³ TeV, k=31 with mass scale 2.6×10⁸ TeV,... The dynamics is expected to be so fast that the cosmic rays arriving here are nucleons or gamma rays, which interact with atmosphere and can create hadrons of the scaled up versions of the ordinary hadron physics. Simple estimates show that most events would be due to M₈₉ hadrons and highest energy cosmic rays above GKZ bound having energy about 10¹¹ GeV could create M₆₁ hadrons.

In the previous posting I told about evidence by Fermi for gamma ray pairs which could result in annihilations of particles with 130 GeV to two gamma rays. A possible TGD based interpretation of particles could is as charged counterparts of M₈₉ pions predicted to have this mass by naive scaling argument. Neutral M₈₉ pion would be just the Higgs candidate with mass 125 GeV, which does not seem to behave like a descent Higgs should behave despite the fact that theoreticians strongly encourage it to keep at least the name "Higgs" and title "God particle". Also an "octave" of M₈₉ neutral pion predicted to be possible by p-adic length scale hypothesis can be considered as an explanation but now the mass would be 260 GeV and 10 GeV larger than what experimental finding suggests. These particles could be dark in TGD sense (having Planck constant equal to a multiple of its standard value) but I am not aware of any constraint forcing this.

Standard view about dark matter is in difficulties

Also the standard view about galactic dark matter is in difficulties. The assumption is that galactic dark matter forms a spherical halo around the galaxy: with a suitable distribution this would explain constant velocity distribution of distant stars. Sometime ago NASA reported that Fermi telescope does not find support for dark matter in this sense in small faint galaxies that orbit our own.

Another blow against standard view came now. A team using the MPG/ESO 2.2-metre telescope at the European Southern Observatory's La Silla Observatory, along with other telescopes, has mapped the motions of more than 400 stars up to 13,000 light-years from the Sun. Also in this case the signature would have been the gravitational effects of dark matter. No evidence for dark matter has been found in this volume. The results will be published in an article entitled "Kinematical and chemical vertical structure of the Galactic thick disk II. A lack of dark matter in the solar neighborhood," by Moni-Bidin et al. to appear in The Astrophysical Journal.

Also these findings support the TGD based model for galactic dark matter (to be carefully distinguished from dark matter as large hbar phases appearing in much smaller amounts and essential for life in TGD inspired quantum biology). TGD based model for the galactic dark matter postulates that the dominating contribution is along long magnetic flux tubes resulting from these during cosmic expansion and containing galaxies around them like pearls in a necklace.

The distribution of dark matter would be concentrated around this string rather than forming a spherical halo around galaxy. This would give rise to a gravitational acceleration behaving like 1/ρ , where ρ is transversal distance from the string, explaining constant velocity spectrum for distant stars. The killer prediction is that galaxies could move along the string direction freely. Large scale motions difficult to understand in standard cosmology has been indeed observed. It has been also known for a long time that galaxies tend to concentrate on linear structures

Sad to say, that again there is impenetrable communication barrier involved. People calling themselves professionals read only articles published in so called respected journals. If the article is by a theoretician he must be a name. Lubos tells about the war between experimenters who find experimental evidence for the existence of something which they interpret as dark matter and other experimenters who find evidence for its absence. Situation would become much more interesting if some-one would be ready to ask whether something in the basic thinking might be wrong in this kind of situation! Physics cannot be a war! In more peaceful circumstances, fresh views and new ideas could flourish. One example of a fresh view is that in TGD Universe the dark matter candidates claimed by DAMA could be created in atmospheric collisions of cosmic rays from the direction of galactic center rather than coming from the galactic center (see this)!

It is ironic that at the period when electronic communications have transformed the world to a village, the deepest problems in modern physics seem to be due to the impossibility to communicate! Professional ego is an extremely effective cognitive immune system!

Has Fermi detected dark matter?

Resonaances reports about a possible dark matter signal at Fermi satellite (see this). Also Lubos has a posting about the finding and mentions that the signficance is 3.3 sigma.

The proposed dark matter interpretation for the signal would be pair of monochromatic photons with second one detected at Earth. The interpretation would be that dark matter particles with mass m nearly at rest in galactic center annihilate to a pair of photons so that one obtains a pair of photons with energy equal to the cm energy which is in a good approximation the sum E= 2× m for the masses of the particles. The mass value would be around m=130 GeV for if the final state involves only 2 photons.

In TGD framework I would consider as a first guess a pion like state decaying to two photons with standard coupling given by the coupling to the "instanton density" E• B of electromagnetic field. The mass of this particle would be 260 GeV, in reasonable approximation 2 times the mass m=125 GeV of the Higgs candidate.

1. Similar coupling was assumed to explain also the CDF anomaly (see this). The anomaly would have been produced by tau-pions which are pionlike states formed by pairs of colored excitations of tau and its antiparticle (or possibly their super-partners). What was remarkable that the mass had three values coming as powers of two: M=2^k× 2m(τ), k=0,1,2. The interpretation in terms of p-adic length scale hypothesis would be obvious: also the octaves of the basic state are there. The constraint from intermediate gauge boson decay widths requires that these states are dark in TGD sense and therefore correspond to a non-standard value of Planck constant coming as an integer multiple of the standard value.
   
   
2. Also the explanation of the findings of Pamela requires octaves of tau-pion produced in Earth's atmosphere (see this).
   
   
3. Even ordinary pion should have 2-adic octaves. But doesn't this kill the hypothesis? We "know" that pion does not have any octaves! Maybe not, there is recent evidence for satellites of ordinary pion with energy scale of 40 MeV interpreted in terms of IR Regge trajectories assignable to the color magnetic flux tubes assignable to pion (see this). There has been several wrong alarms about Higgs: at 115 GeV, 125 GeV, 145 GeV at least. Could it be that there there is something real behind these wrong alarms: the scale for IR Regge trajectories would be about 20 GeV now!
   
   

So: could the dark matter candidates with mass around 260 GeV correspond to the first octave of M₈₉ pion with mass around 125 GeV, the particle that colleagues want to call Higgs boson although its decay signatures suggest something different?

1. In this case it does not seem necessary to assume that the Planck constant has non-standard value although this is possible.
   
   
2. This particle should be produced in M₈₉ strong interactions in the galactic center. This would require the presence of matter consisting of M₈₉ nucleons emitting these pions in strong interactions. Galactic center is very exotic place and believed to contain even super-massive black hole. Could this environment accommodate also a scaled up copy of hadron physics? Presumably this would require very high temperatures with thermal energy of order .5 TeV correspond to the mass of M₈₉ proton to make possible the presence of M₈₉ matter. Or could M₈₉ pion be produced in ultrastrong non-orthogonal electric and magnetic fields in the galactic center by the coupling to the instanton density. The needed field strengths would be extremely high. I have indeed proposed long time ago an explanation of very high energy cosmic rays in terms of the decay products of scaled up hadron physics (see "Cosmic Rays and Mersenne primes" of this).
   

One can of course imagine that the photon pair is produced in the annilation of M₈₉ pions with opposite charges via standard electromagnetic coupling. Also the annihilation of M₈₉ spions consisting of squark pair can be considered in TGD framework where squarks could have same mass scale as quarks. In this case mass would be near 125 GeV identified as mass of neutral M₈₉ pion. By scaling up the mass difference 139.570-134.976 MeV of the ordinary charged and neutral pion by the ratio of the pion M₈₉ and M₁₀₇ pion masses equal to 125/140 × 10³ one obtains that the charged M₈₉ pion should have mass equal to 129.6 MeV to be compared with the 130 GeV mass suggested by experimental evidence.

Quantum Mechanics as Quantum Mathematics?

Quantum Mathematics (QM) suggests that the basic structures of Quantum Mechanics (QM) might reduce to fundamental mathematical and metamathematical structures, and that one even consider the possibility that Quantum Mechanics reduces to Quantum Mathematics with mathematician included or expressing it in a concice manner: QM=QM!

The notes below were stimulated by an observation raising a question about a possible connection between multiverse interpretation of quantum mechanics and quantum mathematics. The heuristic idea of multiverse interpretation is that quantum state repeatedly branches to quantum states which in turn branch again. The possible outcomes of the state function reduction would correspond to different branches of the multiverse so that one could save keep quantum mechanics deterministic if one can give a well-defined mathematical meaning to the branching. Could quantum mathematics allow to somehow realize the idea about repeated branching of the quantum universe? Or at least to identify some analog for it? The second question concerns the identification of the preferred state basis in which the branching occurs.

Quantum Mathematics briefly

Quantum Mathematics replaces numbers with Hilbert spaces and arithmetic operations + and × with direct sum ⊕ and tensor product ⊗.

1. The original motivation comes from quantum TGD where direct sum and tensor product are naturally assigned with the two basic vertices analogous to stringy 3-vertex and 3-vertex of Feynman graph. This suggests that generalized Feynman graphs could be analogous to sequences of arithmetic operations allowing also co-operations of ⊕ and ⊗.
   
   
2. One can assign to natural numbers, integers, rationals, algebraic numbers, transcendentals and their p-adic counterparts for various prime p Hilbert spaces with formal dimension given by the number in question. Typically the dimension of these Hilbert spaces in the ordinary sense is infinite. Von Neuman algebras known as hyper-finite factors of type II₁ assume as a convention that the dimension of basic Hilbert space is one although it is infinite in the standard sense of the word. Therefore this Hilbert space has sub-spaces with dimension which can be any number in the unit interval. Now however also negative and even complex, quaternionic and octonionic values of Hilbert space dimension become possible.
   
   
3. The decomposition to a direct sum matters unlike for abstract Hilbert space as it does also in the case of physical systems where the decomposition to a direct sum of representations of symmetries is standard procedure with deep physical significance. Therefore abstract Hilbert space is replaced with a more structured objects. For instance, the expansion ∑_n x_npⁿ of a p-adic number in powers of p defines decomposition of infinite-dimensional Hilbert space to a direct sum ⊕_n x_n⊗ pⁿ of the tensor products x_n⊗ pⁿ. It seems that one must modify the notion of General Coordinate Invariance since number theoretic anatomy distinguishes between the representations of space-time point in various coordinates. The interpretation would be in terms of cognition. For instance, the representation of Neper number requires infinite number of pinary digits whereas finite integer requires onlya finite number of them so that at the level of cognitive representations general coordinate invariance is broken.
   

   Note that the number of elements of the state basis in pⁿ factor is pⁿ and m∈ {0,...,p-1} in the factor x_n. Therefore the Hilbert space with dimension pⁿ>x_n is analogous to the Hilbert space of a large effectively classical system entangled with the microscopic system characterized by x_n. p-Adicity of this Hilbert space in this example is for the purpose of simplicity but raises the question whether the state function reduction is directly related to cognition.
   
   

4. On can generalize the concept of real numbers, the notions of manifold, matrix group, etc... by replacing points with Hilbert spaces. For instance, the point (x₁,..,x_n) of Eⁿ is replaced with Cartesian product of corresponding Hilbert spaces. What is of utmost importance for the idea about possible connection with the multiverse idea is that also this process can be also repeated indefinitely. This process is analogous to a repeated second quantization since intuitively the replacement means replacing Hilbert space with Hilbert space of wave functions in Hilbert space. The finite dimension and its continuity as function of space-time point must mean that there are strong constraints on these wave functions. What does this decomposition to a direct sum mean at the level of states? Does one have super-selection rules stating that quantum inteference is possible only inside the direct summands?
   
   
   
5. Could one find a number theoretical counterpart for state function reduction and preparation and unitary time evolution? Could zero energy ontology have a formulation at the level of the number theory as earlier experience with infinite primes suggest? The proposal was that zero energy states correspond to ratios of infinite integers which as real numbers reduce to real unit. Could zero energy states correspond to states in the tensor product of Hilbert spaces for which formal dimensions are inverses of each other so that the total space has dimension 1?
   

Unitary process and state function reduction in ZEO

The minimal view about unitary process and state function reduction is provided by ZEO.

1. Zero energy states correspond to a superposition of pairs of positive and negative energy states. The M-matrix defining the entanglement coefficients is product of Hermitian square root of density matrix and unitary S-matrix, and various M-matrices are orthogonal and form rows of a unitary U-matrix. Quantum theory is square root of thermodynamics. This is true even at single particle level. The square root of the density matrix could be also interpreted in terms of finite measurement resolution.
   
   
2. It is natural to assume that zero energy states have well-defined single particle quantum numbers at the either end of CD as in particle physics experiment. This means that state preparation has taken place and the prepared end represents the initial state of a physical event. Since either end of CD can be in question, both arrows of geometric time identifiable as the Minkowski time defined by the tips of CD are possible.
   
   
3. The simplest identification of the U-matrix is as the unitary U-matrix relating to each other the state basis for which M-matrices correspond to prepared states at two opposite ends of CD. Let us assume that the preparation has taken place at the "lower" end, the initial state. State function reduction for the final state means that one measures the single particle observables for the "upper" end of CD. This necessarily induces the loss of this property at the "lower" end. Next preparation in turn induces localization in the "lower" end. One has a kind of time flip-flop and the breaking of time reversal invariance would be absolutely essential for the non-triviality of the process.
   

The basic idea of Quantum Mathematics is that M-matrix is characterized by Feynman diagrams representing sequences of arithmetic operations and their co-arithmetic counterparts. The latter ones give rise to a superposition of pairs of direct summands (factors of tensor product) giving rise to same direct sum (tensor product). This vision would reduce quantum physics to generalized number theory. Universe would be calculating and the consciousness of the mathematician would be in the quantum jumps performing the state function reductions to which preparations reduce.

Note that direct sum, tensor product, and the counterpart of second quantization for Hilbert spaces in the proposed sense would be quantum mathematics counterpart for set theoretic operations, Cartesian product and formation of the power set in set theory.

ZEO, state function reduction, unitary process, and quantum mathematics

State function reduction acts in a tensor product of Hilbert spaces. In the p-adic context to be discussed n the following x_n⊗ pⁿ is the natural candidate for this tensor product. One can assign a density matrix to a given entangled state of this system and calculate the Shannon entropy. One can also assign to it a number theoretical entropy if entanglement coefficients are rationals or even algebraic numbers, and this entropy can be negative. One can apply Negentropy Maximization Principle to identify the preferred states basis as eigenstates of the density matrix. For negentropic entanglement the quantum jump does not destroy the entanglement.

Could the state function reduction take place separately for each subspace x_n⊗ pⁿ in the direct sum ⊕_n x_n⊗ pⁿ so that one would have quantum parallel state function reductions? This is an old proposal motivated by the many-sheeted space-time. The direct summands in this case would correspond to the contributions to the states localizable at various space-time sheets assigned to different powers of p defing a scale hierarhcy. The powers pⁿ would be associated with zero modes by the previous argument so that the assumption about independent reduction would reflect the super-selection rule for zero modes. Also different values of p-adic prime are present and tensor product between them is possible if the entanglement coefficients are rationals or even algebraics. In the formulation using adeles the needed generalization could be formulated in a straightforward manner.

How can one select the entangled states in the summands x_n⊗ pⁿ? Is there some unique choice? How do unitary process and state function reduction relate to this choice? Could the dynamics of Quantum Mathematics be a structural analog for a sequence of state function reductions taking place at the opposite ends of CD with unitary matrix U relating the state basis for which single particle states have well defined quantum numbers either at the upper or lower end of CD? Could the unitary process and state function reduction be identified solely from the requirement that zero energy states correspond to tensor products Hilbert spaces, which correspond to inverses of each other as numbers? Could the extension of arithmetics to include co-arithmetics make the dynamics in question unique?

What multiverse branching could mean?

Could QM allow to identify a mathematical counterpart for the branching of quantum states to quantum states corresponding to preferred basis? Could one can imagine that a superposition of states ∑ c_nΨ_n in a direct summand x_n⊗ pⁿ is replaced by a state for which Ψ_n belong to different direct summands and that branching to non-intefering sub-universes is induced by the proposed super-selection rule or perhaps even induces state function reduction? These two options seem to be equivalent experimentally. Could this decoherence process perhaps correspond to the replacement of the original Hilbert space characterized by number x with a new Hilbert space corresponding to number y inducing the splitting of x_n⊗ pⁿ? Could the interpretation of finite integers x_n and pⁿ as p-adic numbers p₁≠ p induce the decoherence?

This kind of situation is encountered also in symmetry breaking. The irreducible representation of a symmetry group reduces to a direct sum of representations of a sub-group and one has in practice super-selection rule: one does not talk about superpositions of photon and Z⁰. In quantum measurement the classical external fields indeed induce symmetry breaking by giving different energies for the components of the state. In the case of the factor x_n⊗ pⁿ the entanglement coefficients define the density matrix characterizing the preferred state basis. It would seem that the process of branching decomposes this state space to a direct sum 1-D state spaces associated with the eigenstates of the density matrix. In symmetry breaking superposition principle holds true and instead of quantum superposition for different orientations of "Higgs field" or magnetic field a localization selecting single orientation of the "Higgs field" takes place.

Could state function reduction be analogous process? Could non-quantum fluctuating zero modes of WCW metric apper as analogs of "Higgs fields". In this picture quantum superposition of states with different values of zero modes would not be possible, and state function reduction might take place only for entanglement between zero modes and non-zero modes.

The replacement of a point of Hilbert space with Hilbert space as a second quantization

The fractal character of the Quantum Mathematics is what makes it a good candidate for understanding the self-referentiality of consciousness. The replacement of the Hilbert space with the direct sum of Hilbert spaces defined by its points would be the basic step and could be repeated endlessly corresponding to a hierarchy of statements about statements or hierarchy of n^th order logics. The construction of infinite primes leads to a similar structure.

What about the step leading to a deeper level in hierarchy and involving the replacement of each point of Hilbert space with Hilbert space characterizing it number theoretically? What could it correspond at the level of states?

1. Suppose that state function reduction selects one point for each Hilbert space x_n⊗ pⁿ. The key step is to replace this direct sum of points of these Hilbert spaces with direct sum of Hilbert spaces defined by the points of these Hilbert spaces. After this one would select point from this very big Hilbert space. Could this point be in some sense the image of the Hilbert space state at previous level? Should one imbed Hilbert space x_n⊗ pⁿ isometrically to the Hilbert space defined by the preferred state x_n⊗ pⁿ so that one would have a realization of holography: part would represent the whole at the new level. It seems that there is a canonical manner to achieve this. The interpretation as the analog of second quantization suggest the identification of the imbedding map as the identification of the many particle states of previous level as single particle states of the new level.
   
   
2. Could topological condensation be the counterpart of this process in many-sheeted spacetime of TGD? The states of previous level would be assigned to the space-time sheets topologically condensed to a larger space-time sheet representing the new level and the many-particle states of previous level would be the elementary particles of the new level.
   
   
3. If this vision is correct, second quantization performed by theoreticians would not be a mere theoretical operation but a fundamental physical process necessary for cognition! The above proposed unitary imbedding would imbed the states of the previous level as single particle states to the new level. It would seem that the process of second quantization, which is indeed very much like self-reference, is completely independent from state function reduction and unitary process. This picture would conform with the fact that in TGD Universe the theory about the Universe is the Universe and mathematician is in the quantum jumps between different solutions of this theory.
   

Returning to the motivating question: it seems that the endless branching of the states in multiverse interpretation cannot correspond to a repeated second quantization but could have interpretation as a decoherence identifiable as delocalization in zero modes. If state function is allowed, it corresponds to a localization in zero modes analogous to Higgs mechanism. The Quantum Mathematics realization for a repeated second quantization would represent a genuinely new kind of process which does not reduce to anything already known.

How to build a quantum computer from magnetic flux tubes?

Magnetic flux tubes play a key role in TGD inspired model of quantum biology. Could the networks of magnetic flux tubes containing dark particles with large hbar in macroscopic quantum states and carrying beams of dark photons define analogs of electric circuits? This would be rather cheap technology since no metal would be needed for wires. Dark photon beams would propagate along the flux tubes representing the analogs of optical cables and make possible communications with maximal signal velocity.

I have actually made much more radical proposal in TGD inspired quantum biology. According to this proposal, flux tube connections are dynamical and can be changed by reconnection of two magnetic flux tubes. The signal pathways A→ C and B→ D would be transformed to signal pathways to A→ D and B→ C by reconnection. Reconnection actually represents a basic stringy vertex. The contraction of magnetic flux tubes by a phase transition changing Planck constant could be fundamental in bio-catalysis since it would allow distant molecules connected by flux tubes to find each other in the molecular crowd.

DNA as a topological quantum computer is the idea that I have been developing for 5 years or so. I have concentrated on the new physics realization of braids and devoted not much thought to how the quantum computer problems might run in this framework. I was surprised to realize how little I know about what happens in even ordinary computation. Instead of going immediately to Wikipedia I take the risk of publicly making myself fool and try to use my own brain.

1. What can one learn from ordinary computer programs

One could begin with the question what happens in classical computation. How the program is realized and how it runs? The notion of Turing machine represents an extreme abstraction mentioning nothing about the technical side and does not help much in attempts to answer these questions. Turing paradigm also assumes that program is a temporal sequence of operations. These operations could however correspond to a linear spatial sequences and inputs and outputs in this case would correspond to boundary values at the ends of the linear structure. This requires that the dynamics is such that evolution in spatial direction is analogous to a deterministic time evolution. In this case it is much easier to imagine biological realizations of quantum computer programs in TGD inspired bio-world.

To develop concrete ideas, one can start from the picture provided by ordinary computer program.

1. Programs consist of temporal/spatial sequences of commands and commands represent basic functions from which one can build more complex functions by the composition of functions having some numbers of input and output arguments. The eventual output variable can be expressed by printing of a piece of text or as an image in the computer screen. Each step in the program corresponds to a composition of functions: f_n+1= g_n+1 o f_n. There is some minimal set of primitive/prime functions from which one builds up more complex functions by composition.

2. How this is realized at the level of hardware? One can assume that the basic functions are at some fixed places in the computer memory having addresses given by integers represented as bit sequences. This address represents the command - a name of the function. The names for input variables and output variables are bit sequences giving the addresses of the places containing the values of these variables. Program is a sequence of commands represented as bit sequences giving the address of the function to be computed at a given step and the addresses of inputs and outputs. As the processing unit reads the command, it generates/activates connections from the addresses of inputs to the address representing the function and from this address to the addresses of outputs.

   Essentially the challenge is to reconnect, build/activate connections. An interesting question is whether learning as strengthening of synaptic connections is one particular example of this process.

3. How the sequence of bits representing command address is realized? As the processing unit reads the address of command it should automatically create/activate a connection from this address to the command address. The connections from the processing unit to the addresses could exist physically as wirings.

4. It is not necessary that program is dynamical so that the inputs and outputs would be initial and final values of variables. Inputs and outputs could also correspond to values of variables at the ends of a linear structure. In topological quantum computation space-like entanglement would represent superposition of input-output pairs characterizing a function as a rule with instances represented as instances appearing in the superposition.

If this picture is roughly correct, re-connection would be the basic process. Reconnection is the basic process for magnetic flux tubes and ADP↔ ATP has been assigned to this process with ATP molecule serving as a relay activating the flux tube connection. Maybe ADP-ATP process, which is usually seen as a basic step of metabolism, could be seen as the core step for quantum computation performed by living matter. One expects that the presence ATP makes the rule represented by negentropic quantum entanglement conscious.

2. Quantum computation magnetic flux tubes as connections

Consider now quantum computation could take place in a circuitry having magnetic flux tubes as wires and some bio-molecules of groups of them as units defining prime functions. DNA as topological quantum computer could be taken as a starting point. The outcome of quantum computation is determined statistically as ensemble average so that a large number of copies of the program should be present and realized in terms of groups of cells or molecules connected by braidings if the quantum computation is space-like. This option seems more natural than time-like quantum computation realized as a 2-D liquid flow of lipids in the lipid layers of the cell membrane.

2.1 The hardware

Consider first the hardware of topological quantum computation using space-like braids.

1. Magnetic flux tubes would represent the wires along which inputs and outputs travel in the case of classical computation or dynamical quantum computation. In the case of space-like topological quantum computation entanglement is between the ends of the flux tubes.

2. Variables could be represented in many manners. For space-like quantum computations they could correspond to spin states of dark electrons at flux tubes or to polarization states of dark electrons at the flux tubes. In the original model of DNA as topological quantum computer quarks and antiquarks where proposed as a representation of genetic codons: also this quite science fictive option could make sense in TGD Universe since TGD predicts scaled versions of QCD like dynamics and presence of elementary particles in several p-adic scales and in scales dictated by value of Planck constant for given p-adic length scale.

   The spin states of electron pair has been proposed as one possible representation of the 4 genetic codons. Quantum variables would be represented by qubit sequences and the measurement of qubit would give a bit sequence characterizing the classical value of the variable. Bio-molecules would be natural places for storing the values of the variables. For dynamical computations the values of variables could be transmitted using dark photons.

3. There would exist basic processing units calculating the prime functions from which more complex functions would be obtained as composites. Basic units could correspond to bio-molecules. In the case of classical computation the inputs to molecules and outputs from them would travel along the flux tubes. In quantum computation these signals could be used to control the initial values of the variables. Molecules could also serve as gates for quantum computation.

2.2 Representation of programs

The basic program units in the case of quantum computation would be represented by braidings.

1. If the ends of braid strands are able to move freely when needed, it becomes possible to re-write programs. Lipid layers of cell membrane can be in liquid crystal state so that these are ideal for this purpose. The time-like braiding resulting from lipid flow and representing running topological quantum computation program would induce space-like braiding representing space-like topological quantum computation or a rule. A particular quantum computer program represented as space-like braiding of the flux tubes would result as liquid crystal melts for a moment and freezes again.

   The process in which proteins covered by ordered water analogous to ice temporarily melt and form aggregates is basic process induced by the feed of energy to the cellular system and could be compared to cellular summer. This process could mean quite generally molecular re-programming induced by the flow of cellular water inducing molecular flows inducing re-braidings. The braiding would also store the highlights of the cellular summer to cellular memory! This could be also seen learning by a modification of various quantum computer programs.

2. Negentropic entanglement is highly suggestive and would conform with the idea that the rule represented by entanglement represents conscious information or information which can become conscious. The process of becoming conscious information could involve ATP→ ADP and de-activating the flux tube and destroy the information. Time-like braiding represented by liquid flow would modify space-like braiding.

   It is not quite clear whether the information is conscious when negentropic entanglement (and ATP) is present - as Bohm's notion of active information would suggest - or when ATP is transformed to ADP and connection becomes passive. Negentropic entanglement can be stable with respect to NMP so that the presence of ATP could mean period of conscious experience - negentropic entanglement could be analogous to active information.

   TGD based model for the memory recall by sending negative energy signals to geometric past suggests that the absorption of negative energy photon transforms ATP to ADP. Conscious experience is regenerated in the geometric now where the negative energy signal came from - perhaps by transforming ADP to ATP by using the negative resulting by sending of negative energy signal! Conscious reading would be actually memory recall and analogous to teleportation? The destruction of the representation of memory in the geometric past would have interpretation in terms of no-cloning theorem.

3. Static realizations of the programs are easier to imagine since no temporal codes are needed for the transfer of bits. An attractive idea is that the computations are represented by static entanglements for linear structures and that time-like braiding allows to modify the programs.

2.3 The realization of program

The program would be basically a sequence of address lists. Address list would contain the address of the function to be performed and the addresses of the input molecules and output molecules. How to represent the address physically?

1. The simplest manner to realize this would use existing flux tubes connecting the processing unit to all possible input and output addresses as well as command addresses, and activate those flux tubes to which input and output data are assigned and reconnect them to the flux tubes connecting processing unit to the unit representing the function. The processing unit would have flux tubes coming from all possible inputs, going to all possible outputs, flux tubes going to places representing functions and coming from these places. Processing unit would be like a relay station or old fashioned telephone center whose sole purpose would be to create connections by reconnecting flux tubes. ATP molecule would be probably involved with the activation and - allowing a sloppy language - one could say that communication line becomes conscious when ATP is attached to it.
   1. Addressing would be just selection of activated molecules and analogous to that used in telephone network or computer network connected by cables. This would require static flux tube network and flux tubes could be either active or passive. In passive state flux tubes could be short-cut by a reconnection with hydrogen bond so that the ends of cut flux tube would end up to water molecules. This is however not necessary. Activation in absence of the short cut would involve reconnection of a flux tube with a flux tube connecting two parts of ATP - possibly hydrogen bond again- so that ATP becomes part of the flux tubes. If also short cut is involved, the strands coming to the two water molecules reconnect and generate hydrogen bond and flux tube to which ATP would attach in the proposed manner. As ATP is used it transforms to ADP and de-attaches from the flux tube.

   2. One can imagine also a dynamical addressing based on the generation of magnetic flux tubes between inputs and submodules. The computational process could be still space-like. The first manner to realize dynamical addressing would be by attaching to the ends of dynamical flux tubes biomolecules, which bind to specific receptors. Receptor mechanism would allow to connect distant cells to each other and build a magnetic flux tube connection between them. Computational unit specialized to run a specific program could excrete biomolecules binding to the input and output receptors: this program would realized function in terms of space-like entanglement. Glands emit hormones binding to receptors and various glands could in principle serve as computational units. Various information molecules bind very selectively and this might also relate to quantum space-like computations.

   3. Second mechanism of dynamical addressing would use dark photons. In this case resonant interaction selecting the target would replace the receptor mechanism. In this kind of situation one can claim that flux tubes are un-necessary, one can use just resonance to build connection to a desired place just as one does in radio communications. Of course, topological light rays could be accompanied by flux tubes. For instance, DNA nucleotide could attach by flux tube to its conjugate in distant DNA molecule and if the connection is based on resonance only similar nucleotide sequences could connect with each other. I have discussed this kind of mechanism in a model for remote replication of DNA based on the experimental work by Peter Gariaev and his group. The resonance mechanism could also make possible to establish flux tube connections and the quantum computation could be a static operation.

2. DNA as topological quantum computer vision gives some idea about how the computer program could be realized as a spatial linear structure.
   1. Program would be a sequence of topological quantum computations. Given topological quantum computation would be represented by a braiding of flux tubes connecting DNA nucleotides with the lipid molecules of the inner lipid layer. Program would correspond to a linear sequence of cells with the outer lipid layer connected to the DNA of the second cell.

   2. Lipid flows at given lipid layer could be used to rewrite programs and the programs could respond to the changes in environment in this manner: this would require that the lipid layer is in liquid crystal state during the period when program is changed. Also nerve pulse patterns would induce these flows. Programs would also represent memories as rules realized as quantum abstractions or as quantum functions.

   3. The program would "run" in the spatial direction. The selection of active input and output variables would be by acting the connection from molecule in question by attaching ATP as a relay through which the reconnected flux tube would traverse. This would be also part of the writing of the program. The superposition of entangled inputs and outputs could be seen as a quantum superposition of classical programs assigning outputs to inputs. Also microtubule-lipid layer braiding suggested also to play a key role in the realization of memories could give rise to similar space-like quantum computation representing rules.

   4. The effective 2-dimensionality implied by strong form of holography implied in turn by strong form of general coordinate invariance means that the physics depends on partonic 2-surfaces and 4-D tangent space data at them. This suggests that the dynamics on space-like 3-surfaces and light-like orbits of partonic 2-surfaces is fixed by a process analogous to gauge selection. Does just this effective gauge symmetry make possible to write quantum computer programs? Already ordinary deterministic computer program means selection of one particular dynamics from several alternative options suggesting that strict determinism is broken.

3. What could be the role of bio-catalysis in the computation? Bio-catalysis is a central part of the biological information processing and it would not be surprising if the catalysts connected by flux tubes to substrate molecules were involved with the computations. An attractive idea is that various information molecules binding to receptors involved with bio-control (neurotransmitters, hormones, etc...) are involved with building the flux tube connections between cells. These bio-molecules could carry the ends of flux tubes to special places for which receptors serve as addresses and in this manner build hardware for topological quantum computation involving inputs and outputs in distant parts of the body. The final output could be transformed to controlled gene expression. Quite generally, catalysts bind very selectively and could play a role similar that played by information molecules in building up the quantum computer programs.

4. One can imagine also purely classical computation based on catalytic mechanism probably allowing generalization to quantum case. The idea is that computer program - understood now as dynamical structure - is analogous to what happens in fairy tale in which hero finds a key which fits to a lock of a room containing a key which... There exists a beautiful realization of classical computation in terms of chemical concentrations using DNA. The output of given reaction representing computational step appears in the next reaction provide the system contains additional participating molecules, which could be both substrate molecules and catalysts. The program could be represented as concentrations of molecules needed at intermediate steps and lock-to-key mechanism guarantees that they are performed in the correct temporal order. Inputs and output molecules could be connected by flux tubes to bio-molecules which bind to specific receptors associated with the molecule representing the particular subprogram. This would automatically create a large number of classical computations proceeding in fixed order, maybe even quantum computations.

For a pdf version of this article see How to build a quantum computer from magnetic flux tubes?.

What could be the counterpart of T-duality in TGD framework?

Stephen Crowell sent me a book of Michel Lapidus about zeros of Riemann zeta and also about his own ideas in this respect. The book has been written in avery lucid manner and looks very interesting. The big idea is that the T-duality of string models could correspond to the functional equation for Riemann zeta relating the values of zeta at different sides of the critical line. T-duality is formulated for strings in space M^d× S¹ or its generalization replacing S¹ with higher-dimensional torus and generalized to fractal strings. Duality states that the transformation R→ 1/R with suitable unit for R defined by string tension is a duality: the physics for these different values of R is same. Intuitively this is due to the fact that the contributions of the string modes representing n-fold winding and those representing vibrations labelled by integer n are transformed to each other in the transformation R→ 1/R.

Lapidus is a mathematician and mathematicians often do not care too much about the physical meaning of the numbers. For a physicist like me it is extremely painful to type the equation R→ 1/R without explicitly explaining that it should actually read as R→ R₀²/R, where R₀ is length unit, which must represent fundamental length scale remaining invariant under the duality transformation. Only after this physicist could reluctantly put R₀=1 but still would feel himself guilty of unforgivable sloppiness. R₀=1 simplifies the formulas but one must not forget that there are three scales involved rather than only two. The question inspired by this nitpicking is how the physics in the length scales R₁ and R relates to the physics in length scale R. Are dualities - or perhaps holography like relations in question- so that T-duality would follow from these dualities?

Could one replace winding number with magnetic charge and T-duality with canonical identification?

How could one generalize T-duality to TGD framework? One should identify the counterpart of the winding number, the three fundamental scales, and say something about the duality transformation itself.

1. In TGD Universe partonic 2-surfaces are the basic object. Partonic 2-surface is not strings and the only reasonable generalization for winding number is as Kähler magnetic charge representing the analog of winding of the partonic 2-surface around magnetically charged 2-sphere of CP₂. Magnetic charge tells how many times partonic 2-surface wraps around the homologically non-trivial geodesic sphere with unit magnetic charge. If the generalization of T-duality holds true, one would expect that the contributions of the oscillations and windings of the partonic two-surface to ground state energy must be transformable to each other by the counterpart of the transformation R→ R₀²/R - or something akin to that. Also less concrete and more general interpretations are possible, and below the most plausible interpretation will be considered.

2. The duality R→ R_0^2/R= R₁ gives R_0 as a geometric mean R_0=(RR₁)^1/2 of the scales R and R₁. What are these three length scales in TGD Universe? The obvious candidate for R is CP₂ size scale. p-Adic mass calculations imply that the primary p-adic length scale L_p,1= p^1/2R is of order of Compton length of the elementary particle characterized by the p-adic prime p. The secondary p-adic length scale L_p,2= pR in turn defines the size scale of causal diamond (CD) assignable to the magnetic body of the elementary particle characterized by prime p. For instance, for electron this scale corresponds to .1 seconds, a fundamental biological time scale.

   One indeed has L_p,1= (L_p,2R)^1/2, and CP₂ scale and CD length scale are dual to each other if T-duality holds true. Therefore the duality would relate physics at CP₂ scale - counterpart of Planck length in TGD framework - and in biological scales and would have direct relevance to quantum biology. One has an infinite hierarchy of p-adic length scales and each of them would give rise to one particular instance of the T-duality. Adeles would provide appropriate formulation of T-duality in TGD framework. The corresponding mass scales would be hbar/R, hbar/p^1/2R and hbar/pR. The third scale corresponds to a scale, which for electron corresponds to the 10 Hz frequency in the case of photons. The duality would suggest that the physics associated with the frequencies in EEG scale related to the communications from the biological body to magnetic body is dual to the physics in CP₂ scale.

   Note that one cannot exclude alternative variants of T-duality. In particular, Planck scale and CP₂ length scale as candidates R₁ and R could be considered.

3. What is the interpretation of these three length scales? CP₂ length scale corresponds naturally to the size scale of wormhole contacts. They are Euclidian regions of space-time surface and represent lines of generalized Feynman graphs. Both general arguments and the construction of elementary bosons forces to assign to these regions braid strands playing a role of Euclidian strings. Parallel translation along the strands is essential in the construction of fermionic bilinears as invariant under general coordinate transformations and gauge transformations. The ends of these strands carry fermion and anti-fermion numbers. The counterpart of string tension involved appearing in stringy mass formula implied by super-conformal invariance is indeed determined by R and p-adic thermodynamics leads to a detailed and successful calculations for elementary particle masses using only p-adic thermodynamics, super-conformal invariance, and p-adic length scale hypothesis as basic assumptions.

4. The wormhole throats carrying fermion number are Kähler magnetic monopoles and the wormhole must be accompanied by a second wormhole throat carrying opposite magnetic charge and also a neutrino pair neutralizing the weak isospin so that weak massivation takes place. The end of the flux tube containing the neutrino pair is virtually non-existent at low energies. The length scale for this string must correspond to Compton length for elementary particle given essentially by primary p-adic length scale L_p,1. The more restrictive assumption that this length scale corresponds to the Compton length of weak bosons looks un-necessarily restrictive and looks also un-natural.

5. The excitations with mass scale hbar/pR would correspond to excitations assignable to entire CD, maybe assignable to the flux tubes of the magnetic bodies of elementary particles defining also string like objects but in macroscopic scales. For electron the scale is of order of the circumference of Earth. This dynamics would naturally correspond to the dynamics in Minkowskian space-time regions. The dynamics at intermediate length scale would be intermediate between the Euclidian and Minkowskian dynamics and reduce to that for light-like orbits of partonic 2-surfaces with metric intermediate between Minkowskian and Euclidian.

6. A natural interpretation for T-duality in this sense is in terms of strong form of holography. The interior dynamics at length scale R resp. pR assigned to Euclidian resp. Minkowskian regions of space-time surface corresponds by holography to the dynamics of light-like orbits of partonic 2-surfaces identified as wormhole throats. Therefore the dynamics in Euclidian and Minkowskian regions are dual to each other. Therefore T-duality in TGD sense would follow from the possibility of having both Euclidian and Minkowskian holography. Strong form of holography in turn reduces to strong form of General Coordinate Invariance, which has turned out to be extremely powerful principle in TGD framework.

Is the physics of life dual to the physics in CP₂ scale?

The duality of life with elementary particle physics at CP₂ length scale - the TGD counterpart of Planck scale - looks rather far-fetched idea. There is however already earlier support for this idea.

1. p-Adic physics is physics of cognition, and one can say that living systems are in the algebraic intersection of real and p-adic worlds: the intersection of cognition and matter. Canonical identification maps p-adic physics to real physics. This map takes p-adic integers which are small in p-adic sense to larger integers in real sense and thus maps long real scales to short real scales. Clearly this map is jhighly analogous to the T-duality. p-Adic length scales are indeed explicitly related with the above identification of the T-duality so that canonical identification might be involved with T-duality.

   If this interpretation is correct, cognitive p-adic representations in long real length scales would give representations for the physics in short length scales. EEG range of frequencies allowing communication to the magnetic bodies is absolutely essential for brain function. CDs would correspond to the real physics scale associated with the cognitive representations. These cognitive representations are indeed exactly what our science is building so that T-duality would make also scientist as a part of the big vision!

2. The model for dark nucleons as three quark states led to one of the greatest surprises of my professional life. Under rather general conditions the three quark states for nucleon are in one-one correspondence with the DNA, RNA, tRNA codons, and aminoacids for vertebrate genetic code and there is natural physical correspondence between DNA triplets and aminoacids. This suggests that genetic code is realized at the level of hadrons and that living matter is a kind of emulation for it, or that living matter is representation for matter at hadron level. This leads to rather far reaching speculations about biological evolution - not as random process - but a process analogous R&D applied in industry. New genes would be continually tested at the level of dark matter and the modifications of genome could be carried out if there is a transcription process transforming dark DNA to ordinary DNA.

3. The secondary p-adic mass scale of electron corresponds to the 10 Hz frequency, which defines a fundamental biorhythm. Also to current quark masses, which are actually not so well-known but are in MeV range, one can assign biologically interesting time scales in millisecond range. This suggests that all elementary particles induce physics in macroscopic time scales via their CD:s containing their magnetic bodies.

The unavoidable and completely crazy looking question raised by T-duality is whether there is intelligent life in the Euclidian realm below the CP₂ length scale - inside the lines of generalized Feynman graphs. This kind of possibility cannot be avoided if one takes holography absolutely seriously. In purely mathematical sense TGD suggests even stronger form of holography based on the notion of infinite primes. In this holography the number theoretic anatomy of given space-time point is infinitely complex and evolves. The notion of quantum mathematics replacing numbers by Hilbert spaces representing ordinary arithmetics in terms of direct sum and tensor product suggest the same. Space-time point would be in this picture its own infinitely complex Universe - the Platonia.

Could one get expression for Kähler coupling strength from restricted form of modular invariance?

The contributions to the exponent of the vacuum functional, which is proportional to Kähler action for preferred extremal, are real resp. imaginary in Euclidian resp. Minkowskian regions. Under rather general assumptions (weak form of electric-magnetic duality defining boundary conditions at wormhole throats plus additional intuitively plausible assumption) these contributions are proportional to the same Chern-Simons term but with possibly different constant of proportionality.

These terms sum up to a Chern-Simons term with a coefficient analogous to the complex inverse gauge coupling

τ=θ/2π + i4π/g²_K .

The real part would correspond to Kähler function coming from Euclidian regions defining the lines of generalized Feynman diagrams and imaginary part to Minkowskian regions. There are could arguments suggesting that With the conventions that I have used θ/2π is counterpart for 1/α_K and there are good arguments that it corresponds to finte structure constant in electron length scale. Furthermore, T-duality would suggest that τ is proportional to 1+i so that one would have

θ/2π = 4π/g²_K .

This condition would fit nicely with the fact that Chern-Simons contributions from Minkowskian and Euclidian regions are identical. If this equation holds true the modular transformations must reduce to those leaving this relationship invariant and can only permute the complex and real parts and thus leave τ invariant. One could also interpret this value of τ as physically especially interesting representation and assign to all values of τ related by modular transformation an isotropy group leaving it fixed. All other physically equivalent values would be obtained as SL(2,Z) orbit of this value.

The counterpart of T-duality should somehow relate dynamics in Minkowskian and Euclidian regions and this raises the question whether it corresponds to τ→ iτ and is represented by some duality transformation

τ→ (aτ+b)/(cτ+d) ,

where (a,b;c,d) defines a unimodular matrix (ad-bc=1) with integer elements, that is in SL(2,Z). The electric-magnetic duality τ→ -1/τ and the shift τ→ τ+1 are the generators of this group. It is not however quite clear whether they can be regarded as gauge symmetries in TGD framework. If they are gauge symmetries, then the critical values of Kähler coupling strength defined as fixed points of coupling constant evolution must form an orbit of SL(2,Z). It could be also that modular symmetry is broken to a subgroup of SL(2,Z) and this subgroup leaves τ invariant in the case of minimal symmetry.

1. τ→ iτ would permute Euclidian and Minkowskian regions with each other and is therefore a candidate for the T-duality. This condition cannot be satisfied in generic case but one can ask whether for some special choices of τ these transformations could generate a non-trivial sub-group of modular transformations. This subgroup

   To see whether this is the case let us write explicitly the condition τ→ iτ:

   (aτ +b)/(cτ+d)= θ/2π+ i/α_K , α_K= g²_K/4π .

   The condition allows to solve τ as

   τ= ((a-id)/2c)× [1+ε₁(1+ 4ibc/(d-ia)²)^1/2)] , ε₁=+/- 1 .

2. For

   d=ε a , ε=+/- 1

   implying a²-bc=1, the solution simplifies since the argument of square root is real. One has

   τ= (a/2c)(1-ε i)[1+ε₁ (1-ε (a-1/a)^1/2)^1/2] .

   The imaginary and real parts of τ are identical: this might allow an interpretation in terms of the fact that Chern-Simons terms from two regions are identical (normal derivatives are however discontinuous at wormhole throat). Certainly this is a rather strong prediction.

3. Does this mean that SL(2,C) is broken down to the 4-element isotropy group generated by this transformation? If so, a the condition just deduced could allow to deduce additional constraints on the value of Kähler coupling strength, which is in principle fixed by the criticality condition to have only finite number of values? By the earlier arguments - related to p-adic mass calculations and the heuristic formula for the gravitational constant - the value of Kähler coupling strength is in a good approximation equal to fine structure constant at electron length scale:

   α_K =g_K²/4π≈ α , 1/α≈ 137.035999084 .

4. One obtains the following estimate for a/2c from the estimate for α_K by considering the imaginary part of τ:

   (a/2c)[1+ε₁ (1-ε(a-1/a)^1/2)^1/2]= 1/α_K .

   At the limit a→ ∞ one has

   (a/2c)[[1+ε₁ (1-ε)^1/2 ]= 1/α_K .

   The simplest option at this limit corresponds to ε=1 giving a/2c≈ 137.035999084 .

   Note that a/2c=137 is not allowed by determinant condition so that the deviation of α_K from 1/137 is predicted. One must have a>137× 2c≥ 2× 137. This implies

   1+ε₁(1-(a-1/a)^1/2)^1/2 =1+.0026 ε₁ .

   By expanding the square root in first order to Taylor series one obtains the condition

   (a/2c)× (1+ε₁/2^3/2× 137 c)≈ 1/α_K .

   For large enough values of a and c it is possible to have arbitrary good approximation to fine structure constant. Note that the integers a and c cannot have common factors since this together with determinant condition a²-bc=1 would lead to contradiction.

For a pdf version of this article see What could be the counterpart of T-duality in TGD framework? or the chapter Miscellaneous topics of "Quantum TGD as Infinite-Dimensional Geometry".

About the construction of mesons and elementary bosons in TGD Universe

It looks somewhat strange to talk about the construction of mesons and elementary bosons in the same sentence. In TGD framework the construction recipes are however structurally identical so that it is perhaps sensible to proceed from mesons to elementary bosons. Therefore I will first consider the construction of meson like states relevant for the TGD based model of hadrons, in particular for the model of the pion of M₈₉ hadron physics possibly explaining the 125 GeV state for which LHC finds evidence. The more standard interpretation is as elementary spin 0 boson, which need not however have anything to do with Higgs.

One can construct also spin 0 states and their super-partners in TGD framework and at this stage it is not clear whether some internal consistency argument could exclude them from the spectrum. Amusingly, the construction for meson like states and elementary spin 0 states obey very similar mathematics. At this moment it is not possible to select between M₈₉ and elementary spin 0 boson as an explanation for 125 GeV state. What is however clear that this analogy of Higgs has also charged companions and that the couplings to fermions are not those predicted by standard model since particle massivation is not caused by Higgs vacuum expectation value but described by p-adic thermodynamics.

As an unexpected by-product construction gives insight to the difference between (π,η) and (K,Kbar) assignable to 3+1 and 2+2bar multiplets of strong isospin symmetry group SU(2). In QCD framework both the relation of this symmetry to color group and the organization of meson states to these 3+1 and 2+2bar are poorly understood. The construction reveals that these representations in TGD framework correspond directly to the u(2) subalgebra of su(3) and its complement and that this u(2) subalgebra can be assigned with strong isospin. Therefore strong isospin group is in well defined sense subgroup of color group but acting in weak isospin spin degrees of freedom! This example shows once again how good theory relates seemingly completely unrelated things to each other.

Construction of meson like states in TGD framework

The challenge is how translate attributes like scalar and pseudo-scalar making sense at M⁴ level to statements making sense at the level of M⁴× CP₂.

In QCD the view about construction of pseudo-scalar mesons is roughly that one has string like object having quark and antiquark at its ends, call them A and B. The parallel translation of the antiquark spinor from A to B is needed in order to construct gauge invariant object of type ‾ΨOΨ, where O characterizes the meson. The parallel translation implies stringy non-locality. In lattice QCD this string correspond to the edge of lattice cell. For a general meson O is "charge matrix" obtained as a combination of gamma matrices (γ₅ matrix for pseudo-scalar), polarization vectors, and isospin matrices.

This procedure must be generalized to TGD context. In fact a similar procedure applies also in the construction of gauge bosons possible Higgs like states since also in this case one must have general coordinate invariance and gauge invariance. Consider as an example pseudo-scalars.

1. Pseudo-scalars in M⁴ are replaced with axial vectors in M⁴× CP₂ with components in CP₂ direction. One can say that these pseudo-scalars have CP₂ polarization representing the charge of the pseudo-scalar meson. One replaces γ₅ with γ₅× O_a where O_a=O_a^kγ_k is the analog of ε^kγ_k for gauge boson. Now however the gamma matrices are CP₂ gamma matrices and O_a^k is some vector field in CP₂. The index a labels the isospin components of the meson.

2. What can one assume about O_a at the partonic 2-surfaces? In the case of pseudo-scalars pion and η (or vector mesons ρ and ω with nearly the same masses) one should have four such fields forming isospin triplet and singlet with large mass splitting. In the case of kaon would should have also 4 such fields but with almost degenerate masses. Why such a large difference between kaon and (π,η) system? A plausible explanation is in terms of mixing of neutral pseudo-scalar mesons with vanishing weak isospin mesons raising the mass of η but one might dream of alternative explanations too.
   1. Obviously O_a:s should form strong isospin triplets and singlets in case of (π,η) system. In the case of kaon system they should form strong isospin doublets. The group in question should be identifiable as strong isospin group. One can formally identify the subgroup U(2)⊂ SU(3) as a counterpart of strong isospin group. The group SO(3)⊂ SU(3) defines second candidate of this kind. These subgroups correspond to two different geodesic spheres of S². The first gives rise to vacuum extremals of Kähler action and second one to non-vacuum extremals carrying magnetic charge at the partonic 2-surface. Cosmic strings as vacuum extremals and cosmic strings as magnetically charged objects are basic examples of what one obtains. The fact that partonic 2-surfaces carry Kähler magnetic charge strongly suggests that U(2) option is the only sensible one but one must avoid too strong conclusions.

   2. Could one identify O_a as Killing vector fields for u(2)⊂ su(3) or for its complement and in this manner obtain two kinds of meson states directly from the basic Lie algebra structure of color algebra? For u(2) one would obtain 3+1 vector fields forming a representation of u(2) decomposing to a direct sum of representations 3 and 1 of U(2) having interpretation in terms of π and η the symmetry breaking is expected to be small between these representations. For the complement of u(2) one would obtain doublet and its conjugate corresponding to kaon like states. Mesons states are constructed from the four states U_iDbar_j, barU_iD_j, U_ibarU_j, D_ibarD_j. For i=j one would have u(2) and for i≠ j its complement.

   3. One would obtain a connection between color group and strong isospin group at the level of meson states and one could say that mesons states are not color invariants in the strict sense of the world since color would act on electroweak spin degrees of freedom non-trivially. This could relate naturally to the possibility to characterize hadrons at the low energy limit of theory in terms of electroweak quantum numbers. Strong force at low energies could be described as color force but acting only on the electroweak spin degrees of freedom. This is certainly something new not predicted by the standard model.

3. Covariant constancy of O_a at the entire partonic 2-surface is perhaps too strong a constraint. One can however assume this condition only at the the braid ends.
   1. The holonomy algebra of the partonic 2-surface is Abelian and reduces to a direct sum of left and right handed parts. For both left- and right-handed parts it reduces to a direct sum of two algebras. Covariant constancy requires that the induced spinor curvature defining classical electroweak gauge field commutes with O_a. The physical interpretation is that electrowak symmetries commute with strong symmetries defined by O_a. There would be at least two conditions depending only on the CP₂ projection of the partonic 2-surface.

   2. The conditions have the form

      F^ABj^a_B=0 ,

      where a is color index for the sub-algebra in question and A,B are electroweak indices. The conditions are quadratic in the gradients of CP₂ coordinates. One can interpret F^AB as components of gauge field in CP₂ with Abelian holonomy and j^a as electroweak current. The condition would say that the electroweak Lorentz force acting on j^a vanishes at the partonic 2-surface projected to CP₂. This interpretation looks natural classically. The conditions are trivially satisfied at points, where one has j^a_B=0, that is at the fixed points of the one-parameter subgroups of isometries in question. O_a would however vanish identically in this case.

   3. The condition F^ABj^a_B=0 at all points of the partonic 2-surface looks un-necessary strong and might fail to have solution. The reason is that quantum classical correspondence strongly suggests that the color partial waves of fermions and planewaves associated with 4-momentum are constant along the partonic surface. The additional condition F^ABj^a_B=0 allows only a discrete set of solutions.

      A weaker form of these conditions would hold true for the braid ends only and could be used to identify them. This conforms with the notion of finite measurement resolution and looks rather natural from the point of view of quantum classical correspondence. Both forms of the conditions allows SUSY in the sense that one can add to the fermionic state at partonic 2-surface a covariantly constant right-handed neutrino spinor with opposite fermionic helicity.

   4. These conditions would be satisfied only for the operators O_a characterizing the meson state and this would give rise to symmetry breaking relating to the mass splittings. Physical intuition suggests that the constraint on the partonic 2-surface should select or at least pose constraints on the maximum of Kähler function. This would give the desired quantum classical correlation between the quantum numbers of meson and space-time surface.

4. The parallel translation between the ends connecting the partonic 2-surfaces at which quark and antiquark reside at braid ends is along braid strand defining the state of string like object at the boundary of CD. These stringy world sheets are fundamental structures in quantum TGD and a possible interpretation is as singularity of the effective covering of the imbedding space associated with the hierarchy of Planck constans and due to the vacuum degeneracy of Kähler action implying that canonical momentum densities correspond to several values for the gradients of imbedding space coordinates. The parallel translation is therefore unique once the partonic 2-surface is fixed. This is of outmost importance for the well-definedness of quantum states. Obviously this state of affairs gives an additional "must" for braids.

The construction recipe generalizes trivially to scalars. There is however a delicate issue associated with the construction of spin 1 partners of the pseudo-scalar mesons. One must assign to a spin 1 meson polarization vector using ε^kγ_k as an additional factor in the "charge matrix" slashed between fermion and antifermion. If the charge matrix is taken to be Q_a=ε^kγ_k j^a_kΓ^k, it has matrix elements only between quark and lepton spinors. The solution of the problem is simple. The triplet of charge matrices defined as Q_a=ε^kγ_k D_kj^a_lΣ^kl transforms in the same manner as the original triplet under U(2) rotations and can be used in the construction of spin 1 vector mesons.

Generalization to the construction of gauge bosons and spin 0 bosons

The above developed argument generalizes with trivial modifications to the construction of the gauge bosons and possible Higgs like states as well as their super-partners.

1. Now one must form bi-linears from fermion and anti-fermion at the opposite throats of the wormhole contact rather than at the ends of magnetic flux tube. This requires braid strands along the wormhole contact and parallel translation of the spinors along them. Hadronic strings are replaced with the TGD counterparts of fundamental strings.

2. For electro-weak gauge bosons O corresponds to the product ε^k γ^k Q_i, where Q_i is the charge matrix associated with gauge bosons contracted between both leptonic and quark like states. For gluons the charge matrix is of form Q_A= ε_kγ^k H_A, where H_A is the Hamiltonian of the corresponding color isometry.

3. One can also consider the possibility of charge matrices of form Q_A=ε^kγ_k D_kj^A_lΣ^kl, where j^A is the Killing vector field of color isometry. These states would compose to representations of u(2)⊂ u(3) to form the analogs of (ρ,ω) and (K^*,Kbar^*) system in CP₂ scale. This is definitely something new.

4. In the case of spin zero states polarization vector is replaced with polarization in CP₂ degrees of freedom represented by one of the operators O_a already discussed. One would obtain the analogs of (π,η) and (K,kenooverlineK) systems at the level of wormhole contacts. Higgs mechanism for these does not explain fermionic masses since p-adic thermodynamics gives the dominant contributions to them. It is also difficult to imagine how gauge bosons could eat these states and what the generation of vacuum expectation value could mean mathematically. Higgs mechanism is essentially 4-D concept and now the situation is 8-dimensional.

5. At least part of spin zero states corresponds to polarizations in CP₂ directions for the electroweak gauge bosons. This would mean that one replaces ε_kγ^k with j_a^kΓ_k, where j_a is Killing vector field of color isometry in the complement of u(2)⊂ su(3). This would give four additional polarization states. One would have 4+2=6 polarization just as one for a gauge field in 8-D Minkowski space. What about the polarization directions defined by u(2) itself? For the Kähler part of electroweak gauge field this part would give just the (ρ,ω) like states already mentioned. Internal consistency might force to drop these states from consideration.

The nice aspect of p-adic mass calculations is that they are so general: only super-conformal invariance and p-adic thermodynamics and p-adic length scale hypothesis are assumed. The drawback is that this leaves a lot of room for the detailed modeling of elementary particles.

1. Lightest mesons are lowest states at Regge trajectories and also p-adic mass calculations assign Regge trajectories in CP₂ scale to both fermions and bosons.

2. It would be natural to assign the string tension with the wormhole contact in the case of bosons and identifiable in terms of the Kähler action assignable to the wormhole contact modelable as piece of CP₂ type vacuum extremal and having interpretation in terms of the action of Kähler magnetic fields.

3. Free fermion has only single wormhole throat. The action of the piece of CP₂ type vacuum extremal could give rise to the string tension also now. One would have something analogous to a string with only one end, and one can worry whether this is enough. The magnetic flux of the fermion however enters to the Minkowskian region and ends up eventually to a wormhole throat with opposite magnetic charge. This contribution to the string tension is however expected to be small being proportional to 1/S, where S is the thickness of the magnetic flux tube connecting the throats. Only if the magnetic flux tube remains narrow, does one obtain the needed string tension from the Minkowskian contribution. This is the case if the flux tube is very short. It seem that the dominant contribution to the string tension must come from the wormhole throat.

4. The explanation of family replication phenomenon (see this) based on the genus of wormhole throat works for fermions if the the genus is same for the two throats associated with the fermion. In case of bosons the possibility of different genera leads to a prediction of dynamical SU(3) group assignable to genus degree of freedom and gauge bosons should appear also in octets besides singlets corresponding to ordinary elementary particles. For the option assuming identical genera also for bosons only the singlets are possible.

5. Regge trajectories in CP₂ scale indeed absolutely essential in p-adic thermodynamics in which massless states generate thermal mass in p-adic sense. This makes sense in zero energy ontology without breaking of Poincare invariance if CD corresponds to the rest system of the massive particle. An alternative way to achieve Lorentz invariance is to assume that observed mass squared equals to the thermal expectation value of thermal weight rather than being thermal expectation for mass squared.

It must be emphasized that spin 0 states and exotic spin 1 states together with their super-partners might be excluded by some general arguments. Induced gauge fields have only two polarization states, and one might argue that that same reduction takes place at the quantum level for the number of polarization states which would mean the elimination of F_LbarF_R type states having interpretation as CP₂ type polarizations for gauge bosons. One could also argue that only gauge bosons with charge matrices corresponding to induced spinor connection and gluons are realized. The situation remains open in this respect.

For background and more details see the chapter New Particle Physics Predicted by TGD: Part I or the short article The particle spectrum predicted by TGD and TGD based SUSY.