Nima's lecture about twistor uprising

Lubos saved my Sunday by giving a link to an excellent talk by Nima Arkani-Hamed about the latest twistorial breakthroughs. Lubos talks about "minirevolution" but David Gross uses a more appropriate expression "uprising". I would prefer to speak about revolution inducing at the sociological level a revolt. One must give up QFT in fixed space-time and string theory, and replace them with a theory whose name Nima guesses to be just "T". I disagree;-). More about this at the end of the posting. Stay tuned;-).

I wrote last autumn a chain of postings inspired by my noble attempts to understand the articles of Nima and collaborators. I certainly failed at the level of technical details but I think that I managed to abstract the essentials from TGD point of view. By combining twistorial ideas with the basic visions of TGD- in particular zero energy ontology and bosonic emergence- I ended up with very nice general picture. I give a list of my postings in chronological order below. The titles might give some idea how the ideas about the connection of twistors and TGD evolved (I had of course pondered the problem already earlier).

1. What could be the generalization of Yangian symmetry?

2. Exact Yangian symmetry, non-trivial scattering amplitudes, no IR singularities: only a dream?

3. M⁴×CP₂ from pairs of twistors and twistorial formulation of TGD

4. A comment about exact Yangian symmetry realized in terms of bound states of partons

5. A comment about formulation of TGD in the product of twistor space and its dual

6. Yangian symmetry, twistors, and TGD

7. Unitarity in TGD variant of Grassmannian twistor approach

This activity led to a new chapter Yangian Symmetry, Twistors, and TGD of "Towards M-Matrix" giving a more organized view about the final outcome.

For some time ago Lubos told about the latest articles from Nima and collaborators: A Note on Polytopes for Scattering Amplitudes and Local Integrals for Planar Scattering Amplitudes. Unfortunately, I did not have time to try to read the articles so that my frustrated reaction was just If I only had time.

Soon after this Lubos gave a link to a video in which Witten talked about knot invariants. This talk was very inspiring and led to TGD based vision about how to calculate invariants of braids, braid cobordisms, and 2-knots in TGD framework and the idea that TGD could be seen as symplectic QFT for calculating these invariances among other things. Much of work was just translation of the basic ideas involved to TGD framework.

One crucial observation was that one can assign to the symplectic group of δM⁴₊ × CP₂ gerbe gauge potentials generalizing ordinary gauge potentials in terms of which one can define infinite number of classical 2-fluxes allowing to generalize Wilson loop to a Wilson surface. Most importantly, a unique identification for the decomposition of space-time surface to string world sheets identified as singularities of induce gauge fields and partonic 2-surfaces emerged and one can see the two decompositions as dual descriptions. TGD as almost topological QFT concretized to a symplectic QFT for knots, braids, braid cobordisms, and 2-knots. These ideas are document in the following two postings and also in the brand new chapter Knots and TGD of "TGD: Physics as Infinite-Dimensional Geometry".

1. Witten's talk about knot invariants

2. Witten's physical view about Khovanov homology translated to TGD framework

I did not realize the obvious connection with twistor approach as I wrote the articles and new chapter.

In his rather energetic lecture Nima emphasized how the Yangian symmetry originally discovered in 2-D QFTs, algebraic geometry, twistor theory, and string theory fuse to something bigger called "T". I realized that the twistorial picture developed in the earlier postings integrates nicely with the braidy vision inspired by Witten's talk and that one could understand in TGD framework why twistor description, Yangian symmetry of 2-D integrable systems, and algebraic geometry picture are so closely related. In particular, the dual conformal symmetries of twistor approach could be understood in terms of duality between partonic 2-surfaces and string world sheets expressing the strong form of holography. Also a generalization for the dual descriptions provided by super Wilson loop and ordinary scattering amplitude in N=4 SUSY in terms of Wilson sheets suggests itself among many other things. Also a rather obvious solution to the problem posed by non-planar diagrams to twistor approach suggests itself. Planar diagrams are simply not present and parton-string duality and huge symmetries of TGD give good reasons for why this should be the case.

1. The declaration of revolution by Nima from TGD point of view

At first look Nima's program is a declaration of revolution against all that has been sacred to us (do not count me: I started my personal revolution about 33 years ago;-). Nima dooms space-time, Nima wants to get rid of QFT, Nima does not even care about unitarity, Nima wants to throw Feynman diagrams to paper basket (as the Europe's worst Feynman-graphic designer I think that much hotter place would be in order for a mathematical recipe which has produced so much suffering). Nima does not even respect string theory and sees it only as one particular- possibly inept- manner to describe the underlying simplicity.

1. In many respects I agree with Nima about the fate of space-time of QFT. I however see Nima's view a little bit exaggerated: one can perhaps compute scattering amplitudes without Minkowski space but one cannot translate the results of computations to the language of experiments without bringing in frequencies and wavelengths, classical fields, and therefore also space-time. Quantum classical correspondence: this is needed and this brings space-time unavoidably into the picture. Space-time surface serves as a dynamical correlate for quantum dynamics- generalized Bohr orbit required by General Coordinate Invariance and strong form of holography. The enormously important implication is absence of Feynman graphs in ordinary sense since their is no path integral over space-time surface but just single surface: the preferred extremal of Kähler action is enough (forgetting the delicacies caused by the failure of classical determinism in standard sense for Kähler action allowing to realize also the space-time correlates of quantum jump sequences).

2. Nima uses black hole based arguments to demonstrate that local observables are not operationally defined in neither gravitational theories nor quantum field theories and concludes that space-time is doomed. What would remain would be 4-D space-time regarded as a boundary of higher dimensional space-time (AdS/CFT correspondence). I think that this is quite too complex and that the reduction in degrees of freedom is much more radical: the landscape misery is after all basically due to the exponential inflation in the number of degrees of freedom due to the fatal mistake of making 10-D target space dynamical.

   What remains in TGD are boundaries of space-time surfaces at the upper and lower ends of causal diamonds CD×CP₂ (briefly CD) and wormhole throats at which the signature of induced metric changes from Euclidian to Minkowskian (recall that Euclidian regions represent generalized Feynman diagrams). CD is essentially a representation of Penrose diagram which fits nicely with twistor approach. Strong form of holography implies that partonic 2-surfaces (or dual string world sheets) and 4-D tangent space data a them are enough as basic particle physics objects. The rest of space-time is needed to realized quantum classical correspondence essential for quantum measurement theory.

   The basic message of TGD is that quantum superpositions of space-time surfaces are relevant for physics in all scales. Particles are the dynamical space-time quanta. There is however higher-dimensional space-time which is fixed and rigid M⁴×CP₂ and is needed for the symmetries of the theory and guarantees the Kähler geometric existence of the world of classical worlds (WCW). This simplifies the situation enormously: instead of 10- or 11-D dynamical space-time one has just 4-D space-time and 2-D surfaces plus 4-D tangent space data. Holography is what we experience it to be: we see only 2-D surfaces. And physics is experimental science although some super string theorists might argue something else!

3. Nima argues also that fields are doomed too. I must say that I do not like this Planck length mysticism: it assumes quite too much and in TGD framework something new emerge already in CP₂ scale about 10⁴ longer than Planck scale. According to Nima all this pain with Feynman diagrams would be due to the need to realize unitary representations of Poincare group in terms of fields. For massless particles one is forced to assume gauge invariance to eliminate the unphysical polarizations. Nima sees gauge invariance as the source of all troubles. Here I do not completely agree with Nima. The unitary time evolution in fixed space-time translated to the path integral over classical fields is what leads to the combinatorial nightmare of summing over Feynman diagrams and plagues also φ⁴ theory. Amusingly, as Nima emphasizes all this has been known for 60 years. It is easy to understand that the possibility to realize unitarity elegantly using Feynman diagrams led to the acceptance of this approach as the only possible one.

   In TGD framework the geometry of sub-manifolds replaces fields: the dynamics of partonic 2-surfaces identified as throats of light-like wormhole contacts containing fermions at them gives rise to bosons as bound states of fermions and antifermions. There is no path integral over space-time surfaces, just functional integral over partonic 2-surfaces so that path integral disappears (one cannot however exclude the possibility that the reduction of functional integrand to Chern-Simons action could allow the imaginary unit transforming functional integral to path integral). In zero energy ontology this means that incoming states are bound states of massless fermions and antifermions at wormhole throats and virtual states consist also of massless fermions but without the bound state constraint. This means horribly strong kinematic constraints on vertices defined by partonic 2-surfaces and UV finiteness and IR finiteness are automatic outcome of the theory. Massivation guaranteeing IR finiteness is consistent with massless-ness of fundamental particles since massive states are bound states of massless particles.

4. Nima talks also about emergence as something fundamental and claims that also space-time emerges. In TGD framework emergence has very concrete meaning. All particles are bound states of massless fermions and the additional purely bosonic degrees of freedom correspond to vibrational degrees of freedom for partonic 2-surfaces.

What is lacking from the program of Nima the visions about physics as the geometry of worlds of classical worlds (WCW) and physics as generalized number theory forming the core of TGD. This is what makes the higher-D imbedding space unique and allows the geometrization of quantum physics and identification of standard model symmetries as number theoretical symmetries. Infinite-dimensional geometry is unique just from the requirement that it exists!

2. Basic results of twistor approach from TGD point of view

The basic ideas of twistor approach are remarkably consistent with the basic picture of TGD.

2.1 Only on mass-shell amplitudes appear in the recursion formula

What is striking that the recursion formula of Nima and collaborators for the integrands of the planar amplitudes of N=4 SUSY involve only on mass shell massless particles in the role of intermediate states. This is in sharp conflict with not only Feynman diagrammatic intuition but also with the very path integral ideology motivated by the need to realize unitary time development.

As already mentioned, in ZEO (zero energy ontology) all states- both on mass shell and off mass shell are composites of massless states assigned to 2-D partonic surfaces. Path integral is indeed replaced with generalized Bohr orbits and one obtains only very few generalized Feynman diagrams. What remains is functional integral over 3-surfaces, or even less over partonic 2-surfaces with varying tangent space data. It was already noticed that reduction to Chern-Simons action allows also to consider path integral option.

A further simplification is that as a result of the dynamics of preferred extremals many particle states correspond to discrete sets of points at partonic 2-surfaces serving as the ends of orbits of braid strands and possibly also 2-knots and functional integral involves integral over different configurations of these points. The physical interpretation is as a realization of finite measurement resolution as a property of dynamics itself. The string word sheets are uniquely identified as inverse images under imbedding map of space-time surface to H=M⁴× CP₂ of homologically non-trivial geodesic sphere of CP₂ defining homological magnetic monopole. Holography in its strongest sense states that all information about non-trivial 2-homology if space-time surface and knottedness of the string world sheets is coded to the data at partonic 2-surfaces. For details see the chapter Knots and TGD of "TGD: Physics as Infinite-Dimensional Geometry".

2.2 Twistors and algebraic geometry connection emerge naturally in TGD framework

H=M⁴×CP₂ and the reduction of all on mass shell states to bound states of massless states imply that twistor approach is the natural description of scattering amplitudes in TGD framework (see Yangian Symmetry, Twistors, and TGD of "Towards M-Matrix").

What is new that one must convolute massless theories in the sense that opposite throats of CP₂ sized wormhole contacts carry massless states. This allows to get rid of IR divergencies and realize exact Yangian symmetry by a purely physical mechanism making particle states massive as described in Yangian Symmetry, Twistors, and TGD.

An important implication is that even photon, gluons, and graviton have small masses and that in TGD framework all components of Higgs field are eaten by electroweak gauge bosons. Also gluons have colored scalar and pseudo-scalar counterparts and already now there are some hints at LHC for pseudo-scalar gluons. The discovery of Higgs can of course kill this idea anytime.

The connection with twistors allows to understand how algebraic geometry of projective spaces emerges in TGD framework and one indeed ends up to an alternative formulation of quantum TGD with space-time surfaces in H replaced with holomorphic 6-surfaces of CP₃×CP₃ which are sphere bundles and there effectively 4-D. The equations determining the 6-surfaces are dictated by rather general constraints (see this).

2.3 Dual descriptions in terms of QFT and strings

The connections of N=4 SUSY with 2-D integrable systems and the possibly of both stringy and QFT descriptions characterized by dual conformal symmetries giving rise to Yangian invariance reduce in TGD framework to the duality between descriptions based on string world sheets and partonic 2-surfaces.

1. The connection with string description emerges from the basic TGD in the sense that one can localize the solutions of the modified Dirac equation at braid strands located at the light-like 3-D wormhole throats. Similar localization to string world sheets defined in the above described manner holds true in space-time interior. The solutions of the modified Dirac equation localized to braid strands (and to string world sheets in space-time interior) are characterized by what I called pseudo momenta not directly identifiable as momenta. The natural identification is as the region momenta of the twistor approach. Recall that the twistorialization of region momenta leads to the momentum twistor approach making dual conformal invariance manifest.

2. The strange looking localization of fermions at braid strands makes mathematically sense only because the classical dynamics of preferred extremals reduces to hydrodynamics such that the flow parameters for flow lines integrate to global coordinates. So called Beltrami flows are in question and mean that preferred extremals have interpretation as perfect fluid flows for which dissipation is minimal. This property implies also the almost topological QFT property of TGD meaning that Kähler action reduces to Chern-Simons action localized at light-like wormhole throats and space-like 3-surfaces at the ends of CDs.

3. The mathematical motivation on braid strands comes from the fact that this allows to avoid delta functions in the anticommutators of fermionic oscillator operators at partonic 2-surfaces and therefore also the basic quadratic divergences of quantum field theories. The number of oscillator operators is finite or at least countable and the loss of complete locality is in terms of finite measurement resolution. The larger the number of braid points selected at partonic 2-surface, the larger the number string world sheets and the higher the complexity of space-time surface. This obviously means a concrete realization of holography. The oscillator algebra has interpretation as SUSY algebra with arbitrarily large N fixed by the number of braid points. This SUSY symmetry is dynamical and badly broken. For right handed neutrino the breaking is smallest but also in this case the mixing of left- and right handed M⁴ chiralities in modified Dirac equation implies non-conservation of R-parity as well as particle massivation and also the absence of lightest stable SUSY partner.

4. The big difference between TGD and string models is that super generators do not correspond to Majorana spinors: this is indeed impossible for M⁴×CP₂ since it would mean non-conservation of baryon and lepton numbers. I believed for a long time that stringy propagators emerge from TGD and the long standing painful question was what about stringy propagator defined by the inverse 1/G of the hermitian super generator in string models. In TGD 1/G cannot define stringy propagator since G carries fermion number. The reduction of strings to pairs of massless particles saves the situation and ordinary massless propagator for the counterparts of region momenta gives well defined propagators for on mass shell massless states! Stringy states reduce to bound states of massless particles in accordance with emergence philosophy. Nothing is scared these days!

2.4 Connection with integrable 2-D discrete systems

Twistor approach has revealed a striking connection between 2-D integrable systems and N=4 SUSY. For instance, one can calculate the anomalous dimensions of N=4 SUSY from an integrable model for spin chain in 2 dimensions without ever mentioning Feynman diagrams.

The description in terms of partonic 2-surfaces mean a direct connection with braids appearing in 2-D integrable thermodynamical systems and the description in terms of string world sheets means connection with integrable quantum field theories in 2-D Minkowski space. Both theories involve Yangian symmetry for which there exists a hierarchy of non-local conserved charged. Super-conformal invariance and its dual crucial for Yangian symmetry correspond to partonic 2-surfaces and string world sheets. The symmetry algebra is extended dramatically. In N=4 SUSY one has Yangian of conformal algebra of M⁴. In TGD this algebra is generalized to include the super Kac-Moody algebra associated with isometries of the imbedding space, the super-conformal variant of the symplectic algebra of δ M⁴×CP₂, and also conformal transformations of M⁴ mapping given boundary of CD to itself.

This allows also to understand and generalize the duality stating that QFT amplitudes for N=4 SUSY have interpretation as supersymmetric Wilson loops in dual Minkowski space. The ends of braid strands indeed define Wilson loops. In TGD framework work one must however generalize Wilson loops to Wilson sheets and the circulations of gauge potentials are replaced with fluxes of gerbe gauge potentials associated with the symplectic group of δM⁴₊× CP₂. As noticed, dual conformal symmetries correspond to duality of partonic 2-surfaces and string world sheets implies by the 2-D holography for string world sheets.

3. Could planar diagrams be enough in the theory transcending N=4 SUSY?

Twistor approach as it appears in N=4 SYM is of course not the final solution.

1. N=4 SUSY is not enough for the purposes of LHC.
2. The extremely beautiful Yangian symmetry fails as one performs integration to obtain the scattering amplitudes and generates IR singularities. ZEO provides an elegant solution to this problem by replacing physical on mass shell particles with bound states of massless particles. Also string like objects emerge as this kind of states.

3. Only planar diagrams allow to assign to assign to the sum of Feynman diagrams a single integrand defining the twistor diagram. Something definitely goes wrong unless one is able to treat the non-planar diagrams. The basic problem is that one cannot assign common loop momentum variables to all diagrams simultaneously and this is due to the tricky character of Feynman diagrams. It is difficult to integrate without integrand!

The easy-to-guess question is whether the sum over the non-planar diagrams vanishes or whether they are just absent in a theory transcending N=4 SUSY and QFTs. Let N denote the number of colors of the SUSY. For N → ∞ limit with g²N fixed only planar diagrams survive in this kind of theory and one obtains a string model like description as conjectured long time ago by 't Hooft. This argument led later to AdS/CFT duality.

The stringy diagrams in TGD framework could correspond to planar diagrams of N=4 QFT. Besides this one would have s functional integral over partonic 2-surfaces.

1. The description would be either in terms of partonic 2-surfaces or string world sheets with both determined uniquely in terms of a slicing of space-time surface with physical states characterized in terms of string world sheets in finite measurement resolution.

2. N → ∞ limit could in TGD framework be equivalent with two replacements. The color group with the infinite-D symplectic group of δM⁴₊× CP₂ and symplectic group and isometry group of H are replaced with their conformal variants.
3. Could g²N =constant be equivalent with the use of hyper-finite factors of type II₁ for which the trace of the unit matrix equals to 1 instead of N=∞. These factors characterize the spinor structure of WCW identifiable in terms of Clifford algebra defined by infinite-D fermionic oscillator algebra defined by second quantized fermions at partonic 2-surfaces.

4. Motives and twistors

Nima mentions at the end of his talk motives. I know about this abstract branch of algebraic geometry only that it is an attempt to build a universal cohomology theory , which in turn is an algebraic approach to topology allowing to linearize highly non-linear situations encountered typically in algebraic geometry where topology is replaced with holomorphy which is must more stringent property and allows richer structures.

1. Physics as generalized number theory vision involving also the fusion of real and p-adic number fields to a larger super structure brings algebraic geometry to the core of TGD. The partonic 2-surfaces allowing interpretation as inhabitants of the intersection of real and p-adic worlds serve as correlates for living matter in TGD Universe. They are algebraic surfaces allowing in preferred coordinates a representation in terms of polynomials with rational coefficients. Motives would be needed to understand the cohomology of these surfaces. One encounters all kinds of problems such as counting the number of rational points in the intersection of p-adic and real variants of the surface and for algebraic surfaces this reduces to the counting of rational points for real 2-surface about which algebraic geometers know a lot of. Surfaces of form xⁿ+ yⁿ+ zⁿ=0 for n≥ 3 appearing in Fermat's theorem are child's play since they allow only origin as a common point but the situation is not so easy in the general case;-).

2. As cautiously concluded in Knots and TGD, the intersection form for string world sheets defines a representation of the second relative homology of space-time surface and by Poincare duality also second cohomology. "Relative" is with respect to ends of space-time (including restaurants) at the boundaries of CDs and light-like wormhole throats. The intersection form characterizing the collection of self-intersection points at which the badly treated braid strands are forced to go through each other is almost enough to characterize connected 4-manifolds topologically by Donaldson theorem.

3. String world sheets define a violent unknotting procedure based on reconnections for braid strands- basic stringy vertex for closed strings- and in this manner knot invariant in the same manner as the recursion allowing to calculate the value of Jones polynomial for a given knot. Quantum TGD gives as a by-product rise to a symplectic QFT describing braids, their cobordisms, and 2-knots. It would not be surprising if the M-matrix elements would have also interpretation as symplectic covariants providing information about the topology of the space-time surface. The 2-braid theory associated with space-time surface would also characterize its topology just as ordinary knots can characterize topology of 3-manifolds.

To sum up, TGD suggests a surprisingly stringy but at the same time incredibly simple generalization of string model in which the discoveries made possible by the twistor approach to N=4 SUSY find a natural generalization. Nima has realized that much more than a mere discovery of computational recipes is involved and indeed talks about T-theory. Maybe he forgot two important letters "G" and "D" following the lonely "T" and completing it to holy trinity;-). I forgive this little lapsus linguae: Nima had a horrible hurry in trying to communicate as much as possible during single Nima's academic hour (100 minutes in standard time units;-)).

Non-Standard Numbers and TGD

I had opportunity to read articles of Elemer Rosinger about possible physical applications of non-standard numbers and it was natural to compare these numbers with the generalization of real numbers inspired by the notion of infinite primes. This lead to the idea of writing a commentary about the articles.

• How Far Should the Principle of Relativity Go?
• Quantum Foundations: Is Probability Ontological?
• Group Invariant Entanglements in Generalized Tensor Products
• Heisenberg Uncertainty in Reduced Power Algebras
• Surprising Properties of Non-Archimedean Field Extensions of the Real Numbers
• No-Cloning in Reduced Power Algebras

I have a rather rudimentary knowledge about non-standard numbers and my comments are very subjective and TGD centered. I however hope that they might tell also something about Rosinger's work. My interpretation of the message of articles relies on associations with my own physics inspired ideas related to the notion of number. I divide the articles to physics related and purely mathematical ones. About the latter aspects I am not able to say much.

The construction of ultrapower fields (generalized scalars) is explained using concepts familar to physicist using the close analogies with gauge theories, gauge invariance, and with the singularities of classical fields. Some questions related to the physical applications of non-standard numbers are discussed including interpretational problems and the problems related to the notion of definite integral. The non-Archimedean character of generalized scalars is discussed and compared with that of p-adic numbers.

Rosinger considers several physical ideas inspired by ultrapower fields including the generalization of general covariance to include the independence of the formulation of physics on the choice of generalized scalars, the question whether generalized scalars might allow to understand the infinities of quantum field theories, and the question whether the notion of measurement precision could realized in terms of scale hierarchy with levels related by infinite scalings. These ideas are commented in the article by comparison to p-adic variants of these ideas.

Non-standard numbers are compared with the numbers generated by infinite primes. It is found that the construction of infinite primes, integers, and rationals has a close similarity with construction of the generalized scalars. The construction replaces at the lowest level the index set Λ=N of natural numbers with algebraic numbers A, Frechet filter of N with that of A, and R with unit circle S¹ represented as complex numbers of unit magnitude. At higher levels of the hierarchy generalized -possibly infinite and infinitesimal- algebraic numbers emerge. This correspondence maps a given set in the dual of Frechet filter of A to a phase factor characterizing infinite rational algebraically so that correspondence is like representation of algebra. The basic difference between two approaches to infinite numbers is that the counterpart of infinitesimals is infinitude of real units with complex number theoretic anatomy: one might loosely say that these real units are exponentials of infinitesimals.

With motivations coming from quantum computation, Rosinger discusses also a possible generalization of the notion of entanglement allowing to define it also for what could be regarded as classical systems. Entanglement is also number theoretically very interesting notion. For instance, for infinite primes and integers the notion of number theoretical entanglement emerges and relates to the physical interpretation of infinite primes as many particles states of second quantized super-symmetry arithmetic QFT. What is intriguing that the algebraic extension of rationals induces de-entanglement. The de-entanglement corresponds directly to the replacement of a polynomial with rational coefficients with a product of the monomials with algebraic roots in general.

For details see the new chapter Non-Standard Numbers and TGD of "Physics as a Generalized Number Theory".

Second top quark related anomaly from CDF

Both Jester and Lubos tell about top quark related anomaly in proton-antiproton collisions at Tevatron reported by CDF collaboration. The anomaly has been actually reported already last summer but has gone un-noticed. For more detailed data see this.

What has been found is that the production rate for jet pairs with jet mass around 170 GeV, which happens to correspond to top quark mass, the production cross section is about 3 times higher higher than QCD simulations predict. 3.44 sigma deviation is in question meaning that its probability is same as for the normalized random variable x/σ to be larger than 3.44 for Gaussian distribution

exp(-(x/σ)²/2)/(2πσ²)^1/2.

Recall that 5 sigma is regarded as so unprobable fluctuation that one speaks about discovery. If top pairs are produced by some new particle, this deviation should be seen also when second top decays leptonically meaning a large missing energy of neutrino. There is however a slight deficit rather than excess of these events.

One can consider three interpretations.

1. The effect is a statistical fluke. But why just at the top quark mass?

2. The hadronic signal is real but there is a downwards fluctuation reducing the number of leptonic events slightly from the expected one. In the leptonic sector the measurement resolution is poorer so that this interpretation looks reasonable. In this case the decay of some exotic boson to top quark pair could explain the signal. Below this option will be considered in more detail in TGD framework and the nice thing is that it can be connected to another top quark related anomaly reported by CDF for few weeks ago.

3. Both effects are real and the signal is due to R-parity violating 3-particle decays of gluinos with mass near to top quark mass. This is the explanation proposed in the paper of Perez and collaborators.

   This option could make sense also in TGD framework where R-parity is violated by the same -purely TGD based- mechanism which forces massivation of particles at the fundamental level. Super partners are created by adding covariantly constant right-handed neutrinos and antineutrinos to the states. The dynamics of the modified Dirac equation however mixes right-handed and left-handed neutrinos so that R-parity is not conserved.

   Addition: The three-body decay would be in TGD framework sg→ st+tbar (or t+stbar) → t+tbar+ νbar (or t+tbar+ ν). Only the other member of top pair would produce neutrino and the charged lepton accompanying it in electroweak decays would be absent so that the signature is unique. t is accompanied by ν_Rbar and tbar by ν_R and gluino by ν_Rbar at fermionic wormhole throat or by ν_R at antifermionic wormhole throat also gluino for which both throats are sfermionic is possible.

   The three-body decay would be in TGD framework sg→ st+tbar (or t+stbar) → t+tbar+ νbar (or t+tbar+ ν). Only the other member of top pair would produce neutrino and the charged lepton accompanying it in electroweak decays would be absent so that the signature is unique. t is accompanied by ν_Rbar and tbar by ν_R and gluino by ν_Rbar at fermionic wormhole throat or by ν_R at antifermionic wormhole throat also gluino for which both throats are sfermionic is possible.

   This effect is one of the basic signatures of quantum TGD and due to the fact that gamma matrices appearing in Dirac equation are different from those appearing in standard quantum field theories. Both induced gamma matrices (defined as projections of the imbedding space gamma matrices to the space-time surface) and modified gamma matrices (defined as contractions with imbedding space gamma matrices of canonical momentum densities defined by action) are mixtures of M⁴ and and CP₂ gamma matrices. As a consequence, M⁴ chiralities get mixed and only H=M⁴ × CP₂-chirality is conserved and corresponds to the separate conservation of baryon and lepton numbers.

   It must be emphasized that the mixing is a direct signature for the space-time as a 4-surface identitification and distinguishes sharply and at very general level between TGD and competing theories.

Consider now the second option in which one would have heavy new boson decaying to top quark pair. The following short argument is modified from that appearing in the earlier posting.

1. Already earlier both Jester and Lubos told that CDF sees 3.4 sigma top quark pair asymmetry in proton-antiproton collisions. The asymmetry would be roughly five times larger than predicted by QCD. The asymmetry requires that the quark-antiquark pair annihilating to top quark pair can do so by coupling not only to gluons and to a new boson which has axial or partially axial coupling so that interference term would produce the asymmetry.

2. Axial vector color octet with rather strange couplings to quarks and leptons was suggested by the experimenters as an explanation of the finding. I do not however see any deep reason -correct me if I am wrong!- for why one could not consider also pseudo-scalar octet. TGD indeed predicts that all gauge bosons should be accompanied by scalars and pseudo-scalars with same quantum numbers: also gluons. Scalars should be eaten to give the third polarization to gauge bosons. Maybe the coupling to a pseudo-scalar variant of color octet Higgs could give rise to a contribution interfering with the contribution of spin zero virtual gluons and in this manner give rise to the asymmetry. Maybe there is simple objection but I am not able to invent it now. More complex option would be color octet excitation of Z⁰.

3. The decays of pseudo-scalar gluon to top quark pairs might also explain the above described anomaly since the coupling would be strong so that at least orders of magnitude would be correct.

4. Addition: Jester reports new data about the strange top-pair forward-backward asymmetry. For top pairs with invariant mass above 450 GeV the asymmetry is claimed by CDF to be stunningly large 48+/-11 per cent. 3 times more often top quarks produced in qqbar annihilation prefer to move in the direction of q. If true this would favor color octet excitations of Z⁰as the most natural explanation since the asymmetry would be not only due to the interference of vector and axial vector exchanges but also due to the inherent parity breaking of colored Z⁰ couplings. The effect would provide further support for the identification of color quantum numbers in terms of color partial waves rather than as spin like quantum numbers. The earlier support comes from the evidence for colored excitations of leptons.

   Addition: After a badly slept night I have come to new thoughts about the possible explanation of the effect. What is so weird (really weird when one begins to think the numbers!) that the outgoing top quark (t) remembers the direction of motion of quark q before annihilation to intermediate gluon which it should by the basic definition of annihilation diagram. For any exchange diagram the situation would be totally different: consider only Coulomb scattering! The quark q of the first proton would scatter from the quark of the second proton and transform to top quark in the scattering and keep its direction of motion in good approximation since small angle exchanges dominate due to the propagator factor. Flavor changing exchange diagrams are however not possible in the standard model world since the only flavor changing are charged weak currents and their contribution is negligible.

   In the new physics inspired by TGD situation is however different! The identification of family replication phenomenon in terms of genus of the wormhole throats (see this) predicts that family replication corresponds to a dynamical SU(3) symmetry with gauge bosons belonging to the octet and singlet representations. Ordinary gauge bosons would correspond besides the familar singlet representation also to exotic octet representation for which the exchanges induce neutral flavor changing currents in the case of gluons and neutral weak bosons and charge changing ones in the case of charged gauge bosons. The exchanges of the octet representation for gluons would explain the anomaly! Also electroweak octet could of of course contribute. Note that this mechanism would explain both anomalies associated with top quark production.

   What is fantastic is that LHC will soon allow to decide whether this explanation is correct!

The pleasant manner to solve fundamental problems of science

Templeton foundation has published the list of large grant awardees: the awardees are supposed to solve the problem of time with the help of 100,000 dollars per awardee on the average. There is also a meeting with the title Setting Time Aright beginning from Bergen with a luxury cruise for an invited elite and continuing as a closed meeting in Copenhagen. There are similar conferences in Iceland and Azores. I do not which are the fundamental problems to be solved in Azores but it must be an enjoyable experience.

The goals of Templeton foundation are noble but sharing money to rich people most of whom have long time ago finished their active period in science is not the manner to achive these goals. What on Earth people who have a regular research funding can do with 100,000 extra dollars: how it helps them to develop their views about time? Do these dollars make them somehow smarter? Sad to say but all this sounds like a corruption: these good-willed people with extra money have become victims of academic sharks.

Templeton could do a lot of good by founding journals which would allow to publish real frontier science censored out from so called respected journals. Templeton could organize open conferences instead of closed meetings of the economic and power elite of science. Templeton could perhaps give a small financial support for scientists living below the poverty threshold. But no big money for individuals since this brings unavoidably to the scheme the people who can sniff the odour of easy money from distance. Real progress in science is made by genuine explorers, not by rich people spending time at luxury cruises.

A good example of an activity leading to a real progress in science with very little money are the journals founded by Huping Hu. Rubbish cannot be published but the threshold for publishing is otherwise low so that referees cannot misuse their power. Everything is based on voluntary work of individuals. No money for anyone since this would bring in wrong kind of people and lead to corruption. Just the possibility to publish, to discuss, and co-operate to do science as a real exploration.

DNA Decipher Journal is launched

Huping Hu - see QuantumBrain.org & QuantumMind.org - has done a wonderful favor for those scientists able to represent new visions about physics, biology, consciousness, and also the spiritual aspects of existence by founding four journals dedicated to these issues.

I had given long time ago given up the attempts to publish anything about TGD in so called respected journals. Then I was asked to do this in a journal, which is rapidly gaining respectability. For an outsider it is not perhaps not easy to what this meant for me personally. I enjoy again the basic right of scientist- to publish the fruits of the life work, which has required total devotion and lasted already for more than three decades in my own case! It seems that science as a genuine exploration continues despite the degeneration of most of the academic research to mere career building.

The fourth journal in the series is DNA Decipher Journal devovted to issues related to biology. DNA Decipher Journal has just published its Inaugural Issue entitled "Decipherment of the Secrets of DNA" to be found at http://dnadecipher.com.

Also I have the honor to contribute articles to the inaugural issue.

• Two articles are devoted to the model of DNA as topological quantum computer based on the braiding of magnetic flux tubes connecting DNA nucleotides and lipids of nuclear and cell membranes.

• There is a commentary of the dramatic findings of Montagnier group about water memory in terms of TGD based model of water memory and the three exotic representations of genetic code suggested by TGD. The findings allow to see the representations of the codes as necessary parts of a bigger structure involving the interaction of visible and dark matter.

• There is also an article together with Peter Gariaev about how DNA could be able to generate holograms. If the proposed interpretation is correct, it is possible to photograph the magnetic body of DNA sample containing dark matter identified as phases with large value of Planck constant.

The table of contents is below.

DNA Decipher Journal Vol 1, No 1 (2011): Inaugural Issue: Decipherment of the Secrets of DNA

Table of Contents
http://dnadecipher.com/index.php/ddj/issue/view/1

Editorial

Decipherment of the Secrets of DNA
Huping Hu, Maoxin Wu

Articles

Principles of Linguistic-Wave Genetics
Peter Gariaev, Mark J. Friedman, Ekaterina A. Leonova-Gariaeva

DNA as Basis for Quantum Biocomputer
Peter Gariaev, Peter J. Marcer, Katherine A. Leonova-Gariaeva, Uwe Kaempf, Valeriy D. Artjukh

Model for the Findings about Hologram Generating Properties of DNA
Peter Gariaev, Matti Pitkanen

The Tree of Life: Tangled Roots and Sexy Shoots: Tracing the genetic pathway from the first Eukaryotes to Homo sapiens
Chris King

DNA as Topological Quantum Computer: Part I
Matti Pitkanen

DNA as Topological Quantum Computer: Part II
Matti Pitkanen

Commentary

DNA & Water Memory: Comments on Montagnier Group s Recent Findings
Matti Pitkanen

Witten's physical view about Khovanov homology translated to TGD framework

Khovanov homology generalizes the Jones polynomial as knot invariant. The challenge is to find a quantum physical construction of Khovanov homology analous to the topological QFT defined by Chern-Simons action allowing to interpret Jones polynomial as vacuum expectation value of Wilson loop in non-Abelian gauge theory.

Witten's approach to Khovanov homology relies on fivebranes as is natural if one tries to define 2-knot invariants in terms of their cobordisms involving violent un-knottings. Despite the difference in approaches it is very useful to try to find the counterparts of this approach in quantum TGD since this would allow to gain new insights to quantum TGD itself as almost topological QFT identified as symplectic theory for 2-knots, braids and braid cobordisms.

An essentially unique identification of string world sheets and therefore also of the braids whose ends carry quantum numbers of many particle states at partonic 2-surfaces emerges if one identifies the string word sheets as singular surfaces in the same manner as is done in Witten's approach. Even more, the conjectured slicings of preferred extremals by 3-D surfaces and string world sheets central for quantum TGD can be identified uniquely. The slicing by 3-surfaces would be interpreted in gauge theory in terms of Higgs= constant surfaces with radial coordinate of CP₂ playing the role of Higgs. The slicing by string world sheets would be induced by different choices of U(2) subgroup of SU(3) leaving Higgs=constant surfaces invariant.

Also a physical interpretation of the operators Q, F, and P of Khovanov homology emerges. P would correspond to instanton number and F to the fermion number assignable to right handed neutrinos. The breaking of M⁴ chiral invariance makes possible to realize Q physically. The finding that the generalizations of Wilson loops can be identified in terms of the gerbe fluxes ∫ H_A J supports the conjecture that TGD as almost topological QFT corresponds essentially to a symplectic theory for braids and 2-knots.

I do not bother to type the details but give a link to the article Could one generalize braid invariant defined by vacuum expectation of Wilson loop to an invariant of braid cobordisms and of 2-knots?. See also the new chapter Knots and TGD of "TGD: Physics as Infinite-Dimensional Geometry".

More about water memory

The findings of Montagnier and his team could mean a revolution comparable to that sparked by the Origin of Species so that they deserve another posting (and maybe even more;-)). During these years I have learned that there are two kinds of theoreticians. Theoreticians of first kind require that theory explains facts and if not gives up the theory. Theoreticians of second kind require that facts fit the theory and bravely reject the facts that do not.

Also Lubos has commented the findings of HIV Nobelist Montagnier's group and the comments of Lubos reflect excellently the distinction between the theoretician of second kind and good experimentalist. Good experimentalist is ready to accept the reality and this is why discoveries are possible. The theoretician of second kind "knows" before hand what is possible and what is not and tells it aloud and authoritatively. Ironically, Lubos is ready to accept M-theory as the only possible theory of everything despite the fact that there is not a single thread of empirical evidence supporting it but refuses to consider the possibility that biology would be something more than just complex chemistry. Our luck is that there are also theoreticians who are ready to consider the possibility that they do not yet know everything.

Questions

Montagnier's work gives support for water memory in terms of representations of some molecules in terms of water molecules or some nano-structures present in water treated in the same manner as the homeopathic remedies are made. What is really amazing is that the representations seem also to realize genetic code. The experimental arrangement stimulates several questions which I try to answer in the framework of TGD inspired model of biology.

1. The presence of 7 Hz magnetically induced oscillation seems to be necessary for the presence of the effect. What is the role of this radiation whose frequency is not far from the lowest Schumann resonance frequency with nominal value of 7.83 Hz. Recall that this frequency is in the lowest approximation determined by the radius of Earth of alone. The wave length of 7 Hz photons is slightly larger than the circumference of Earth. Could it be that a temporal pattern associated with a single period of 7 Hz oscillation could code for DNA codons. The energies involved are of course ridiculously small as compared to the thermal energy at room temperature and quantal effects are exclued in standard quantum theory.

2. How water could represent some biologically relevant aspects of molecules? For what kind of molecules this representation does exist? What are the roles of mechanical agitation and dilution in the generation of water memory? Does the 7 Hz frequency near the lowest Schumann resonance frequency relate to this somehow?

3. How water and electromagnetic radiation could represent genetic code?

Some key ideas of TGD inspired quantum biology

TGD helps to imagine possible answers to these questions. The identification of dark matter as a hierarchy of phases with large Planck constant and the notion of magnetic body (see for instance this - both deriving naturally from basic quantum TGD- are the key notions.

1. The basic vision is that magnetic body communicates with biological body and controls it by using a generalized variant of EEG consisting of fractal hierarchy of dark photons corresponding to a hierarchy of values of Planck constant (see this) E=hf with large Planck constant implying that even ELF photons can have thermal energies above thermal energy. This is the essential element in the model for the effects of ELF frequencies on vertebrate brain. The transformation of dark photon to a bunch of ELF photons or single high energy photon would be basic mechanisms transforming dark photons to ordinary ones. Biophotons would be dark photons transformed to single dark photon. EEG would represent outcome consting of a bunch of ELF photons.

2. TGD suggests that dark DNA, RNA,... and even dark aminoacids could have a key role in biological evolution providing kind of virtual world realization of biomolecules. This would make possible controlled evolution analogous to the research and development carried out in industry (see this). This is in conflict with the vision of standard biology according to which the planning of travel phone would be a process in which one throws some random collection of electronic components to a hat and looks whether a travel phone emerges from the hat after sufficiently long waiting period.

   Biological R&D would require that transcription and translation process have dark counterparts. Also the transcription of dark DNA to ordinary DNA and vice versa and even more general processes should be possible. If the water containing ordinary DNA contains its dark variant able by its darkness to leak through the filters used in the experimental situation studied by Montagnier and collaborators, the dark DNA could be able to cheat the polymerase protein so that it interprets dark DNA as a genuine DNA template and starts to generate ordinary DNA. If the magnetic flux tubes coding for DNA are all that is relevant for this, this mechanism would not depend whether the ends of flux tubes contain real or dark DNA.

3. The dark magnetic flux tubes connecting bio-molecules make it possible for them to recognize and find each other in the dense soup of biomolecules. The reduction of Planck constant for the flux tube brings the bio-molecules near to each other so that catalytic reaction becomes possible. The reconnection process for flux tubes is also in an essential role and involved with ADP-ATP process and would provide elegant realization of codes.

Representations for the genetic code in TGD

TGD suggest several non-standard representations of the genetic code.

1. Temporal patterns of electromagnetic radiation with some carrier frequency is one possibility. Gariaev's work suggests that temporal patterns of polarization directions of radiation could code for DNA sequences with each nucleotide corresponding to a definite change of polarization direction (see the discussion in here). This would mean a hierarchy of realizations of the code corresponding to different frequency scales with period of radiation defining the duration of the code word.

2. The TGD inspired model for DNA as topological quantum computer suggests a realization of codons in terms of u, and d quarks and their antiquarks at the ends of magnetic flux tubes connecting DNA nucleotides to lipids of nuclear or cellular membranes. TGD indeed predicts the possibility of several fractally scaled up copies of hadron physics with different mass scales and also dark variants of ordinary hadron physics with the Compton lengths of quarks scaled up while keeping mass scales the same. Entire fractal hierarchy of representations corresponding to carrier frequencies of dark photons could be realized.

3. One of the most amazing predictions of TGD comes from the model of dark nucleons. The states of dark nucleons are in 1-1 correspondence with DNA, RNA, tRNA, and aminocids and vertebrate genetic code is realized naturally as dark nuclear strings analogous to ordinary nuclei which are also nuclear strings in TGD based model of nuclei. The representation could be based on triplets of magnetic flux tubes with quarks at ends correlating with the genetic code words defined by the states of dark nuclei just like the representation of DNA in DNA as tqc model. A natural guess would be that the size scale of dark nucleon is same as the size scale of single DNA triplet.

What is the role of 7 Hz radiation?

7 Hz is near the frequency of the lowest Schumann resonance representing collective oscillation of the Earth's magnetic field and one can wonder about its role in the experiment of Montagnier and collaborators.

1. 7 Hz need not provide a representation for genetic code although it could do so. A possible role is as the provider of bio-rhythm and as a possible source of energy in the case that dark photons with energy above thermal energy are in question. TGD inspired theory of consciousness predicts what I call self hierarchy and one can speak about gene expression at the level of organism and even population. Schumann resonance would naturally couple with living matter and couple the magnetic bodies of living systems to the magnetic body of Earth- magnetic Mother Gaia one might say. Flux tubes within flux tubes would be simplest representation for the coupling making possible frequency modulation and also amplitude modulation. Frequency modulation is especially interesting and the song of whales provides a possible concrete example of underlying frequency modulation. The model for hologram generating properties of DNA suggests that the dark photons assignable to 7 Hz radiation pump energy to build up hologrammic representations of DNA.

2. Cyclotron resonances for ions in the Earth's magnetic field are in 1-100 Hz range and it has been known from seventies that electromagnetic fields in this frequency range have effects of vertebrate brain. These effects look very quantal and correspond to cyclotron frequencies which is .2 Gauss- 2/5 of the nominal value of the Earth's magnetic field. Also the authors of Montagnier article suggest that cyclotron resonances of ions are involved and in TGD inspired model for living body in terms of magnetic bodies cyclotron resonances are in a key role. Cyclotron frequencies could provide a coupling of biologically important ions to Schumann resonance if the flux tubes involved can vary their thickness so that the strength of magnetic field varies by flux conservation.

3. VLF frequencies above kHz seem to take this role in water memory. The wave lengths and corresponding layers of magnetic bodies are still enormous as compared to that of DNA.

How water could represent molecules?

The TGD inspred mode a model for how how water could represent at least some aspects of at least some molecules is based on earlier ideas (see this) plus some ideas inspired by the findings of Montagnier's group and by the role of ordered water in the self-organization of biomolecules.

1. The basic idea is that the magnetic body of the molecules represents biologically relevant aspects of molecule in the sense that the cyclotron radiations generated by the magnetic body is responsible for biological control and also receives signals from part of organism in some length and time scales. The mechanical agitation of water involves in the process generating water memories implies that the magnetic bodies of some molecules just drop to water. This is enough for the mimicry of the biomolecules by water.

2. Water interacts strongly with polar (hydrophilic) molecules so that the polarity of the molecules in question is expected to be very relevant for the process. Polar molecules are covered by a hydrogen bonded layer of ordered water molecules analogs to ice covering. This molecular ice freezes various biomolecules to standard configuration and the feed of energy freezes the ice cover so that processes like protein folding and formation of their aggregates which is central element in the reaction of living matter to external perturbations becomes possible. The natural idea is that the polar molecules having hydrogen bonds with water layer dictate to high degree the structure of the magnetic body.

3. The mechanical agitation of water could feed the energy needed to induce the splitting of the hydrogen bonds of a polar molecule so that the ice coating to which the magnetic body of the molecule would drop out. The process would be similar to the reaction of biomolecule to external influence. This magnetic body would represent the molecule in terms of cyclotron frequencies and behave as a real molecule as far as the effects caused by cyclotron frequencies are considered. Basically a symbolic representation of the biomolecule would be in question.

This mechanism is obviously very general and the prediction is that water remembers the presence of molecules with polar regions and do not distinguish between molecules with different non-polar regions. These non-polar regions are hydrophobic and tend to be shielded from water. Protein folding is one example of this shielding.

How the magnetic bodies could represent genetic code?

The intriguing finding that about 1/4 of hydrogen atoms of water behave effectively like dark matter in attosecond time scale was one of the first findings motivating the development of ideas about dark matter as large hbar phases and is also of crucial importance for the model of water memory. The TGD based explanation is that dark hydrogen atoms correspond to dark protons with Compton size of order atom size at least. The varying fraction of this phase would explain the large number of anomalies related to the thermodynamics of water.

The proposal is that the splitting of hydrogen bond transforms the hydrogen or at least the proton of hydrogen to a dark nucleon. The states of dark nucleons would correspond to multiplets assignable to DNA, RNA, tRNA, and aminocids. If the state of dark nucleon corresponds to the quark assignable to the end of magnetic flux tube one has a representation of the genetic code in terms of dark nuclear string consisting of protons glued to form (TGD indeed leads to a model of nuclei as nucleon sequencies connected by color magnetic bonds).

How transcription and translation type processes could be realized for dark DNA and how dark DNA and DNA could transform to each other?

Reconnection of magnetic flux tubes allows to imagine a very simple model for how DNA is coded to dark DNA and vice versa. As a matter fact, the process applies to very general class of processes defining a pairing of biomolecules. All that is needed is that the quark pair at the ends of the flux tube to some degree dictates which molecules can form. One can actually imagine a generalization of the genetic code applying to much more general molecules than molecules involved with the genetic code if this mechanism involves dark nucleons at the ends of the magnetic flux tubes involves.

1. Assume that the nucleotides of dark DNA and conjugate molecules are connected by flux tubes having quark and antiquark at their ends that u, d and their antiquarks correspond in one-one manner to DNA nucleotides so that coding results. Suppose that similar coding takes place for dark DNA in the sense that dark DNA code work is connected by three flux tubes to its conjugate for corresponding dark aminoacid. Assume that both dark and ordinary DNA nucleotides can be be connected to their conjugates by relatively long flux tubes (large hbar) and that they can be also accompanied by short-circuited flux loops. Assume again that genetic code mapping codons to quarks is realized. Similar short circuited closed flux loops could be possible for aminoacids and RNA.

2. Assume that a reconnection for long flux tube connecting nucleotides and their conjugates and for nucleotide flux loop is possible if corresponding quarks are same so that the assignement realizes genetic code. For instance, a reconnection in the middle of flux tubes connecting dark DNA and its conjugate would generate an ordinary DNA sequence. If this sequence binds to DNA strand and if the reverse of the reconnection process occurs after that, dark DNA sequence becomes coded ordinary DNA sequence. Obviously much more general processes of this kind are possible and are relatively independent of what is at the ends of the flux tubes so that genetic coded would permeate whole biology and determined selection rules of reaction involving all kinds of polar molecules.

What is the role of dilution and agitation?

I have discussed these questions also in the chapter Homeopathy in Many-sheeted Space-time of the book Biosystems as Conscious Holograms. The following discussion involves new ideas inspired by the findings of Montagnier's group.

The role of dilutions in the generation of water memories looks like a mystery and provides strongest weapon for a simple-minded sketic and one can make only guesses in this respect. The situation does not distinguish between DNA and other molecules which water is able to represent. All these molecules could correspond to dark molecules resulting when the hydrogen bonds connecting polar molecule to its water coating split if above ideas are on a right track. Consider now the questions.

1. Is the dilution necessary in order that the magnetic flux tubes of the molecular magnetic expected to have size of order 100 nm in the solution do not overlap? This would mean that the density of dark DNA in the experiments of Montagnier would be rather low in the experimental situation, maybe something like 1 DNA sequence per volume of cell nucleus. Can so low density explain the effects of polymerase in the experiment of Montagnier's team? Could the critical dilution be the dilution above which the 7 Hz radiation is able to serve as a metabolic resource?

2. Could it be that the density of dark molecules is actually much higher than the dilution would suggest? This would require replication of dark molecules, which is indeed quite conceivable if dark molecules define a life form preceiding ordinary DNA. The mechanical agitation could provide the metabolic energy for the dark molecules. Dark molecules could also be part of time in lethargic state and wake up only when energy is feeded and replicate just as biomolecules are ice-covered and wake up only when external perturbation feeds energy and induces self-organization. But why would be critical dilution required? Why the density of ordinary molecules must be so small? This is difficult to understand.

3. Is it the number of dilutions and agitations which matters rather than the density of the ordinary molecules in the final situation? Could the sequence of dilutions induce an evolutionary process analogous to a sequence of environmental catastrophes posing evolutionary pressures and leading to rapid evolution of dark DNA variant able to replicate and survive? Could each mechanical agitation induce quantum phase transitions increasing the value of Planck constant for the flux tubes inducing evolutionary leps and increasing the size scle of the corresponding magnetic body? Could the associated feed of metabolic energy also induce a replication of the dark molecules so that one would have a population with a density much higher than that of the ordinary molecules in the final situation? Whether the number of agitation-dilution processes matters instead of final density of molecules could be tested by using different initial values for the density.

4. Could the limiting wave length correspond to 7 Hz cyclotron frequency for some relevant ion and therefore also to an evolutionary step making possible to couple to magnetic Mother Gaia. The condition that the frequency corresponds to energy above thermal energy at room temperature would require hbar/hbar₀ ==r ≈ 10¹¹. The number of the dilutions should be so high that the molecular evolution reaches this level. As a matter fact, the work of Cyril Smith [Smith] with water memory suggests that r= 2× 10¹¹ is very special in the sense that a transformation for low frequency photons to high frequency photons with f_h/f_l= 2× 10¹¹ seems to be involved with water memory. Does the maximal value of hbar for the flux tubes correlate with the dilution factor or the number of dilutions inducing 10-fold reduction of density each? Does 10-fold reduction of density correspond to reduction factor which still allows the population to fill the sample completely for a reasonable amount of mechanical agitation?

5. Cyclotron radiation of dark photons from the magnetic body of dark DNA transforming to ordinary VLF photons serves as a signature for its presence. In the abstract of their article Montagnier group reports following.

   Electromagnetic signals of low frequency have been shown to be durably produced in aqueous dilutions of the Human Imunodeficiency Virus DNA. In vivo, HIV DNA signals are detected only in patients previously treated by antiretroviral therapy and having no detectable viral RNA copies in their blood. We suggest that the treatment of AIDS patients pushes the virus towards a new mode of replication implying only DNA, thus forming a reservoir insensitive to retroviral inhibitors. Implications for new approaches aimed at eradicating HIV infection are discussed.

   "New mode of replication" would correspond in TGD framework to replication of magnetic bodies of DNA representing genes as dark nucleon sequences and would allow HIV DNA to survive despite the treatment.

The idea about rapid micro-evolution taking place in human time scale for the magnetic bodies is as radical as it is fascinating but is in principle testable. I have considered alternative explanations but they are not so simple as this one. I do not of course believe that attitudes in biological sciences would be mature for testing this kind of ideas. Big changes in the world view are painful and take place slowly and existing theoretical hegemony is the worst obstacle in the progress.

I am however an optimist: a real understanding what makes HIV so resistant to ordinary treatments making possible to develop better cures of it could force even skeptics to accept the facts.

[Smith] C. Smith (2001),Learning From Water, A Possible Quantum Computing Medium. Talk in CASYS'2001, 5th international conference on Computing Anticipating Systems held in Liege, Belgium, August 13-18. Abstract book published by Chaos.

For a more organized summary of TGD based model for the findings of Montagnier's team see the article DNA Waves and Water at my homepage.

Does water represent genetic code?

Stealth Skater sent me an interesting popular article with title Scorn over claim of teleported DNA in the latest New Scientist. The furor is about the article DNA waves and water by L. Montagnier, J. Aissa, E. Del Giudice, C. Lavallee, A. Tedeschi, and G. Vitiello, which has not yet been even published.

Already "DNA waves and water" is enough to induce a deep growl from the throat of a hard-nosed skeptic, and the words "homeopathy" and "water memory" are the signals which transform even civilized skeptic to a raging blood hound. Water memory at gene level is indeed what the article is about. What makes the situation so problematic is that Montagnier is HIV Nobelist so that it is not so easy to dismiss the work as has been done routinely for all work related to water memory since the days of Benveniste and before.

The story began when Benveniste found evidence for water memory. Water solution of biomolecules was diluted so that there was no trace about the molecules. What Benveniste and collaborators claimed was that the treated water is however somehow able to represent the biologically relevant properties of molecules so that its action on some biomolecules can be the same as that of the original molecules. This could obviously explain the claimed effects of homeopathy.

Benveniste got a label of fraudster in a scientific investigation led by the magician James Randi (true, this is what the standards of skeptic science sadly often are!). The work of Benveniste has been however continued behind the scenes and it has been for a long time to possible to reproduce the effects of biologically active molecules by using only the low frequency electromagnetic spectrum of these molecules which suggest that biological signalling relies on low frequency em radiation. Skeptics have simply dismissed all this research.

That genes have electromagnetic representation have been also claimed by Peter Gariaev and his collaborators for long time ago. For TGD inspired explanations for the findings of Gariaev see this and this: the latter link is an article written in collaboration with Peter Gariaev and will be published in the first issue of DNADJ journal during this month.

The claim of Montagnier's team is that the radiation generated by DNA affects water in such a manner that it behaves as if it contained the actual DNA. A brief summary of experiment of Montagnier and collaborators is in order.

1. Two test tubes containing 100 bases long DNA fragments were studied. Both tubes were subjected to 7 Hz electromagnetic radiation. Earth's magnetic field was eliminated to prevent its possible inteference (the cyclotron frequencies of Earth's magnetic field are in EEG range and one of the family secrets of biology and neuroscience since eventies is that cyclotron frequencies in magnetic fields have biological effects on vertebrate brain). The frequencies around 7 Hz correspond to cyclotron frequencies of some biologically important ions in the endogenous magnetic field of .2 Tesla explaining the findings. This field is 2/5 of the nominal value of the Earth's magnetic field.

2. What makes the situation so irritating for skeptics who have been laughing for decades for homepathy and water memory is that the repeated dilution process used for the homeopathic remedies was applied to DNA in the recent case. The dilution containing no detectable amounts DNA (dilution factor was 10^-12) was placed in second test tube whereas the first test tube contained 100 bases long DNA in the original concentration.

3. After 16 to 18 hours both tubes were subjected to polymerase chain reaction (PCR), which builds DNA from its basic building bricks using DNA polymerase enzyme. What is so irritating that DNA was generated also in the test tube containing the highly diluted water. Water seems to be able to cheat the polymerase by mimicking the presence of the actual DNA serving in the usual situation as a template for builing copies of DNA. One could also speak about the analog quantum teleportation.

In TGD inspired quantum biology the representations of genes in terms of temporal patterns of em radiation are in central role (see this) . TGD leads to a concrete model for water memory in terms of the magnetic body of biomolecule whose cyclotron frequency pattern codes for the biological effects of the molecule. Water memory means that water can build magnetic bodies mimicking those of biomolecules or perhaps steal them in the process of dilution which involves the shaking of the solution.

TGD suggest also another representation of the genetic code in terms of dark nucleons, which could be highly relevant for the realization of water memory in terms of a dark portion of water for which there exist empirical evidence. This dark portion would also explain the numerous anomalies of water. It became as a total surprise that the states of dark nucleons correspond in natural manner to DNA, RNA, tRNA, and aminoacids. DNA would define only one particular representation of the genetic code, which in the primary form would be realized at elementary particle level and that there could exists many representations of DNA. Also the model for DNA as topological quantum computer proposes a non-standard representation of the code.

The existence of a multitude of representations of the code would not be too surprising when one realizes that the information processing performed by computers involves endless variety of different representations of various codes. The problem is about attitudes: the dogma that biology is nothing but chemistry is what is being challenged and we love dogmas because they liberate us from the burden of using our own brains.

Witten's talk about knot invariants

Lubos gave a link to a recent talk of Witten about knots and quantum physics. While listening the lecture one senses the enormous respect and -I dare say- love that the audience feels towards this silently talking genius completely free of all what might be called ego. Warmly recommended.

Witten manages to explain in rather comprehensible manner both the construction recipe of Jones polynomial and the idea about how Jones polynomial emerges from topological quantum field theory as a vacuum expectation of so called Wilson loop defined by path integral with weighting coming from Chern-Simons action. Witten also tells that during the last year he has been working with an attempt to understand in terms of quantum theory the so called Khovanov polynomial associated with a much more abstract link invariant whose interpretation and real understanding remains still open.

This kind of talks are extremely inspiring and lead to a series of questions unavoidably culminating to the frustrating "Why I do not have the brain of Witten making perhaps possible to answer these questions?". This one must just accept. In the following I summarize some thoughts inspired by the associations of the talk of Witten with quantum TGD and with the model of DNA as topological quantum computer. In my own childish manner I dare believe that these associations are interesting and dare also hope that some more brainy individual might take them seriously.

An idea inspired by TGD approach which also main streamer might find interesting is that the Jones invariant defined as vacuum expectation for a Wilson loop in 2+1-D space-time generalizes to a vacuum expectation for a collection of Wilson loops in 2+2-D space-time and could define an invariant for 2-D knots and for cobordisms of braids analogous to Jones polynomial. As a matter fact, it turns out that a generalization of gauge field known as gerbe is needed and that in TGD framework classical color gauge fields defined the gauge potentials of this field. Also topological string theory in 4-D space-time could define this kind of invariants. Of course, it might well be that this kind of ideas have been already discussed in literature. As reader have noticed, the posting has gradually evolved during last days as I have noticed elementary errors and inaccuracies. My apologies for possible inconvenience.

1. Some TGD background

What makes quantum TGD interesting concerning the description of braids and braid cobordisms is that braids and braid cobordisms emerge both at the level of generalized Feynman diagrams and in the model of DNA as a topological quantum computer.

1.1 Time-like and space-like braidings for generalized Feynman diagrams

1. In TGD framework space-times are 4-D surfaces in 8-D imbedding space. Basic objects are partonic 2-surfaces at the two ends of causal diamonds CD (intersections of future and past directed light-cones of 4-D Minkowski space with each point replaced with CP_2). The light-like orbits of partonic 2-surfaces define 3-D light-like 3-surfaces identifiable as lines of generalized Feynman diagrams. At the vertices of generalized Feynman diagrams incoming and outgoing light-like 3-surfaces meet. These diagrams are not direct generalizations of string diagrams since they are singular as 4-D manifolds just like the ordinary Feynman diagrams.

   By strong form of holography one can assign to the partonic 2-surfaces and their tangent space data space-time surfaces as preferred exrtremals of Kähler action. This guarantees also general coordinate invariance and allows to interpret the extremals as generalized Bohr orbits.

2. One can assign to the partonic 2-surfaces discrete sets of points carrying quantum numbers. As a matter fact, these sets of points seem to emerge from the solutions of the Chern-Simons Dirac equation rather naturally. These points define braid strands as the partonic 2-surface moves and defines a light-like 3-surface as its orbit as a surface of 4-D space-time surface. In the generic case the strands get tangled in time direction and one has linking and knotting giving rise to a time-like braiding.

3. Also space-like braidings are possible. One can imagine that the partonic 2-surfaces are connected by space-like curves defining TGD counterparts for strings and that in the initial state these curves define space-like braids whose ends belong to different partonic 2-surfaces. Quite generally, the basic conjecture is that the preferred extremals define orbits of string-like objects with their ends at the partonic 2-surfaces. One would have slicing of space-time surfaces by string world sheets one one hand and by partonic 2-surface on one hand. This string model is very special due to the fact that the string orbits define what could be called braid cobordisms representing which could represent unknotting of braids. String orbits in higher dimensional space-times do not allow this topological interpretation.

1.2 Dance metaphor

Time like braidings induces space-like braidings and one can speak of time-like or dynamical braiding and even duality of time-like and space-like braiding. What happens can be understood in terms of dance metaphor.

1. One can imagine that the points carrying quantum numbers are like dancers at parquettes defined by partonic 2-surfaces. These parquettes are somewhat special in that it is moving and changing its shape: dancers like me would probably get sea sick at this kind of parquette.

2. Space-like braidings means that the feet of the dancers at different parquettes are connected by threads. As the dance continues, the threads connecting the feet of different dancers at different parquettes get tangled so that the dance is coded to the braiding of the threads. Time-like braiding induce space-like braiding. One has what might be called a cobordism for space-like braiding transforming it to a new one.

1.3 DNA as topological quantum computer

The model for topological quantum computation is based on the idea that time-like braidings defining topological quantum computer programs. These programs are robust since the topology of braiding is not affected by small deformations.

1. The first key idea in the model of DNA as topological quantum computer is based on the observation that the lipids of cell membrane form a 2-D liquid whose flow defines the dance in which dancers are lipids which define a flow pattern defining a topological quantum computation. Lipid layers assignable to cellular and nuclear membranes are the parquettes. This 2-D flow pattern can be induced by the liquid flow near the cell membrane or in case of nerve pulse transmission by the nerve pulses flowing along the axon. This alone defines topological quantum computation.

2. In DNA as topological quantum computer model one however makes a stronger assumption motivated by the vision that DNA is the brain of cell and that information must be communicated to DNA level wherefrom it is communicated to what I call magnetic body. It is assumed that the lipids of the cell membrane are connected to DNA nucleotides by magnetic flux tubes defining a space-like braiding. It is also possible to connect lipids of cell membrane to the lipids of other cell membranes, to the tubulins at the surfaces of microtubules, and also to the aminoadics of proteins. The spectrum of possibilities is really wide.

   The space-like braid strands would correspond to magnetic flux tubes connecting DNA nucleotides to lipids of nuclear or cell membrane. The running of the topological quantum computer program defined by the time-like braiding induced by the lipid flow would be coded to a space-like braiding of the magnetic flux tubes. The braiding of the flux tubes would define a universal memory storage mechanism and combined with 4-D view about memory provides a very simple view about how memories are stored and how they are recalled.

2. Could braid cobordisms define more general braid invariants?

Witten says that one should somehow generalize the notion of knot invariant. The above described framework indeed suggests a very natural generalization of braid invariants to those of braid cobordisms reducing to braid invariants when the braid at the other end is trivial. This description is especially natural in TGD but allows a generalization in which Wilson loops in 4-D sense describe invariants of braid cobordisms.

2.1 Difference between knotting and linking

Before my modest proposal of a more general invariant some comments about knotting and linking are in order.

1. One must distinguish between internal knotting of each braid strand and linking of 2 strands. They look the same in the 3-D case but in higher dimensions knotting and linking are not the same thing. Codimension 2 surfaces get knotted in the generic case, in particular the 2-D orbits of the braid strands can get knotted so that this gives additional topological flavor to the theory of strings in 4-D space-time. Linking occurs for two surfaces whose dimension d₁ and d_2 satisfying d₁+d_2= D-1, where D is the dimension of the imbedding space.

2. 2-D orbits of strings do not link in 4-D space-time but do something more radical since the sum of their dimensions is D=4 rather than only D-1=3. They intersect and it is impossible to eliminate the intersection without a change of topology of the stringy 2-surfaces: a hole is generated in either string world sheet. With a slight deformation intersection can be made to occur generically at discrete points.

2.2 Topological strings in 4-D space-time define knot cobordisms

What makes the 4-D braid cobordisms interesting is following.

1. The opening of knot by using brute force by forcing the strands to go through each other induces this kind of intersection point for the corresponding 2-surfaces. From 3-D perspective this looks like a temporary cutting of second string, drawing the string ends to some distance and bringing them back and gluing together as one approaches the moment when the strings would go through each other. This surgical operation for either string produces a pair of non-intersecting 2-surfaces with the price that the second string world sheet becomes topologically non-trivial carrying a hole in the region were intersection would occur. This operation relates a given crossing of braid strands to its dual crossing in the construction of Jones polynomial in given step (string 1 above string 2 is transformed to string 2 above string 1).

2. One can also cut both strings temporarily and glue them back together in such a manner that end a/b of string 1 is glued to the end c/d of string 2. This gives two possibilities. The first corresponds to reconnection and second to a permanent splitting of second string and is not of interest now- it can however happen in the vertices of generalized Feynman diagrams. Reconnection appears as the second operation in the construction of Jones invariant besides the operation putting the string above the second one below it or vice versa. Reconnection represents the analog of trouser vertex for closed strings replaced with braid strands.

3. These observations suggest that stringy diagrams describe the braid cobordisms and a kind of topological open string model in 4-D space-time could be used to construct invariants of braid cobordisms. The dynamics of strand ends at the partonic 2-surfaces would partially induce the dynamics of the space-like braiding. This dynamics need not induce the un-knotting of space-like braids and simple string diagrams for open strings are enough to define a cobordism leading to un-knotting. The holes needed to realize the crossover for braid strands would contribute to the Wilson loop an additional term corresponding to the rotation of the gauge potential around the boundary of the hole (non-integrable phase factor). In Abelian case this gives simple commuting phase factor.

Note that braids are actually much more closer to the real world than knots since a useful strand of knotted structure must end somewhere. The abstract closed loops of mathematician floating in empty space are not very useful in real life albeit mathematically very convenient as Witten notices. Also the braid cobordisms with ends of a collection of space-like braids at the ends of causal diamond are more practical than 2-knots in 4-D space. Mathematician would see these objects as analogous to surfaces in relative homology allowed to have boundaries if they located at fixed sub-manifolds. Homology for curves with ends fixed to be on some surfaces is a good example of this. Now these fixed sub-manifolds would correspond to space-like 3-surfaces at the ends CDs and light-like wormhole throats at which the signature of the induced metric changes and which are carriers of elementary particle quantum numbers.

2.3. Invariants of braid cobordisms and 2-knots as vacuum expectations of Wilson loops in 4-D space-time

The interpretation of string world sheets in terms of Wilson loops in four-dimensional space-time is very natural so that Witten's a original identification of the Jones polynomial as vacuum expectation for a Wilson loop in 2+1-D space might be replaced with a vacuum expectation for a collection of Wilson loops in 3+1-D space-time and would characterize in the general case (multi-)braid cobordism rather than braid. If the braid at the lower or upped boundary is trivial, braid invariant is obtained. The intersections of the Wilson loops would correspond to the violent un-knotting operations and the boundaries of the resulting holes give an additional Wilson loop. An alternative interpretation is as the analog of Jones polynomial for 2-D knots in 4-D space-time generalizing Witten's theory. This description looks completely general and does not require TGD at all.

Suppose that the space-like braid strands connecting partonic 2-surfaces at given boundary of CD and light-like braids connecting partonic 2-surfaces belonging to opposite boundaries of CD form connected closed strands. The collection of closed loops can be identified as boundaries of Wilson loops and the expectation value is defined as the product of traces assignable to the loops. The definition is exactly the same as in 2+1-D case.

Is this generalization of Wilson loops enough to describe 2-knots? In the spirit of the proposed philosophy one could ask whether there exist two-knots not reducible to cobordisms of 1-knots whose knot invariants require cobordisms of 2-knots and therefore 2-braids in 5-D space-time. Could it be that dimension D=4 is somehow very special so that there is no need to go to D=5? This might be the case since for ordinary knots Jones polynomial is very faithful invariant.

Innocent novice could try to answer the question in the following manner. Let us study what happens locally as the 2-D closed surface in 4-D space gets knotted.

1. In 1-D case knotting reduces to linking and means that the first homotopy group of the knot complement is changed so that the imbedding of first circle implies that the there exists imbedding of the second circle that cannot be transformed to each other without cutting the first circle temporarily. This phenomenon occurs also for single circle as the connected sum operation for two linked circles producing single knotted circle demonstrates.
2. In 2-D case the complement of knotted 2-sphere has a non-trivial second homotopy group so that 2-balls have homotopically non- equivalent imbeddings, which cannot be transformed to each other without intersection of the 2-balls taking place during the process. Therefore the description of 2-knotting in the proposed manner would require cobordisms of 2-knots and thus 5-D space-time surfaces. However, since 3-D description for ordinary knots works so well, one could hope that the generalization the notion of Wilson loop could allow to avoid 5-D description altogether. The generalized Wilson loops would be assigned to 2-D surfaces and gauge potential A would be replaced with 2-gauge potential B defining a three-form F= dB as the analog of gauge field.
3. This generalization of bundle structure known as gerbe structure has been introduced in algebraic geometry (see this and this) and studied also in theoretical physics. 3-forms appear as analogs of gauge fields also in the QFT limit of string model. Algebraic geometer would see gerbe as a generalization of bundle structure in which gauge group is replaced with a gauge groupoid. Essentially a structure of structures seems to be in question. For instance, the principal bundles with given structure group for given space defines a gerbe. In the recent case the space of gauge fields in space-time could be seen as a gerbe. Gerbes have been also assigned to loop spaces and WCW can be seen as a generalization of loop space. Lie groups define a much more mundane example about gerbe. The 3-form F is given by F(X,Y,Z)= B(X,[Y,Z]) , where B is Killing form and for U(n) reduces to (g^-1dg)³. It will be found that classical color gauge fields define gerbe gauge potentials in TGD framework in a natural manner.

3. TGD inspired theory of braid cobordisms and 2-knots

In the sequel the considerations are restricted to TGD and to a comparison of Witten's ideas with those emerging in TGD framework.

3.1 Weak form of electric-magnetic duality and duality of space-like and time-like braidings

Witten notices that much of his work in physics relies on the assumption that magnetic charges exist and that rather frustratingly, cosmic inflation implies that all traces of them disappear. In TGD Universe the non-trivial topology of CP_2makes possible Kähler magnetic charge and inflation is replaced with quantum criticality. The recent view about elementary particles is that they correspond to string like objects with length of order electro-weak scale with Kähler magnetically charged wormhole throats at their ends. Therefore magnetic charges would be there and LHC might be able to detect their signatures if LHC would get the idea of trying to do this.

Witten mentions also electric-magnetic duality. If I understood correctly, Witten believes that it might provide interesting new insights to the knot invariants. In TGD framework one speaks about weak form of elecric magnetic duality. This duality states that Kähler electric fluxes at space-like ends of the space-time sheets inside CDs and at wormhole throats are proportional to Kähler magneic fluxes so that the quantization of Kähler electric charge quantization reduces to purely homological quantization of Kähler magnetic charge.

The weak form of electric-magnetic duality fixes the boundary conditions of field equations at the light-like and space-like 3-surfaces. Together with the conjecture that the Kähler current is proportional to the corresponding instanton current this implies that Kähler action for the preferred extremal sof Kähler action reduces to 3-D Chern-Simons term so that TGD reduces to almost topological QFT. This means an enormous mathematical simplification of the theory and gives hopes about the solvability of the theory. Since knot invariants are defined in terms of Abelian Chern-Simons action for induced Kähler gauge potential, one might hope that TGD could as a by-product define invariants of braid cobordisms in terms of the unitary U-matrix of the theory between zero energy states and having as its rows the non-unitary M-matrices analogous to thermal S-matrices.

Electric magnetic duality is 4-D phenomenon as is also the duality between space-like and time like braidings essential also for the model of topological quantum computation. Also this suggests that some kind of topological string theory for the space-time sheets inside CDs could allow to define the braid cobordism invariants.

3.2 Could Kähler magnetic fluxes define invariants of braid cobordisms?

Can one imagine of defining knot invariants or more generally, invariants of knot cobordism in this framework? As a matter fact, also Jones polynomial describes the process of unknotting and the replacement of unknotting with a general cobordism would define a more general invariant. Whether the Khovanov invariants might be understood in this more general framework is an interesting question.

1. One can assign to the 2-dimensional stringy surfaces defined by the orbits of space-like braid strands Kähler magnetic fluxes as flux integrals over these surfaces and these integrals depend only on the end points of the space-like strands so that one deform the space-like strands in an arbitrarily manner. One can in fact assign these kind of invariants to pairs of knots and these invariants define the dancing operation transforming these knots to each other. In the special case that the second knot is un-knot one obtains a knot-invariant (or link- or braid- invariant).

2. The objection is that these invariants depend on the orbits of the end points of the space-like braid strands. Does this mean that one should perform an averaging over the ends with the condition that space-like braid is not affected topologically by the allowed deformations for the positions of the end points?

3. Under what conditions on deformation the magnetic fluxes are not affect in the deformation of the braid strands at 3-D surfaces? The change of the Kähler magnetic flux is magnetic flux over the closed 2-surface defined by initial non-deformed and deformed stringy two-surfaces minus flux over the 2-surfaces defined by the original time-like and space-like braid strands connected by a thin 2-surface to their small deformations. This is the case if the deformation corresponds to a U(1) gauge transformation for a Kähler flux. That is diffeomorphism of M⁴ and symplectic transformation of CP_2 inducing the U(1) gauge transformation.

   Hence a natural equivalence for braids is defined by these transformations. This is quite not a topological equivalence but quite a general one. Symplectic transformations of CP_2 at light-like and space-like 3-surfaces define isometries of the world of classical worlds so that also in this sense the equivalence is natural. Note that the deformations of space-time surfaces correspond to this kind of transformations only at space-like 3-surfaces at the ends of CDs and at the light-like wormhole throats where the signature of the induced metric changes. In fact, in quantum TGD the sub-spaces of world of classical worlds with constant values of zero modes (non-quantum fluctuating degrees of freedom) correspond to orbits of 3-surfaces under symplectic transformations so that the symplectic restriction looks rather natural also from the point of view of quantum dynamics and the vacuum expectation defined by Kähler function be defined for physical states.

4. A further possibility is that the light-like and space-like 3-surfaces carry vanishing induced Kähler fields and represent surfaces in M⁴× Y², where Y² is Lagrangian sub-manifold of CP_2 carrying vanishing Kähler form. The interior of space-time surface could in principle carry a non-vanishing Kähler form. In this case weak form of self-duality cannot hold true. This however implies that the Kähler magnetic fluxes vanish identically as circulations of Kähler gauge potential. The non-integrable phase factors defined by electroweak gauge potentials would however define non-trivial classical Wilson loops. Also electromagnetic field would do so. It would be therefore possible to imagine vacuum expectation value of Wilson loop for given quantum state. Exponent of Kähler action would define for non-vacuum extremals the weighting. For 4-D vacuum extremals this exponent is trivial and one might imagine of using imaginary exponent of electroweak Chern-Simons action. Whether the restriction to vacuum extremals in the definition of vacuum expectations of electroweak Wilson loops could define general enough invariants for braid cobordisms remains an open question.

5. The quantum expectation values for Wilson loops are non-Abelian generalizations of exponentials for the expectation values of Kähler magnetic fluxes. The classical color field is proportional to the induced Kähler form and its holonomy is Abelian which raises the question whether the non-Abelian Wilson loops for classical color gauge field could be expressible in terms of Kähler magnetic fluxes.

As already noticed, the description of 2-knots seems to necessitate the generalization of gauge field to 3-form and the introduction of a gerbe structure. This seems to be possible in TGD framework. Classical color gauge fields are proportional to the products B_A= H_AJ of the Hamiltonians of color isometries and of Kähler form and the closed 3-form F_A= dB_A= dH_A∧ J could serve as a colored 3-form defining the analog of U(1) gauge field. What would be interesting that color would make F non-vanishing. The "circulation" h_A= ∫ H_AJ over a closed 2-surface is transforms covariatly under symplectic transformations of CP₂, whose Hamiltonians can be assigned to irreps of SU(3): just the commutator of Hamiltonians defined by Poisson bracket appears in the infiitesimal transformation. One could hope that the expectation values for the exponents of the fluxes of B_A over 2-knots could define the invariants able to catch 2-knottedness in TGD framework. The exponent defining Wilson loop would be replaced with exp(iQ^A h_A), where Q^A denote color charges acting as operators on particles involved.

Quantum expectation means in TGD framework functional integral over the symplectic orbits of partonic 2-surfaces plus 4-D tangent space data assigned to the upper and lower boundaries of CD. Suppose that holography fixes the space-like 3-surfaces at the ends of CD and light-like orbits of partonic 2-surfaces. The braids and the stringy space-time sheets could be fixed using a representation in terms of space-time coordinates so that the representation would be always the same but the imbedding varies as also the values of the exponent of Kähler function, of the Wilson loop, and of its 2-D generalization. The functional integral over symplectic transforms of 3-surfaces implies that Wilson loop and its 2-D generalization varies. The conjectured slicing of space-time surface by string world sheets conforms with the that both Wilson loop and its 2-D generalization are actually fixed by the dynamics. One can ask whether the presence of 2-D analog of Wilson loop has a direct physical meaning bringing into almost topological stringy dynamics associated with color quantum numbers and coding explicit information about space-time interior and topology of field lines so that color dynamics would also have interpretation as a theory of 2-knots.

4. Summing up

Let us summarize the ideas discussed above.

1. Instead of knots, links, and braids one could study knot and link cobordisms, that is their dynamical evolutions concretizable in terms of dance metaphor and in terms of interacting string world sheets. Each space-like braid strand can have purely internal knotting and braid strands can be linked. TGD could allow to identify uniquely both space-like and time-like braid strands and thus also the stringy world sheets more or less uniquely and it could be that the dynamics induces automatically the temporary cutting of braid strands when knot is opened violently so that a hole is generated. Gerbe gauge potentials defined by classical color gauge fields could make also possible to characterize 2-knottedness in symplectic invariant manner in terms of color gauge fluxes over 2-surfaces.

   The weak form of electric-magnetic duality would reduce the situation to almost topological QFT in general case with topological invariance replaced with symplectic one which corresponds to the fixing of the values of non-quantum fluctuating zero modes in quantum TGD. In the vacuum sector it would be possible to have the counterparts of Wilson loops weighted by 3-D electroweak Chern-Simons action defined bythe induced spinor connection.

2. One could also leave TGD framework and define invariants of braid cobordisms and 2-D analogs of braids as vacuum expectations of Wilson loops using Chern-Simons action assigned to 3-surfaces at which space-like and time-like braid strands end. The presence of light-like and space-like 3-surfaces assignable to causal diamonds could be assumed also now.

I decided to check whether the article of Gukov, Scwhartz, and Vafa entitled Khovanov-Rozansky Homology and Topological Strings relies on the primitive topological observations made above. To my surprise this did not seem to be the case. I do not of course understand much about the article but it seems that topological strings in this case are strings in 6-D space rather than 4-D space-time.

What is interesting that twistorial considerations lead to a conjecture that 4-D space-time surfaces in 8-D imbedding space have a dual description in terms of certain 6-D homomorphic surfaces which are sphere bundles in 12-D CP₃× CP₃ and effectively 4-D. This suggests a connection between descriptions based on topological strings in 6-D space and Wilson loops in 4-D space-time. Could it really be that these completely trivial observations of a mad Finnish scientist are not a standard part of knot theory?

Addition. I found from web an article by Dror Bar-Natan with title Khovanov's homology for tangles and cobordisms. The article states that the Khovanov Homology theory for knots and links generalizes to tangles, cobordisms and 2-knots. The articles says nothing explicit about Wilson loops but talks about topological QFTs.

Addition. An article of Witten about his physical approach to Khovanov homology has appeared in arXiv. The article is more or less abracadabra for anyone not working with M-theory but the basic idea is simple. Witten reformulates 3-D Chern-Simons theory as a path integral for N=4 super YM theory in the 4-D half space W×R. This allows him to use dualities and bring in the machinery of M-theory and branes. The basic structure of TGD forces a highly analogous appproach: replace 3-surfaces with 4-surfaces, consider knot cobordisms and also 2-knots, introduce gerbes, and be happy with symplectic instead of topological QFT, which might more or less be synonymous with TGD as almost topological QFT. Symplectic QFT would obviously make possible much more refined description of knots. This posting can be found also as a more organized article Could one generalize braid invariant defined by vacuum expectation of Wilson loop to an invariant of braid cobordisms and of 2-knots?. See also the new chapter Knots andTGD of "TGD: Physics as Infinite-Dimensional Geometry".