Scattering amplitudes in positive Grassmannian: TGD perspective

The quite recent but not yet published proposal of Hamed and his former student Trnka has gained a lot of attention. There is a popular article in Quanta Magazine about their work. There is a video talk by Jaroslav Trnka about positive Grassmannian (the topic is actually touched at the end of the talk but it gives an excellent view about the situation) and a video talk by Nima Arkani-Hamed. One can also find the slides of Trnka .

The basic claim is that the Grassmannian amplitudes reduce to volumes of positive Grassmannians determined by external particle data and realized as polytopes in Grassmannians such that their facets correspond to logarithmic singularities of a volume form in one-one correspondence with the singularities of the scattering amplitude. Furthermore, t the factorization of the scattering amplitude at singularities corresponds to the singularities at facets. Scattering amplitudes would characterize therefore purely geometric objects. The crucial Yangian symmetry would correspond to diffeomorphisms preserving the positivity property. Unitarity and locality would be implied by the volume interpretation. Nima concludes that unitarity and locality, gauge symmetries, space-time, and even quantum mechanics emerge. One can however quite well argue that its the positive Grassmannian property and volume interpretation which emerge. In particular, the existence of twistor structure possible in Minkowskian signature only in M⁴ is absolutely crucial for the beautiful outcome, which certainly can mean a revolution as far as calculational techniques are considered and certainly the new view about perturbation theory should be important also in TGD framework.

The talks inspired the consideration of the possible Grassmannian formulation in TGD framework in more detail and to ask whether positivity might have some deeper meaning in TGD framework.

The vision about what BCFW approach to generalized Feynman diagframs could mean has been fluctuating wildly during last months. The Grassmannian formalism for scattering amplitudes is expected to generalize for generalized Feynman diagrams: the basic modification is due to the possible presence of CP₂ twistorialization and the fact that 4-fermion vertex - rather than 3-boson vertex - and its super counterparts define now the fundamental vertices. Both QFT type BFCW and stringy BFCW can be considered. The recent vision is as follows.

1. Fermions of internal lines are massless in real sense and have unphysical helicity. Wormhole contacts carrying fermion and antifermion at their opposite throats correspond to basic building bricks of bosons. For fermions second throat is empty. The residue integral over the momenta of internal lines replaces fermionic propagator with its inverse and replaces 4-D momentum integration with integration over light-cone using the standard Lorentz invariant integration measure.
   
   
2. 4-fermion vertex defines the fundamental vertex and contains constant proportional to length squared. This is definitely a problem. If all 4-fermion vertices contain one bosonic wormhole contact, one can replace regard the verties effectively as BFF or BBB vertices. The four-fermion coupling constant L² having dimensions length squared can be replaced with 1/p² for the bosonic line: this is the new ingredient allowing to overcome the difficulties assignable to four-fermion vertex.
   
   
3. For QFT type BFCW BFF and BBB vertices would be an outcome of bosonic emergence (bosons as wormhole contacts) and 4-fermion vertex is proportional to factor with dimensions of inverse mass squared and naturally identifiable as proportional to the factor 1/p² assignable to each boson line. This predicts a correct form for the bosonic propagators for which mass squared is in general non-vanishing unlike for fermion lines. The usual BFCW construction would emerge naturally in this picture. There is however a problem: the emergent bosonic propagator diverges or vanishes depending on whether one assumes SUSY at the level of single wormhole throat or not. By the special properties of the analog of N=4 SUSY SUSY generated by right handed neutrino the SUSY cannot be applied to single wormhole throat but only to a pair of wormhole throats.
   
   
   
4. This as also the fact that physical particles are necessarily pairs of wormhole contacts connected by fermionic strings forces stringy variant of BFCW avoiding the problems caused by non-planar diagrams. Now boson line BFCW cuts are replaced with stringy cuts and loops with stringy loops. By generalizing the earlier QFT twistor Grassmannian rules one ends up with their stringy variants in which super Virasoro generators G, G^† and L bringing in CP₂ scale appear in propagator lines: most importantly, the fact that G and G^† carry fermion number in TGD framework ceases to be a problem since a string world sheet carrying fermion number has 1/G and 1/G^† at its ends. The general rules is simple: each line emerging from 4-fermion vertex carries 1/G and 1/G^† as vertex factor. Twistorialization applies because all fermion lines are light-like.
   
   
5. A more detailed analysis of the properties of right-handed neutrino demonstrates that modified gamma matrices in the modified Dirac action mix right and left handed neutrinos but that this happens markedly only in very short length scales comparable to CP₂ scale. This makes neutrino massive and also strongly suggests that SUSY generated by right-handed neutrino emerges as a symmetry at very short length scales so that spartners would be very massive and effectively absent at low energies. Accepting CP₂ scale as cutoff in order to avoid divergent gauge boson propagators QFT type BFCW makes sense. The outcome is consistent with conservative expectations about how QFT emerges from string model type description.
   
   
6. The generalization to gravitational sector is not a problem in sub-manifold gravity since M⁴ - the only space-time geometry with Minkowski signature allowing twistor structure - appears as a Cartesian factor of the imbedding space. A further finding is that CP₂ and S⁴ are the only Euclidian 4-manifolds allowing twistor space with Kähler structure. Since S⁴ does not allow Kähler structure, CP₂ is completely unique just like M⁴. Stringy picture indeed treats gravitons and other elementary particles completely democratically.
   
   
7. The analog of twistorial construction in CP₂ degrees of freedom based on the notion of flag manifold and geometric quantization is proposed. Light-likeness in real sense poses a powerful constraint analogous to constraints posed by moves in the case of SYMs and if volume of a convex polytope dictated by the external momenta and helicities provides a representation of the scattering amplitude, the tree diagrams would give directly the full volume.
   

Perhaps it is not exaggeration to say that the architecture of generalized Feynman diagrams and their connection to twistor approach is now reasonably well-understood. There are of course several problems to be solved. On must feed in p-adic thermodynamics for external particles (here zero energy ontology might be highly relevant). Also the description of elementary particle families in terms of elementary particle functionals in the space of conformal equivalence classes of partonic 2-surface must be achieved.

As both Arkani-Hamed and Trnka state "everything is positive". This is highly interesting since p-adicization involves canonical identification, which is well defined only for non-negative reals without further assumptions. This raises the conjecture that positivity is necessary in order to achieve number theoretical universality.

Addition: Lubos wrote again about amplituhedrons and managed to write something about which I can agree almost whole-heartedly. Not a single mention of superstrings or Peter Woit! I also agree with Lubos about Scott Aaronson's parody: Scott was entertaining but not deep. The twistors and Grassmannians will revolutionize theoretical physics in many manners: my basic bet is that the uniqueness of M⁴×CP₂ from twistorial considerations and positivity conditions (whatever they really mean Minkowskian signature!) as a prerequisite for p-adicization will be at the core of the revolution and make TGD a mathematical "must". Mathematicians might be able to generalize the Grassmannian approach to CP₂ degrees of freedom without much effort.

For details see the chapter Some fresh ideas about twistorialization of TGD or the article with the same title.

Homepage works again

To my surprise I noticed that homepage tgdtheory.com has started to work again. I have received no responses to my queries nor the key making possible the domain transfer. I hope that the page will work until next year: after that domain transfer to a new service provider should be possible since the recent domain name expires. I am sorry for inconvenience.

A little comment about the hierarchy of Planck constants

Originally the hierarchy of Planck constant was assumed to correspond to a book like structure having as pages the n-fold coverings of the imbedding space for various values of n serving therefore as a page number. The pages are glued together along a 4-D "back" at which the branches of n-furcations are degenerate. This leads to a very elegant picture about how the particles belonging to the different pages of the book interact. All vertices are local and involve only particles with the same value of Planck constant: this is enough for darkness in the sense of particle physics. The interactions between particles belonging to different pages involve exchange of a particle travelling from page to another through the back of the book and thus experiencing a phase transition changing the value of Planck constant.

Is this picture consistent with the picture based on n-furcations? This seems to be the case. The conservation of energy in n-furcation in which several sheets are realized simultaneously is consistent with the conservation of classical conserved quantities only if the space-time sheet before n-furcation involves n identical copies of the original space-time sheet or if the Planck constant is h_eff=nh. This kind of degenerate many-sheetedness is encountered also in the case of branes. The first option means an n-fold covering of imbedding space and h_eff is indeed effective Planck constant. Second option means a genuine quantization of Planck constant due to the fact the value of Kähler coupling strength α_K=g_K²/4πhbar_eff is scaled down by 1/n factor. The scaling of Planck constant consistent with classical field equations since they involve α_K as an overall multiplicative factor only.

For background see the chapter "Does TGD predict a spectrum of Planck constants" of "Towards M-matrix".

Note: The new (temporary) address of my homepage is http://www. tgdtheory.fi. The only change is the replacement of "com" with "fi" and one can get from any link to the new address just by replacing "com" with "fi".

Has IceCube detected neutrinos coming from decays of p-adically scaled up copies of weak bosons?

This note was inspired by very interesting posting "Storm in IceCube" by Jester. IceCube is a neutrino detector located at South Pole. Most of the neutrinos detected are atmospheric neutrinos originating from Sun but what one is interested in are neutrinos from astrophysical sources.

1. Last year the collaboration reported the detection for neutrino cascade events, with with energy around 1 PeV=10⁶ GeV. The atmospheric background decreases rapidly with energy and at these energies the detection of a pair of events at these energies corresponds to about 3 sigma. The recent report tells about a broad excess of events (28 events) above 30 TeV: only about 10 are expected from atmospheric neutrinos alone. The flavor composition is consistent with 1:1:1 ratio of the 3 neutrino species as expected for distant sources for which the oscillations during the travel should cause complete mixing. The distribution of the observed events is consistent with isotropy.
   
   
2. There is a dip ranging from .4 PeV to about 1 PeV and the spectrum has probably a sharp cutoff somewhat above 1 TeV. This suggests a monochromatic neutrino line resulting from the decays of some particle decaying to neutrino and some other particle - possibly also neutrino (see this). Astrophysical phenomena with standard model physics are expected to produce smooth power-law spectrum - typically 1/E² - rather than peak. The proposal is that the events around 1 PeV could come from the decay of dark matter particles with energy scale of 2 TeV. The observation of two events gives a bound for the life-time of dark matter particle in question: about 10²¹ years much longer than the age of the Universe. The bound of course depends on what density is assumed for the dark matter.
   
   
3. There is also a continuum excess in the range [.1, .4] PeV. This could result from many-particle decay channels containing more than 2 particles.
   

What says TGD?

1. TGD almost-predicts a fractal hierarchy of hadron physics and weak physics labelled by Mersenne primes M_n=2ⁿ-1. Also Gaussian primes M_G,n= (1+i)ⁿ-1 are possible. M₁₀₇ would correspond to the ordinary hadron physics. M₈₉ would correspond to weak bosons and a scaled up copy of hadron physics, for which there are many indications: in particular, the breaking of perturbative QCD at rather high energies assignable at LHC to proton heavy nucleus collisions. The explanation in terms of AdS/CFT correspondence has not been successful and is not even well-motivated since it assumes strong coupling regime.
   
   
2. The next Mersenne prime is M₆₁ and the first guess is that the observed TeV neutrinos result from the decay of W and Z bosons of scale up copy of weak physics having mass near 1 TeV. The naivest estimate for the masses of these weak bosons is obtained by the naive scaling the masses of ordinary weak bosons by factor 2^(89-61)/2=2¹⁴. For m_W=80 GeV and m_Z=90 GeV one obtains m_W(61)= 1.31 PeV and m_Z(61)= 1.47 PeV. The energy of the mono-chromatic neutrino would be about about .65 PeV and .74 PeV in the two cases. This is in the almost empty range between .4 PeV and 1 PeV and too small roughly by a factor of kenosqrt2.
   

   An improved estimate for upper bound of Z mass is based on the p-adic mass scale m(M₈₉) related to the p-adic mass scale M₁₂₇ of electron by scaling factor 2^(127-89)/2= 2¹⁹ giving m(89)≈ 120 GeV for m_e= (5+X)^1/2m(127) =.51 MeV and X=0 (X≤ 1 holds true for the second order contribution to electron mass). The scaling by the factor 2^(89-61)/2= 2¹⁴ gives m(61)= 1.96 TeV consistent with the needed 2 TeV. The exact value of weak boson mass depends on the value of Weinberg angle sin²(θ_W) and the value of the second order contribution to the mass: m(61) gives upper bound for the mass of Z(61). The model predicts two peaks with distance depending on the value of Weinberg angle of M₆₁ weak physics.
   
   

3. What about the interpretation of the continuum part of anomaly? The proposed interpretation for many-particle decays looks rather reasonable. The simplest possibility is the decay to a pair of light quarks of M₆₁ hadron physics, followed by a decay of quark or antiquark via emission of W boson decaying to lepton-neutrino pair.
   

To sum up, the existence of scaled up copy of weak physics would mean the last nail to the coffin of GUTs predicting huge desert between weak mass scale and the mass scale of leptoquarks having mass scale of order 10^-4 Planck masses. It would also provide a further support for the p-adic length scales hypothesis.

For background see the chapter "New particle physics predicted by TGD: part I".

Note: The new (temporary) address of my homepage is http://www. tgdtheory.fi. The only change is the replacement of "com" with "fi" and one can get from any link to the new address just by replacing "com" with "fi".

Turoks wellcome speech to new physics students

Peter Woit wrote a nice blog posting about Perimeter Institute and the crisis in modern physics. He gave a link to a wellcome speech of Neil Turok for the new students. The tone of the speech was very warm and encouraging and raised optimism. Maybe theoretical physics is not dead discipline yet;-). What made me especially happy that Turok was not telling that superstring theory is the only possible theory as I have seen so many times being told in blogs.

Turok admitted that we have been building enormous number of models during last decades: GUTs, SUSYs, superstring models, loop quantum gravity models, and whatever. The common feature of these models is that they are extremely complicated whereas Nature according to LHC and cosmological data from Planck satellite seems to be very simple. Nowadays theoreticians everywhere in the world are totally confused: their expectations were totally wrong. Turok made some special points which deserve comments.

Does Higgs really have vacuum expectation value?

Turok made some very interesting comments related to the Higgs discovery at LHC. The problem is that the mass of Higgs believed to be dictated by its vacuum expectation value does not correspond to stable vacuum. Vacuum is only meta-stable and can decay to the stable vacuum by quantum tunnelling releasing enormous amount of energy. That Nature would have chosen metastability looks strange.

Turok did not continue with an additional question natural with my background. We have discovered Higgs particle but do we really know that Higgs has vacuum expectation in reality? We do not! Perhaps Higss field exists and has vacuum expectation only in quantum field theoretic approximation to particle physics? Are we perhaps trying to describe Higgs using a language, which has reached its limits of applicability. Is this the source of all this confusion and even despair manifesting as a belief that physics has reached its end?

What if their is no vacuum energy for Higgs? In TGD framework it is very natural to imagine that Higgs couples to fundamental fermions with a gradient coupling proportional to the mass of fermion. The dimensionless coupling of Higgs particle to fermions would be same for all fermions: this is very natural as any theoretician must admit! This would turn around the entire habit of thinking about Higgs. One would have naturality and one would get rid of the metastable vacuum since there would be neither vacuum expectation nor vacuum energy associated with Higgs.

This sounds nice but there is an objection. How can we understand the massivation of fermions and gauge bosons? This is quite a problem in quantum field theory context, and I am convinced that something more general is needed. If one is ready to take TGD seriously, situation changes. In TGD framework physics enjoys a property that I have christened number theoretical universality. As one particular consequence, p-adic number fields as physics of cognition become part of physics. The scope of physics extends to include also cognition and consciousness. It would be wonderful if LHC could force theoreticians to become consciousness theorists!

p-Adic thermodynamics assumes only conformal invariance and predicts with one per cent precision particle masses in terms of CP₂ mass scale (about 10^-4 times Planck mass) and single parameter - the p-adic prime p characterizing the particle. By p-adic length scale hypothesis - to be discussed below - p is near to power of two. Also gauge boson masses can be understood since in TGD framework gauge bosons are bound states of fermion and anti-fermion at opposite throats of wormhole contact.

Do we really understand all about scale and conformal invariance?

Turok talked about scale invariance and its generalization conformal symmetries as something very deep that has not been properly understood yet.

In TGD framework conformal symmetry generalizes to 4-D context since light-like 3-surfaces - the basic objects of TGD Universe - are metrically 2-D and possess extended conformal invariance. Perhaps the most important outcome is an explanation for why space-time dimension must be D=4: only in this dimension one can extend the ordinary 2-D conformal symmetries to even huger group of conformal symmetries.

The gigantic conformal symmetries give excellent hopes that the "world of classical worlds" (WCW) consisting of 4-surfaces has K\"ahler geometry: without these symmetries it fails to exist mathematically. Physics as classical physics for spinor fields in WCW is what TGD does for quantum field theory. More concretely, point like particle is replaced with 3-D surface and its "orbit" has interpretation as particle orbit or space-time depending on what the scale of the observer is.

Mathematically WCW would be union of symmetric K\"ahler manifolds parametrized by zero modes defining "classical" variables as opposed to quantum fluctuating degrees of freedom parametrized by coset spaces of symplectic group assignable to δ M⁴₊× CP₂.

Where do all those scales emerge in a physics without scales?

Scale invariant theories do not possess any scale. Physics is however characterized by preferred scales. Turok sees only Planck length, vacuum energy density, and Higgs scale as fundamental scales. I think that this view is not quite correct: to my opinion - actually conviction - also the mass scales of elementary particles varying in huge range as fundamental scales are "rather" fundamental.

In TGD Universe the really fundamental length scale is of course CP₂ length scale defining a fundamental unit for length measurements: all other not quite so fundamental length scales are proportional to it. Where do the scales of the physics the pop up? Conformal field theories provide a mathematical description of conformal symmetry breaking but this is not enough. The mechanism of conformal symmetry breaking is the basic mystery: what is the the mechanism selecting some preferred scales from continuum? "p-Adic thermodynamics" is the TGD based answer to the question.

p-Adic thermodynamics brings in a completely new element: the condition of number theoretic existence for Boltzmann weights: exp(-E/T) is replaces with p^E/T: and this exists only if one has E/T is positive integer. Both p-adic temperature 1/T and E are quantized to integer values! Energy E corresponds now to eigenvalue of conformal scaling generator L₀ for which the spectrum is apart from vacuum contribution integer valued. Number theoretical existence requires conformal invariance!

p-Adic length scale hierarchy discretizes the continuous scale invariance so that only primes label the mass scales and renormalization group evolution also discretizes. p-Adic length scale hypothesis performs further selection - very much analogous to natural selection but generalized to the level of fundamental physics. It picks up primes near powers of two and Mersenne primes seem to be the best survivors since entire physics can be assigned to them - hadron physics, its lepto-hadronic counterparts, new hadron physics at TeV scale, and 4 Gaussian Mersennes in the length scale range from cell membrane thickness to the size of cell nucleus.

Holography and general coordinate invariance

Turok also mentioned holography as a key principle of future physics but not yet understood. I agree. In TGD framework holography reduces to general coordinate invariance and the strong form of this principle implies holography in strong sense. Quantum physics is almost 2-dimensional. Partonic 2-surface almost fix the physics but only almost: the 4-dimensional tangent space data of space-time surface at partonic 2-surfaces is what is needed. 4-D space-time is also needed to build quantum measurement theory: classical non-quantum fluctuating variables - zero modes in WCW geometry - are necessary since it is these which code for the outcomes of measurements.

WCW spinors and zero energy states as quantum superpositions of Boolean statements

Turok mentioned also the idea that the notion of quantum information might be fundamental and that it might be possible to build fermions and bosons from bits. This however requires identification of space-time as lattice and neither Turok or me can share this assumption. I believe however that there is deep connection between spinors and qubits and thus logic. Spinors basis for N-dimensional space defines Boolean algebra on N-bits. In the case of WCW this corresponds to infinite-D Boolean algebra. WCW spinors correspond to Fock states for second quantized fermions in space-time and Fock state basis correspond to statements of infinite-D Boolean algebra. WCW spinor fields correspond thus to logical statements in quantum Boolean algebra.

Spinor structure is square root of Riemannian metric so that logic, geometry, and quantum theory find each other. In zero energy ontology zero energy states correspond to quantum versions of statements of type A→ B that is quantum superpositions of their instances a→ b. Also 2-adic topology has a direct connection to Boolean algebras: 2-adic number can be regarded as a sequence of infinite number of bits such that the importance of higher bits decreases rapidly by the properties of p-adic topology. Hence binary cutoff becomes excellent approximation.

This connection makes sense also for p-adic topologies: in this case there are some "check bits" involved. For Mersenne prime 2^k-1 the number of bits is k-1 and the remaining statements whose number is 2^k-1-1 correspond to check bits. Mersenne primes allow maximum number of check bits and this could guarantee maximal stability for Boolean statements and thus maximum cognitive survival probability. Could this explain why Mersenne primes have been so successful in number theoretic survival of fittest?

Addition: For Lubos the turn of the tide forced by LHC is a painful event as becomes clear from his ranting: Lubos regresses to the level at which personal insults are meant to be scientific arguments.

Addition: Bee has a nice posting titled "Whatever happened to AdS/CFT and the Quark Gluon Plasma?" about not so successful attempts to apply AdS/CFT correspondence to QCD. The motivation comes from the findings from both RHIC for heavy nucleus collisions and from LHC for proton-heavy nucleus collisions. The findings demonstrate that perturative QCD (pQCD) fails and suggest strongly string like structures as cause of the effects. In this regime pQCD should work quite well in proton-heavy nucleus collisions. The application of AdS/CFT correspondence in turn is sensible in strong coupling regime so that it does not look at all well-motivated: no wonder if the results fail quantitatively and even qualitatively. To my opinion this issue is very important. If pQCD fails in an energy regime where it should work well, one can suspect that some new physics is involved: just this new physics LHC has been desperately trying to find. TGD proposal for this physics is M₈₉ hadron physics, and I have discussed this topic many times in previous postings.

Could photosensitive emulsions make dark matter visible?

The article "Possible detection of tachyon monopoles in photographic emulsions" by Keith Fredericks describes in detail very interesting observations by him and also by many other researchers about strange tracks in photographic emulsions induced by various (probably) non-biological mechanisms and also by the exposure to human hands (touching by fingertips) as in the experiments of Fredericks. That the photographic emulsion itself consists of organic matter (say gelatin) might be of significance.

The findings

The tracks have width between 5 μm-110 μm (horizontal) and 5 μm-460 μm (vertical). Even tracks of length up to at least 6.9 cm have been found. Tracks begin at some point and end abruptly. A given track can have both random and almost linear portions, regular periodic structures (figs 11 and 12), tracks can appear in swarms (fig. 24), bundles (fig. 25), and correlated pairs (fig. 16), tracks can also split and recombine (fig. 32) (here and below "fig." refers to a figure of the article .

Tracks differ from tracks of known particles: the constant width of track implies that electrons are not in question. No delta rays (fast electrons caused by secondary ionization appearing as branches in the track) characteristic for ions are present. Unlike alpha particle tracks the tracks are not straight. In magnetic fields tracks have parabolic portions whereas ordinary charged particle move along spiral. The magnetic field needed to cause spiral structure for charged baryons should be by two orders of magnitude higher than in the experiments. The explanation for parabolic portions in terms of magnetic monopole charge predicts that orbit is in a plane containing magnetic field rather than orthogonal to it. The tracks can split into a group of tracks which come together again. The tracks can have random portions and smoottly curved portions bringing in mind proteins. Track begin suddenly and disappear suddenly.

For particle physicist all these features - for instance constant width - strongly suggest pre-existing structures becoming visible for some reason. The pre-existing structure could of course correspond to something completely standard structures present in the emulsion. If one is ready to accept that biology involves new physics, it could be something more interesting.

Also evidence for cold fusion is reported by the group of Urutskoev. There is evidence for cold fusion in living matter: the fact that the emulsion contains gelatin might relate to this. Here a dark matter based mechanism of cold fusion allowing protons to overcome the Coulomb wall is discussed. Either dark protons or dark nuclei with much larger quantum size than usually would make this possible and protons could end up to the dark nuclei along dark flux tubes. In TGD inspired biology dark protons (large h_eff) with scaled up Compton length of order atomic size are proposed to play key role since their states allow interpretation in terms of vertebrate genetic code.

TGD inspired explanation

Dark matter in TGD based belief system corresponds to a hierarchy of phases of ordinary matter with an effective value h_eff of Planck constant coming as integer multiple of ordinary Planck constant. This makes possible macroscopic quantum phases consisting of dark matter. The flux tubes could carry magnetic monopole flux but the magnetic charge would be topological (made possible by the non-trivial second homology of CP₂ factor of the 8-D imbedding space containing space-times as surfaces) rather than Dirac type magnetic charge.

The TGD inspired identification of tracks could be as images of magnetic flux tubes or bundles of them containing dark matter defining one of the basic new physics elements in TGD based quantum biology. One can imagine two options for the identification of the tracks as "tracks".

1. The primary structures are in the photo-sensive emulsion.
   

   
   

2. The structures in photograph are photographs of dark matter in external world, say structures in human hands or human body or of dark matter at some magnetic body, say at the flux tubes of the magnetic body of the emulsion.
   

The fact that the tracks have been observed in experimental arrangements not involving exposure to human hands, indeed suggests that tracks represent photographs about parts of the magnetic body assignable to the emulsion. For this option the external source would serve only as the source of possibly dark photons.

This would imply a close analogy with the experiments of Peter Gariaev's group interpreted in TGD framework as photographing of the magnetic body of DNA sample (see this). Also here one has an external source of light: the light would be transformed to dark photons in DNA sample, scatter from the dark charged particles at the flux tubes of the magnetic body of DNA sample, and return back transforming to ordinary light and generating the image in the photosensitive emulsion.

A detailed TGD based proposal for the tracks is discussed in the chapter Dark Forces and Living Matter and in the article Could photosensitive emulsions make dark matter visible?.

Addition: Unfortunately, the blog page is not avaible at this moment: the owners of the webhotel lost their interest to hotel business and many people lost their money, homepage, and their data. I lost even more: my lifework disappeared from the web since the owners just for fun (or with some other motivation?) refuse to give a formal permission to move the contents of the homepage to a new physical location with the same domain name (which I of course own). Web ethics does not yet exist.

Addition: The homepage is temporarily at new address obtained from the earlier one by replacing "com" by "fi".

What could 4-fermion twistor amplitudes look like?

4-fermion twistor amplitudes are basic building bricks of twistor amplitudes in TGD framework. What can one conclude about them on basis of N=4 amplitudes? Instead of 3-vertices as in SYM, one has 4-fermion vertices as fundamental vertices and the challenge is to guess their general form. The basis idea is that N=4 SYM amplitudes could give as special case the n-fermion amplitudes and their supersymmetric generalizations

1. Attempt to understand the physical picture

One must try to identify the physical picture first.

1. Elementary particles consist of pairs of wormhole contacts connecting two space-time sheets. The throats are connected by magnetic fluxes running in opposite directions so that a closed monopole flux loop is in question. One can assign to the ordinary fermions open string world sheets whose boundary belong to the light-like 3-surfaces assignable to these two wormhole contacts. The question is whether one can restrict the consideration to single wormhole contact or should one describe the situation as dynamics of the open string world sheets so that basic unit would involve two wormhole contacts possibly both carrying fermion number at their throats.
   

   Elementary particles are bound states of massless fermions assignable to wormhole throats. Virtual fermions are massless on mass shell particles with unphysical helicity. Propagator for wormhole contact as bound state - or rather entire elementary particle would be from p-adic thermodynamics expressible in terms of Virasoro scaling generator as 1/L₀ in the case of boson. Super-symmetrization suggests that one should replace L₀ by G₀ in the wormhole contact but this leads to problems if G₀ carries fermion number. This might be a good enough motivation for the twistorial description of the dynamics reducing it to fermion propagator along the light-like orbit of wormhole throat. Super Virasoro algebra would emerged only for the bound states of massless fermions.
   
   

2. Suppose that the construction of four-fermion vertices reduces to the level of single wormhole contact. 4-fermion vertex involves wormhole contact giving rise to something analogous to a boson exchange along wormhole contact. This kind of exchange might allow interpretation in terms of Euclidian correlation function assigned to a deformation of CP₂ type vacuum extremal with Euclidian signature.
   

   A good guess for the interaction terms between fermions at opposite wormhole contacts is as current-current interaction j^α (x) j_α(y), where x and y parametrize points of opposite throats. The current is defined in terms of induced gamma matrices as ‾ΨΓ^αΨ and one functionally integrates over the deformations of the wormhole contact assumed to correspond in vacuum configuration to CP₂ type vacuum extremal metrically equivalent with CP₂ itself. One can expand the induced gamma matrix as a sum of CP₂ gamma matrix and contribution from M⁴ deformation Γ_α = Γ_α^CP2 + ∂_α m^kγ_k. The transversal part of M⁴ coordinates orthogonal to M²⊂ M⁴ defines the dynamical part of m^k so that one obtains strong analogy with string models and gauge theories.
   

The deformation Δ m^k can be expanded in terms of CP₂ complex coordinates so that the modes have well defined color hyper-charge and isospin. There are two options to be considered.

1. One could use CP₂ spherical harmonics defined as eigenstates of CP₂ scalar Laplacian D². The scale of eigenvalues would be 1/R², where R is CP₂ radius of order 10⁴ Planck lengths. The spherical harmonics are in general not holomorphic in CP₂ complex coordinates ξ_i, i=1,2. The use of CP₂ spherical harmonics is however not necessary since wormhole throats mean that wormhole contact involves only a part of CP₂ is involved.
   
   
2. Conformal invariance suggests the use of holomorphic functions ξ₁^mξ₂ⁿ as analogs of zⁿ in the expansion. This would also be the Euclidian analog for the appearance of massless spinors in internal lines. Holomorphic functions are annihilated by the ordinary scalar Laplacian. For conformal Laplacian they correspond to the same eigenvalue given by the constant curvature scalar R of CP₂. This might have interpretation as a spontaneous breaking of conformal invariance.
   

   The holomorphic basis zⁿ reduces to phase factors exp(inφ) at unit circle and can be orthogonalized. Holomorphic harmonics reduce to phase factors exp(imφ₁)exp(inφ₂) and torus defined by putting the moduli of ξ_i constant and can thus be orthogonalized. Inner product for the harmonics is however defined at partonic 2-surface. Since partonic 2-surfaces represent Kähler magnetic monopoles they have 2-dimensional CP₂ projection. The phases exp(imφ_i) could be functionally independent and a reduction of inner product to integral over circle and reduction of phase factors to powers exp(inφ) could take place and give rise to the analog of ordinary conformal invariance at partonic 2-surface. This does not mean that separate conservation of I₃ and Y is broken for propagator.
   
   

3. Holomorphic harmonics are very attractive but the problem is that they are annihilated by the ordinary Laplacian. Besides ordinary Laplacian one can however consider
   (Conformal Laplacian) defined as
   

   D_c²= -6D²+R
   

   and relating the curvatures of two conformally scaled metrics R denotes now curvature scalar). The overall scale factor and also its sign is just a convention. This Laplacian has the same eigenvalue for all conformal harmonics. The interpretation would be in terms of a breaking of conformal invariance due to CP₂ geometry: this could also relate closely to the necessity to assume tachyonic ground state in the p-adic mass calculations.
   

   The breaking of conformal invariance is necessary in order to avoid infrared divergences. The replacement of M⁴ massless propagators with massive CP₂ bosonic propagators in 4-fermion vertices brings in the needed breaking of conformal invariance. Conformal invariance is however retained at the level of M⁴ fermion propagators and external lines identified as bound states of massless states.
   

2. How to identify the bosonic correlation functions inside wormhole contacts?

The next challenge is to identify the correlation function for the deformation δ m^k inside wormhole contacts.

Conformal invariance suggests the identification of the analog of propagator as a correlation function fixed by conformal invariance for a system defined by the wormhole contact. The correlation function should depend on the differences ξ_i=ξ_i,1-ξ_i,2 of the complex CP₂ coordinates at the points ξ_i,1) and ξ_i,2 of the opposite throats and transforms in a simple manner under scalings of ξ_i. The simplest expectation is that the correlation function is power r^-n, where r²= [ξ₁|²+|ξ₂|² defines U(2) invariant coordinate distance squared. The correlation function can be expanded as products of conformal harmonics or ordinary harmonics of CP₂ assignable to ξ_i,1 and ξ_i,2 and one expects that the values of Y and I₃ vanish for the terms in the expansions: this just states that Y and I₃ are conserved in the propagation.

Second approach relies on the idea about propagator as the inverse of some kind of Laplacian. The approach is not in conflict with the general conformal approach since the Laplacian could occur in the action defining the conformal field theory. One should try to identify a Laplacian defining the propagator for δ m^k inside Euclidian regions.

1. The propagator defined by the ordinary Laplacian D² has infinite value for all conformal harmonics appearing in the correlation function. This cannot be the case.
   
   
2. If the propagator is defined by the conformal Laplacian D_c² of CP₂ multiplied by some numerical factor it gives fro a given model besides color quantum numbers conserving delta function a constant factor nR² playing the same role as weak coupling strength in the four-fermion theory of weak interactions. Propagator in CP₂ degrees of freedom would give a constant contribution if the total color quantum numbers for vanish for wormhole throat so that one would have four-fermion vertex.
   
   
3. One can consider also a third - perhaps artificial option - motivated for Dirac spinors by the need to generalize Dirac operator to contain only I₃ and Y. Holomorphic partial waves are also eigenstates of a modified Laplacian D²_C defined in terms of Cartan algebra as
   

   D²_C== [aY²+bI₃²]/R² ,
   

   where a and b suitable numerical constants and R denotes the CP₂ radius defined in terms of the length 2π R of CP₂ geodesic circle. The value of a/b is fixed from the condition Tr(Y²)=Tr(I₃²) and spectra of Y and I₃ given by (2/3,-1/3,-1/3) and (0,1/2,-1/2) for triplet representation. This gives a/b= 9/20 so that one has
   

   D²_C= ((9/20) Y²+ I₃²]× a/R² .
   

   In the fermionic case this kind of representation is well motivated since fermionic Dirac operator would be Y^k e^A_kγ_A+I₃^k e^A_kγ_A, where the vierbein projections Y^ke^A_k Y^ke^A_k and I₃^ke^A_k of Killing vectors represent the conserved quantities along geodesic circles and by semiclassical quantization argument should correspond to the quantized values of Y and I₃ as vectors in Lie algebra of SU(3) and thus tangent vectors in the tangent space of CP₂ at the point of geodesic circle along which these quantities are conserved. In the case of S² one would have Killing vector field L_z at equator.
   

Two general remarks are in order.

1. That a theory containing only fermions as fundamental elementary particles would have four-fermion vertex with dimensional coupling as a basic vertex at twistor level, would not be surprising. As a matter of fact, Heisenberg suggested for long time ago a unified theory based on use of only spinors and this kind of interaction vertex. A little book about this theory actually inspired me to consider seriously the fascinating challenge of unification.
   
   
2. A common problem of all these options seems to be that the 4-fermion coupling strength is of order R² - about 10⁸ times gravitational coupling strength and quite too weak if one wants to understand gauge interactions. It turns out however that color partial waves for the deformations of space-time surface propagating in loops can increase R² to the square L_p²= pR² of p-adic length scale. For D2_C assumed to serve as an propagator in an effective action of a conformal field theory one can argue that large renormalization effects from loops increase R² to something of order pR².
   

3. Do color quantum numbers propagate and are they conserved in vertices?

The basic questions are whether one can speak about conservation of color quantum numbers in vertices and their propagation along the internal lines and the closed magnetic flux loops assigned with the elementary particles having size given by p-adic length scale and having wormhole contacts at its ends. p-Adic mass calculations predict that in principle all color partial waves are possible in cm degreees of freedom: this is a description at the level of imbedding space and its natural counterpart at space-time level would be conformal harmonics for induced spinor fields and allowance of all of them in generalized Feynman diagrams.

1. The analog of massless propagation in Euclidian degrees of freedom would correspond naturally to the conservation of Y and I₃ along propagator line and conservation of Y and I₃ at vertices. The sum of fermionic and bosonic color quantum numbers assignable to the color partial waves woul be conserved. For external fermions the color quantum numbers are fixed but fermions in internal lines could move also in color excited states.
   
   
2. One can argue that the correlation function for the M⁴ coordinates for points at the ends of fermionic line do not correlate as functions of CP₂ coordinates since the distance between partonic 2-surface is much longer than CP₂ scale but do so as functions of the string world sheet coordinates as stringy description strongly suggests and that stringy correlation function satisfying conformal invariance gives this correlation. One can however couner argue that for hadrons the color correlations are different in hadronic length scale. This in turn suggests that the correlations are non-trivial for both the wormhole magnetic flux tubes assignable to elementary particles and perhaps also for the internal fermion lines.
   
   
3. I₃ and Y assignable to the exchanged boson should have interpretation as an exchange of quantum numbers between the fermions at upper and lower throat or change of color quantum numbers in the scattering of fermion. The problem is that induced spinors have constant anomalous Y and I₃ in given coordinate patch of CP₂ so that the exchange of these quantum numbers would vanish if upper and lower coordinate patches are identical. Should one expand also the induced spinor fields in Euclidian regions using the harmonics or their holomorphic variants as suggested by conformal invariance?
   

   The color of the induced spinor fields as analog of orbital angular momentum would realized as color of the holomorphic function basis in Euclidian regions. If the fermions in the internal lines cannot carry anomalous color, the sum over exchanges trivializes to include only a constant conformal harmonic. The allowance of color partial waves would conform with the idea that all color partial waves are allowed for quarks and leptons at imbedding space level but define very massive bound states of massless fermions.
   
   

4. The 4-fermion vertex would involve a sum over the exchanges defined by spherical harmonics or - more probably - by their holomorphic analogs. For both the spherical and conformal harmonic option the 4-fermion coupling strength would be of order R², where R is CP₂ length. The coupling would be extremely weak - about 10⁸ times the gravitational coupling strength G if the coupling is of order one. This is definitely a severe problem: one would want something like L_p², where p is p-adic prime assignable to the elementary particle involved.
   

   This problem provides a motivation for why a non-trivial color should propagate in internal lines. This could amplify the coupling strength of order R² to something of order L_p²=pR². In terms of Feynman diagrams the simplest color loops are associated with the closed magnetic flux tubes connecting two elementary wormhole contacts of elementary particle and having length scale given by p-adic length scale L_p. Recall that ν_L (ν_R)^c pair or its conjugate neutralizes the weak isospin of the elementary fermion. The loop diagrams representing exchange of neutrino and the fermion associated with the two wormhole contacts and thus consisting of two fermion lines assignable to "long" strings and two boson lines assignable to "short strings" at wormhole contacts represent the first radiative correction to 4-fermion diagram. They would give sum over color exchanges consistent with the conservation of color quantum numbers at vertices. This sum, which in 4-D QFT gives rise to divergence, could increase the value of four-fermion coupling to something of order L_p²= kpR² and induce a large scaling factor of $D^2_C$.
   
   

5. Why known elementary fermions correspond to color singlets and triplets? p-Adic mass calculations provide one explanation for this: colored excitations are simply too massive. There is however evidence that leptons possess color octet excitations giving rise to light mesonlike states. Could the explanation relate to the observation that color singlet and triplet partial waves are special in the sense that they are apart from the factor 1/(1+r²)^1/2 , r²=∑ |ξ_i|² for color triplet holomorphic functions?
   

4. Why twistorialization in CP₂ degrees of freedom?

A couple of comments about twistorialization in CP₂ degrees of freedom are in order.

1. Both M⁴ and CP₂ twistors could be present for the holomorphic option. M⁴ twistors would characterize fermionic momenta and CP₂ twistors to the quantum numbers assignable to deformations of CP₂ type vacuum extremals. CP₂ twistors would be discretized since I₃ and Y have discrete spectrum and it is not at all clear whether twistorialization is useful now. There is excellent motivation for the integration over the flag-manifold defining the choices of color quantization axes. The point is that the choice of conformal basis with well-defined Y and I₃ breaks overall color symmetry SU(3) to U(2) and an integration over all possible choices restores it.
   
   
2. Four-fermion vertex has a singularity corresponding to the situation in which p₁, p₂ and p₁+p₂ assignable to emitted virtual wormhole throat are collinear and thus all light-like. The amplitude must develop a pole as p₃+p₃= p₁+p₂ becomes massless. These wormhole contacts would behave like virtual boson consisting of almost collinear pair of fermion and anti-fermion at wormhole throats.
   

5. Reduction of scattering amplitudes to subset of N =4 scattering amplitudes

N=4 SUSY provides quantitative guidelines concerning the actual construction of the amplitudes.

1. For single wormhole contact carrying one fermion, one obtains two N=2 SUSY multiplets from fermions by adding to ordinary one-fermion state right-handed neutrino, its conjugate with opposite spin, or their pair. The net spin projections would be 0, 1/2 ,1 with degeneracies (1,2,1) for fermion helicity 1/2 and (0,-1/2, -1) with same degeneracies for fermion helicity -1/2. These N=2 multiplets can be imbedded to the N=4 multiplet containing 2⁴ states with spins (1,1/2,0,-1/2,-1) and degeneracies given by (1, 4, 6, 4, 1). The amplitudes in N=2 case could be special cases of N=4 amplitudes in the same manner as they amplitudes of gauge theories are special cases of those of super-gauge theories. The only difference would be that propagator factors 1/p² appearing in twistorial construction would be replaced by propagators in CP₂ degrees of freedom.
   
   
2. In twistor Grassmannian approach to planar SYM one obtains general formulas for n-particle scattering amplitudes with k positive (or negative helicities) in terms of residue integrals in Grassmann manifold G(n,k). 4-particle scattering amplitudes of TGD, that is 4-fermion scattering amplitudes and their super counterparts would be obtained by restricting to N=2 sub-multiplets of full N=4 SYM. The only non-vanishing amplitudes correspond for n=4 to k=2=n-2 so that they can be regarded as either holomorphic or anti-holomorphic in twistor variables, an apparent paradox understandable in terms of additional symmetry as explained and noticed by Witten. Four-particle scattering amplitude would be obtained by replacing in Feynman graph description the four-momentum in propagator with CP₂ momentum defined by I₃ and Y for the particle like entity exchanged between fermions at opposite wormhole throats. Analogous replacement should work for twistorial diagrams.
   
   
3. In fact, single fermion per wormhole throat implying 4-fermion amplitudes as building blocks of more general amplitudes is only a special case although it is expected to provide excellent approximation in the case of ordinary elementary particles. Twistorial approach could allow the treatment of also n>4-fermion case using subset of twistorial n-particle amplitudes with Euclidian propagator. One cannot assign right-handed neutrino to each fermion separately but only to the elementary particle 3-surface so that the degeneration of states due to SUSY is reduced dramatically. This means strong restrictions on allowed combinations of vertices.
   

Some words of critism is in order.

1. Should one use CP₂ twistors everywhere in the 3-vertices so that only fermionic propagators would remain as remnants of M⁴? This does not look plausible. Should one use include to 3-vertices both M⁴ and CP₂ type twistorial terms? Do CP₂ twistorial terms trivialize as a consequence of quantization of Y and I₃?
   
   
2. Nothing has been said about modified Dirac operator. The assumption has been that it disappears in the functional integration and the outcome is twistor formalism. The above argument however implies functional integration over the deformations of CP₂ type vacuum extremals.
   

For details see the new chapter Some fresh ideas about twistorialization of TGD or the article with the same title.

Addition: Unfortunately, the blog page is not avaible at this moment: the owners of the webhotel lost their interest to hotel business and many people lost their money, homepage, and their data when the website ceased to exist. I lost even more: my lifework disappeared from web since the owners decided to do a really dirty trick: just for fun these psychopaths refuse to give a formal permission to have a homepage in a new physical location with the same domain name (, which of course is my property!). I find it incredible that this kind of situation can even develop. Net ethics does not yet exist.

Addition: The homepage is temporarily at new address obtained from the earlier one by replacing "com" by "fi".

About the SUSY generated by covariantly constant right-handed neutrinos

The interpretation of covariantly constant right-handed neutrinos (briefly ν_R in what follows) in M⁴× CP₂ has been a continual head-ache. Should they be included to the spectrum or not. If not, then one has no fear/hope about space-time SUSY of any kind and has only conformal SUSY. First some general observations.

1. In TGD framework right-handed neutrinos differ from other electroweak charge states of fermions in that the solutions of the modified Dirac equation for them are delocalized at entire 4-D space-time sheets whereas for other electroweak charge states the spinors are localized at string world sheets (see this).
   
   
2. Since right-handed neutrinos are in question so that right-handed neutrino are in 1-1 correspondence with complex 2-component Weyl spinors, which are eigenstates of γ₅ with eigenvalue say +1 (I never remember whether +1 corresponds to right or left handed spinors in standard conventions).
   
   
3. The basic question is whether the fermion number associated with covariantly constant right-handed neutrinos is conserved or conserved only modulo 2. The fact that the right-handed neutrino spinors and their conjugates belong to unitarily equivalent pseudoreal representations of SO(1,3) (by definition unitarily equivalent with its complex conjugate) suggests that generalized Majorana property is true in the sense that the fermion number is conserved only modulo 2. Since ν_R decouples from other fermion states, it seems that lepton number is conserved.
   
   
4. The conservation of the number of right-handed neutrinos in vertices could cause some rather obvious mathematical troubles if the right-handed neutrino oscillator algebras assignable to different incoming fermions are identified at the vertex. This is also suggested by the fact that right-handed neutrinos are delocalized.
   
   
5. Since the ν_R:s are covariantly constant complex conjugation should not affect physics. Therefore the corresponding oscillator operators would not be only hermitian conjugates but hermitian apart from unitary transformation (pseudo-reality). This would imply generalized Majorana property.
   
   
6. A further problem would be to understand how these SUSY candidates are broken. Different p-adic mass scale for particles and super-partners is the obvious and rather elegant solution to the problem but why the addition of right-handed neutrino should increase the p-adic mass scale beyond TeV range?
   

If the ν_R:s are included, the pseudoreal analog of N=1 SUSY assumed in the minimal extensions of standard model or the analog of N=2 or even N=4 SUSY is expected so that SUSY type theory might describe the situation. The following is an attempt to understand what might happen. For an earlier attempt see this.

1. Covariantly constant right-handed neutrinos as limiting cases of massless modes

For the first option covariantly constant right-handed neutrinos are obtained as limiting case for the solutions of massless Dirac equation. One obtains 2 complex spinors satisfying Dirac equation n^kγ_ku=0 for some momentum direction n^k defining quantization axis for spin. Second helicity is unphysical: one has therefore one helicity for neutrino and one for antineutrino.

1. If the oscillator operators for ν_R and its conjugate are hermitian conjugates, which anticommute to zero (limit of anticommutations for massless modes) one obtains the analog of N=2 SUSY.
   

2. If the oscillator operators are hermitian or pseudohermitian, one has pseudoreal analog of N=1 SUSY. Since ν_R decouples from other fermion states, lepton number and baryon number are conserved.
   

Note that in TGD based twistor approach four-fermion vertex is the fundamental vertex and fermions propagate as massless fermions with non-physical helicity in internal lines. This would suggest that if right-handed neutrinos are zero momentum limits, they propagate but give in the residue integral over energy twistor line contribution proportional to p^kγ_k, which is non-vanishing for non-physical helicity in general but vanishes at the limit p^k→ 0. Covariantly constant right-handed neutrinos would therefore decouple from the dynamics (natural in continuum approach since they would represent just single point in momentum space). This option is not too attractive.

2. Covariantly constant right-handed neutrinos as limiting cases of massless modes

For the second option covariantly constant neutrinos have vanishing four-momentum and both helicities are allowed so that the number of helicities is 2 for both neutrino and antineutrino.

1. The analog of N=4 SUSY is obtained if oscillator operators are not hermitian apart from unitary transformation (pseudo reality) since there are 2+2 oscillator operators.
   
   
2. If hermiticity is assumed as pseudoreality suggests, N=2 SUSY with right-handed neutrino conserved only modulo two in vertices obtained.
   
   
3. In this case covariantly constant right-handed neutrinos would not propagate and would naturally generate SUSY multiplets.
   

3. Could twistor approach provide additional insights?

Concerning the quantization of ν_R:s, it seems that the situation reduces to the oscillator algebra for complex M⁴ spinors since CP₂ part of the H-spinor is spinor is fixed. Could twistor approach provide additional insights?

As discussed, M⁴ and CP₂ parts of H-twistors can be treated separately and only M⁴ part is now interesting. Usually one assigns to massless four-momentum a twistor pair (λ^a, ξ^a') such that one has p^aa'= λ^aξ^a' ( ξ denotes for "\hat(\lambda)" which html does not allow to express). Dirac equation gives λ^a= +/- (ξ^a')^*, where +/- corresponds to positive and negative frequency spinors.

1. The first - presumably non-physical - option would correspond to limiting case and the twistors λ and ξ would both approach zero at the p^k→ 0 limit, which again would suggest that covariantly constant right-handed neutrinos decouple completely from dynamics.
   
   
2. For the second option one can assume that either λ or ξ^a' vanishes. In this manner one obtains 2 spinors λ_i, i=1,2 and their complex conjugates ξ^a'_i as representatives for the super-generators and could assign the oscillator algebra to these. Obviously twistors would give something genuinely new in this case. The maximal option would give 4 anti-commuting creation operators and their hermitian conjugates and the non-vanishing anti-commutators would be proportional to δ_a,bλ^a_i(λ^b)_j^* and δ_a,bξ^a'_i(ξ^a'_j)^*.
   If the oscillator operators are hermitian conjugates of each other and (pseudo-)hermitian, the anticommutators vanish.
   

An interesting challenge is to deduce the generalization of conformally invariant part of four-fermion vertices in terms of twistors associated with the four-fermions and also the SUSY extension of this vertex.

For details see the new chapter Some fresh ideas about twistorialization of TGD of "Towards M-matrix" or the article with the same title. The homepage is off just now as I told in previous posting but I hope that the situation changes within week or two.

Sorry: problems with homepage again

My homepage was closed suddenly: before this it became impossible to transfer material to it. I am sorry for inconvenience. I cannot get any contact with the service provider : the emails are returned as trash with appropriate warnings. Neither are the questions send to service provider from web page for clients are not answered. I am getting help but it takes some time. This is not the first problematic situation with this particular service provider and I should have changed the service provider for long time ago.

These strange unexpected closure of the webpage is not the first one. Last time this occurred two years ago. Strangely enough that service is still available although I was told for almost two years ago that it will be closed. During these four years I have been forced to change homepage address twice meaning that TGD has disappeared totally from web: Helsinki University was responsible for the first cold shower. This would be the third time unless I manage to get contact with the service provider. If this is possible (and it should be if the service provider follows finnish law) then the contents of the homepage can be transferred to the new service provider without any changes in URLs. Otherwise TGD disappears from web again. I can only hope that plain sense wins and a contact with the service provider becomes possible.

In any case, all this looks very strange to me. As a long time scientific out-of-law in Finland I cannot avoid asking whether these strange events are really co-incidences.

Addition: I have now a new temporary domain name tgdtheory.fi. To get accdess to the file in URL http://www.tgdtheory.com/... it is enough to change "com" to "fi": this leads to temporary homepage tgdtheory.fi.

TGD books and articles are also in viXra.org eprint archive: one can get a partial list about articles and books by giving my name in search. I found however that it is not possible to see all the articles. The technology does not work perfectly.

Articles about TGD are also in the journals of Huping Hu and also in Journal of Nonlocality edited by Lian Sidorov.

Addition: What makes me however happy that even string theorists are finally admitting that string approach has failed: something that I have been patiently telling for more than 30 years in vain and also provided the alternative. Leonard Susskind expressed his changed views quite recently. It is ironic that at the moment of victory I am doomed to disappear web.