Electron as a trefoil or something more general?

The progress in the understanding of the modified Dirac equation led to the conclusion that the mere conservation of em charge in spinorial manner rather than from theorem of Gauss allows to conclude that the solutions of the modified Dirac equation must be localized at string world sheets and partonic 2-surfaces: right-handed neutrino is expection and delocalized into entire space-time surface. Looking more closely the implications of this led to the conclusion that every ordinary elementary fermion is accompanied by closed string. Ordinary elementary bosons can be accompanied by two such strings. If light-like wormhole throat orbots carry several fermions one can have several closed strings. These closed strings can get knotted and braided. I do not bother to type more but attach the
abstract of a little article Electron as a trefoil or something more general?.

There have been suggestions that elementary particle could be braided structure and that standard model quantum numbers could be reduced to topology. In TGD framework this option does not look plausible. The braiding at the level of wormhole throat orbits is however in principle possible but need not be significant for the known elementary particles. In TGD framework elementary particle is identified as a closed Kähler magnetic flux tube carrying monopole magnetic field. This flux tube is accompanied by a closed string representing the end of string world sheet carrying induced spinor field. This string can be homologically non-trivial curve and can also get knotted. Bosons even braiding becomes possible and in the general case knotting, braiding, and non-trivial homology are possible. Therefore an extremely rich topological structure is predicted, which might corresponds to relatively low energy scale. Topological sum for knots and reconnection are the basic topological reactions for these strings and can change the knotting of the string. These reactions represent basic vertices for closed strings so that closed string model could give at least idea about the dynamics of knotting and un-knotting.

For details and background see the chapter Knots and TGD, or the already mentioned article .

Fresh view about hyper-finite factors in TGD framework

For about seven years ago there was a burst of ideas related to the mathematics and physics of TGD. The notions of zero energy ontology, hierarchy of (effective as it turned out) Planck constants, and hyperfinite factors and their inclusions as a realization of finite measurement resolution emerged at that time. I have just looked for the vision about hyper-finite finite factors in the light of wisdom gained during these years. I do not bother to type more but just give the abstract of a little article Fresh view about hyper-finite factors in TGD framework representing some new results: in particular, a more precise view about how the quantum spaces identified as factors spaces of including and included HFF can be defined.

In this article I will discuss the basic ideas about the role of hyper-finite factors in TGD with the background given by a work of more than half decade. First I summarize the input ideas which I combine with the TGD inspired intuitive wisdom about HFFs of type II₁ and their inclusions allowing to represent finite measurement resolution and leading to notion of quantum spaces with algebraic number valued dimension defined by the index of the inclusion.

Also an argument suggesting that the inclusions define "skewed" inclusions of lattices to larger lattices giving rise to quasicrystals is proposed. The core of the argument is that the included HFF of type II₁ algebra is a projection of the including algebra to a subspace with dimension D≤ 1. The projection operator defines the analog of a projection of a bigger lattice to the included lattice. Also the fact that the dimension of the tensor product is product of dimensions of factors just like the number of elements in finite group is product of numbers of elements of coset space and subgroup, supports this interpretation.

One also ends up with a detailed identification of the hyper-finite factors in "orbital degrees of freedom" in terms of symplectic group associated with δ M⁴_+/-× CP₂ and the group algebras of their discrete subgroups define what could be called "orbital degrees of freedom" for WCW spinor fields. By very general argument this group algebra is HFF of type II, maybe even II₁.

For more details see the chapter Was von Neumann Right after All? of "Towards M-Matrix" or the already mentioned article.

What about the relationship of gravitational Planck constant to ordinary Planck constant?

Gravitational Planck constant is given by the expression hbar_gr= GMm/v₀, where v₀<1 has interpretation as velocity parameter in the units c=1. Can one interpret also hbar_gr as effective value of Planck constant so that its values would correspond to multifurcation with a gigantic number of sheets. This does not look reasonable.

Could one imagine any other interpretation for hbar_gr? Could the two Planck constants correspond to inertial and gravitational dichotomy for four-momenta making sense also for angular momentum identified as a four-vector? Could gravitational angular momentum and the momentum associated with the flux tubes mediating gravitational interaction be quantized in units of hbar_gr naturally?

1. Gravitational four-momentum can be defined as a projection of the M⁴-four-momentum to space-time surface. It's length can be naturally defined by the effective metric g^αβ_eff defined by the anticommutators of the modified gamma matrices. Gravitational four-momentum appears as a measurement interaction term in the modified Dirac action and can be restricted to the space-like boundaries of the space-time surface at the ends of CD and to the light-like orbits of the wormhole throats and which induced 4- metric is effectively 3-dimensional.
   
   
2. At the string world sheets and partonic 2-surfaces the effective metric degenerates to 2-D one. At the ends of braid strands representing their intersection, the metric is effectively 4-D. Just for definiteness assume that the effective metric is proportional to the M⁴ metric or rather - to its M² projection: g_eff^kl= K²m^kl.
   

   One can express the length squared for momentum at the flux tubes mediating the gravitational interaction between massive objects with masses M and m as
   

   g^αβ_eff p_αp_β= g^αβ_eff ∂_αh^k∂_βh^l p_kp_l == g_eff^kl p_kp_l = n²hbar²/L² .
   

   Here L would correspond to the length of the flux tube mediating gravitational interaction and p_k would be the momentum flowing in that flux tube. g_eff^kl= K²m^kl would give
   

   p²= n²hbar²/K²L² .
   

   hbar_gr could be identifed in this simplified situation as hbar_gr= hbar/K.
   
   

3. Nottale's proposal requires K= GMm/v₀ for the space-time sheets mediating gravitational interacting between massive objects with masses M and m. This gives the estimate
   

   p_gr =[GMm/v₀] 1/L .
   

   For v₀=1 this is of the same order of magnitude as the exchanged momentum if gravitational potential gives estimate for its magnitude. v₀ is of same order of magnitude as the rotation velocity of planet around Sun so that the reduction of v₀ to v₀≈ 2^-11 in the case of inner planets does not mean that the propagation velocity of gravitons is reduced.
   
   

4. Nottale's formula requires that the order of magnitude for the components of the energy momentum tensor at the ends of braid strands at partonic 2-surface should have value GMm/v₀. Einstein's equations T= κ G+Λ g give a further constraint. For the vacuum solutions of Einstein's equations with a vanishing cosmological constant the value of h_gr approaches infinity. At the flux tubes mediating gravitational interaction one expects T to be proportional to the factor GMm simply because they mediate the gravitational interaction.
   
   
5. One can consider similar equation for gravitational angular momentum:
   

   g^αβ_eff L_αL_β= g_eff^kl L_kL_l = l(l+1)hbar² .
   

   This would give under the same simplifying assumptions
   

   L²= l(l+1)hbar²/K².
   

   This would justify the Bohr quantization rule for the angular momentum used in the Bohr quantization of planetary orbits.
   

Maybe the proposed connection might make sense in some more refined formulation. In particular the proportionality between m^kl_eff= Km^kl could make sense as a quantum average. Also the fact, that the constant v₀ varies, could be understood from the dynamical character of m^kl_eff.

Dark matter hierarchy and fractional quantum Hall effect

I have been updating the texts about the hierarchy of effective Planck constants coming as multiples of ordinary Planck constant.

The original hypothesis was that the hierarchy of Planck constants is real. In this formulation the imbedding space was replaced with its covering space assumed to decompose to a Cartesian product of singular finite-sheeted coverings of M⁴ and CP₂.

Few years ago came the realization that it could be only effective but have same practical implications. The basic observation was that the effective hierarchy need not be postulated separately but follows as a prediction from the vacuum degeneracy of Kähler action. In this formulation Planck constant at fundamental level has its standard value and its effective values come as its integer multiples so that one should write hbar_eff=n hbar rather than hbar= nhbar₀ as I have done. For most practical purposes the states in question would behave as if Planck constant were an integer multiple of the ordinary one. In this formulation the singular covering of the imbedding space became only a convenient auxiliary tool. It is no more necessary to assume that the covering reduces to a Cartesian product of singular coverings of M⁴ and CP₂ but for some reason I kept this assumption.

The formulation based on multi-furcations of space-time surfaces to N branches. For some reason I assumed that they are simultaneously present. This is too restrictive an assumption. The N branches are very much analogous to single particle states and second quantization allowing all 0<n≤ N-particle states for given N rather than only N-particle states looks very natural. As a matter fact, this interpretation was the original one, and led to the very speculative and fuzzy notion of N-atom, which I later more or less gave up. Quantum multi-furcation could be the root concept implying the effective hierarchy of Planck constants, anyons and fractional charges, and related notions- even the notions of N-nuclei, N-atoms, and N-molecules.

I have now reconsidered the model of fractional quantum Hall effect (FQHE) in this picture. The original naive formulation was rather naive and although there were wrong elements involved, I would not accept it as a referee of a journal;-). The crucial phenomenological notion that I missed was composite fermion. I had been too inpatient to learn basic facts summarized in a brilliant manner in Nobel lecture by Horst L. Stromer.

Feeding the notion of composite fermion, one can predict correctly filling fractions. What remains to be explained using the notion of many-sheeted space-time is how composite fermions are realized as bound states of electron and magnetic flux quanta, how fractional charges and fractional braiding statistics emerge, and how it is possible to obtain non-commutative braiding statistics and associated dynamical non-abelian gauge group for which there are indications. Below is the abstract of the updated chapter.

In this chapter I try to formulate more precisely the recent TGD based view about fractional quantum Hall effect (FQHE). This view is much more realistic than the original rough scenario, which neglected the existing rather detailed understanding. The spectrum of ν, and the mechanism producing it is the same as in composite fermion approach. The new elements relate to the not so well-understood aspects of FQHE, namely charge fractionization, the emergence of braid statistics, and non-abelianity of braid statistics.

1. The starting point is composite fermion model so that the basic predictions are same. Now magnetic vortices correspond to (Kähler) magnetic flux tubes carrying unit of magnetic flux. The magnetic field inside flux tube would be created by delocalized electron at the boundary of the vortex. One can raise two questions.
   

   Could the boundary of the macroscopic system carrying anyonic phase have identification as a macroscopic analog of partonic 2-surface serving as a boundary between Minkowskian and Euclidian regions of space-time sheet? If so, the space-time sheet assignable to the macroscopic system in question would have Euclidian signature, and would be analogous to blackhole or to a line of generalized Feynman diagram.
   

   Could the boundary of the vortex be identifiable a light-like boundary separating Minkowskian magnetic flux tube from the Euclidian interior of the macroscopic system and be also analogous to wormhole throat? If so, both macroscopic objects and magnetic vortices would be rather exotic geometric objects not possible in general relativity framework.
   
   

2. Taking composite model as a starting point one obtains standard predictions for the filling fractions. One should also understand charge fractionalization and fractional braiding statistics. Here the vacuum degeneracy of Kähler action suggests the explanation. Vacuum degeneracy implies that the correspondence between the normal component of the canonical momentum current and normal derivatives of imbedding space coordinates is 1- to-n. These kind of branchings result in multi-furcations induced by variations of the system parameters and the scaling of external magnetic field represents one such variation.
   
   
3. At the orbits of wormhole throats, which can have even macroscopic M⁴ projections, one has 1→ n_a correspondence and at the space-like ends of the space-time surface at light-like boundaries of causal diamond one has 1→ n_b correspondence. This implies that at partonic 2-surfaces defined as the intersections of these two kinds of 3-surfaces one has 1→ n_a× n_b correspondence. This correspondence can be described by using a local singular n-fold covering of the imbedding space. Unlike in the original approach, the covering space is only a convenient auxiliary tool rather than fundamental notion.
   
   
4. The fractionalization of charge can be understood as follows. A delocalization of electron charge to the n sheets of the multi-furcation takes place and single sheet is analogous to a sheet of Riemann surface of function z^1/n and carries fractional charge q=e/n, n=n_an_b. Fractionalization applies also to other quantum numbers. One can have also many-electron stats of these states with several delocalized electrons: in this case one obtains more general charge fractionalization: q= ν e.
   
   
5. Also the fractional braid statistics can be understood. For ordinary statistics rotations of M⁴ rotate entire partonic 2-surfaces. For braid statistics rotations of M⁴ (and particle exchange) induce a flow braid ends along partonic 2-surface. If the singular local covering is analogous to the Riemann surface of z^1/n, the braid rotation by Δ Φ=2π, where Φ corresponds to M⁴ angle, leads to a second branch of multi-furcation and one can give up the usual quantization condition for angular momentum. For the natural angle coordinate φ of the n-branched covering Δ φ=2π corresponds to Δ Φ=n× 2π. If one identifies the sheets of multi-furcation and therefore uses Φ as angle coordinate, single valued angular momentum eigenstates become in general n-valued, angular momentum in braid statistics becomes fractional and one obtains fractional braid statistics for angular momentum.
   
   
6. How to understand the exceptional values ν=5/2,7/2 of the filling fraction? The non-abelian braid group representations can be interpreted as higher-dimensional projective representations of permutation group: for ordinary statistics only Abelian representations are possible. It seems that the minimum number of braids is n>2 from the condition of non-abelianity of braid group representations. The condition that ordinary statistics is fermionic, gives n>3. The minimum value is n=4 consistent with the fractional charge e/4.
   

   The model introduces Z₄ valued topological quantum number characterizing flux tubes. This also makes possible non-Abelian braid statistics. The interpretation of this quantum number as a Z₄ valued momentum characterizing the four delocalized states of the flux tube at the sheets of the 4-furcation suggests itself strongly. Topology would corresponds to that of 4-fold covering space of imbedding space serving as a convenient auxiliary tool. The more standard explanation is that Z₄=Z₂× Z₂ such that Z₂:s correspond to the presence or absence of neutral Majorana fermion in the two Cooper pair like states formed by flux tubes.
   

   What remains to be understood is the emergence of non-abelian gauge group realizing non-Abelian fractional statistics in gauge theory framework. TGD predicts the possibility of dynamical gauge groups and maybe this kind of gauge group indeed emerges. Dynamical gauge groups emerge also for stacks of N branes and the n sheets of multifurcation are analogous to the N sheets in the stack for many-electron states.
   

For more details see the chapter Quantum Hall effect and the hierarchy of Planck constants of "p-Adic length scale hypothesis and dark matter hierarchy".

Updated view about the hierarchy of Planck constants

During last years the work with TGD proper has transformed from the discovery of brave visions to the work of clock smith. The challenge is to fill in the details, to define various notions more precisely, and to eliminate the numerous inconsistencies.

Few years has passed from the latest formulation for the hierarchy of Planck constants. The original hypothesis was that the hierarchy is real. Few years ago came the realization that it could be only effective but have same practical implications. The basic observation was that the effective hierarchy need not be postulated separately but follows as a prediction from the vacuum degeneracy of Kähler action. In this formulation Planck constant at fundamental level has its standard value and its effective values come as its integer multiples so that one should write hbar_eff=nhbar rather than hbar= nhbar₀ as I have done. For most practical purposes the states in question would behave as if Planck constant were an integer multiple of the ordinary one.

It seems that the time is ripe for checking whether some polishing of this formulation might be needed. In particular, the work with TGD inspired quantum biology suggests a close connection between the hierarchy of Planck constants and negentropic entanglement. Also the connection with anyons and charge fractionalization has remained somewhat fuzzy. In particular, it seems that the formulation based on multi-furcations of space-time surfaces to N branches is not general enough: the N branches are very much analogous to single particle states and second quantization allowing all 0<n≤ N-particle states for given N rather than only N-particle states looks very natural: as a matter fact, this interpretation was the original one and led to the very speculative and fuzzy notion of N-atom, which I later more or less gave up. Quantum multi-furcation could be the root concept implying the effective hierarchy of Planck constants, anyons and fractional charges, and related notions- even the notions of N-nuclei, N-atoms, and N-molecules.

Basic physical ideas

The basic phenomenological rules are simple and there is no need to modify them.

1. The phases with non-standard values of effective Planck constant are identified as dark matter. The motivation comes from the natural assumption that only the particles with the same value of effective Planck can appear in the same vertex. One can illustrate the situation in terms of the book metaphor. Imbedding spaces with different values of Planck constant form a book like structure and matter can be transferred between different pages only through the back of the book where the pages are glued together. One important implication is that light exotic charged particles lighter than weak bosons are possible if they have non-standard value of Planck constant. The standard argument excluding them is based on decay widths of weak bosons and has led to a neglect of large number of particle physics anomalies.
   
   
2. Large effective or real value of Planck constant scales up Compton length - or at least de Broglie wave length - and its geometric correlate at space-time level identified as size scale of the space-time sheet assignable to the particle. This could correspond to the Kähler magnetic flux tube for the particle forming consisting of two flux tubes at parallel space-time sheets and short flux tubes at ends with length of order CP₂ size.
   

   This rule has far reaching implications in quantum biology and neuroscience since macroscopic quantum phases become possible as the basic criterion stating that macroscopic quantum phase becomes possible if the density of particles is so high that particles as Compton length sized objects overlap. Dark matter therefore forms macroscopic quantum phases. One implication is the explanation of mysterious looking quantal effects of ELF radiation in EEG frequency range on vertebrate brain: E=hf implies that the energies for the ordinary value of Planck constant are much below the thermal threshold but large value of Planck constant changes the situation. Also the phase transitions modifying the value of Planck constant and changing the lengths of flux tubes (by quantum classical correspondence) are crucial as also reconnections of the flux tubes.
   

   The hierarchy of Planck constants suggests also a new interpretation for FQHE (fractional quantum Hall effect) in terms of anyonic phases with non-standard value of effective Planck constant realized in terms of the effective multi-sheeted covering of imbedding space: multi-sheeted space-time is to be distinguished from many-sheeted space-time.
   

   In astrophysics and cosmology the implications are even more dramatic. It was Nottale, who first introduced the notion of gravitational Planck constant as hbar_gr= GMm/v₀, v₀<1 has interpretation as velocity light parameter in units c=1. This would be true for GMm/v₀ ≥ 1. The interpretation of hbar_gr in TGD framework is as an effective Planck constant associated with space-time sheets mediating gravitational interaction between masses M and m. The huge value of hbar_gr means that the integer hbar_gr/hbar₀ interpreted as the number of sheets of covering is gigantic and that Universe possesses gravitational quantum coherence in super-astronomical scales for masses which are large. This changes the view about gravitons and suggests that gravitational radiation is emitted as dark gravitons which decay to pulses of ordinary gravitons replacing continuous flow of gravitational radiation.
   
   

3. Why Nature would like to have large effective value of Planck constant? A possible answer relies on the observation that in perturbation theory the expansion takes in powers of gauge couplings strengths α=g²/4πhbar. If the effective value of hbar replaces its real value as one might expect to happen for multi-sheeted particles behaving like single particle, α is scaled down and perturbative expansion converges for the new particles. One could say that Mother Nature loves theoreticians and comes in rescue in their attempts to calculate. In quantum gravitation the problem is especially acute since the dimensionless parameter GMm/hbar has gigantic value. Replacing hbar with hbar_gr=GMm/v₀ the coupling strength becomes v₀<1.
   

Space-time correlates for the hierarchy of Planck constants

The hierarchy of Planck constants was introduced to TGD originally as an additional postulate and formulated as the existence of a hierarchy of imbedding spaces defined as Cartesian products of singular coverings of M⁴ and CP₂ with numbers of sheets given by integers n_a and n_b and hbar=nhbar₀. n=n_an_b.

With the advent of zero energy ontology, it became clear that the notion of singular covering space of the imbedding space could be only a convenient auxiliary notion. Singular means that the sheets fuse together at the boundary of multi-sheeted region. The effective covering space emerges naturally from the vacuum degeneracy of Kähler action meaning that all deformations of canonically imbedded M⁴ in M⁴×₂ have vanishing action up to fourth order in small perturbation. This is clear from the fact that the induced Kähler form is quadratic in the gradients of CP₂ coordinates and Kähler action is essentially Maxwell action for the induced Kähler form. The vacuum degeneracy implies that the correspondence between canonical momentum currents ∂L_K/∂(∥_αh^k) defining the modified gamma matrices and gradients ∂_α h^k is not one-to-one. Same canonical momentum current corresponds to several values of gradients of imbedding space coordinates. At the partonic 2-surfaces at the light-like boundaries of CD carrying the elementary particle quantum numbers this implies that the two normal derivatives of h^k are many-valued functions of canonical momentum currents in normal directions.

Multi-furcation is in question and multi-furcations are indeed generic in highly non-linear systems and Kähler action is an extreme example about non-linear system. What multi-furcation means in quantum theory? The branches of multi-furcation are obviously analogous to single particle states. In quantum theory second quantization means that one constructs not only single particle states but also the many particle states formed from them. At space-time level single particle states would correspond to N branches b_i of multi-furcation carrying fermion number. Two-particle states would correspond to 2-fold covering consisting of 2 branches b_i and b_j of multi-furcation. N-particle state would correspond to N-sheeted covering with all branches present and carrying elementary particle quantum numbers. The branches co-incide at the partonic 2-surface but since their normal space data are different they correspond to different tensor product factors of state space. Also now the factorization N= n_an_b occurs but now n_a and n_b would relate to branching in the direction of space-like 3-surface and light-like 3-surface rather than M⁴ and CP₂ as in the original hypothesis.

In light of this the working hypothesis adopted during last years has been too limited: for some reason I ended up to propose that only N-sheeted covering corresponding to a situation in which all N branches are present is possible. Before that I quite correctly considered more general option based on intuition that one has many-particle states in the multi-sheeted space. The erratic form of the working hypothesis has not been used in applications.

Multi-furcations relate closely to the quantum criticality of Kähler action. Feigenbaum bifurcations represent a toy example of a system which via successive bifurcations approaches chaos. Now more general multi-furcations in which each branch of given multi-furcation can multi-furcate further, are possible unless on poses any additional conditions. This allows to identify additional aspect of the geometric arrow of time. Either the positive or negative energy part of the zero energy state is "prepared" meaning that single n-sub-furcations of N-furcation is selected. The most general state of this kind involves superposition of various n-sub-furcations.

Basic phenomenological rules of thumb in the new framework

It is important to check whether or not the refreshed view about dark matter is consistent with existent rules of thumb.

1. The interpretation of quantized multi-furcations as WCW anyons explains also why the effective hierarchy of Planck constants defines a hierarchy of phases which are dark relative to each other. This is trivially true since the phases with different number of branches in multi-furcation correspond to disjoint regions of WCW so that the particles with different effective value of Planck constant cannot appear in the same vertex.
   
   
2. The phase transitions changing the value of Planck constant are just the multi-furcations and can be induced by changing the values of the external parameters controlling the properties of preferred extremals. Situation is very much the same as in any non-linear system.
   
   
3. In the case of massless particles the scaling of wavelength in the effective scaling of hbar can be understood if dark n-photons consist of n photons with energy E/n and wavelength nλ.
   
   
4. For massive particle it has been assumed that masses for particles and they dark counterparts are same and Compton wavelength is scaled up. In the new picture this need not be true. Rather, it would seem that wave length are same as for ordinary electron.
   

   On the other hand, p-adic thermodynamics predicts that massive elemenetary particles are massless most of the time. ZEO predicts that even virtual wormhole throats are massless. Could this mean that the picture applying on massless particle should apply to them at least at relativistic limit at which mass is negligible. This might be the case for bosons but for fermions also fermion number should be fractionalized and this is not possible in the recent picture. If one assumes that the n-electron has same mass as electron, the mass for dark single electron state would be scaled down by 1/n. This does not look sensible unless the p-adic length defined by prime is scaled down by this fact in good approximation.
   

   This suggests that for fermions the basic scaling rule does not hold true for Compton length λ_c=hbar_m. Could it however hold for de-Broglie lengths λ= hbar/p defined in terms of 3-momentum? The basic overlap rule for the formation of macroscopic quantum states is indeed formulated for de Broglie wave length. One could argue that an 1/N-fold reduction of density that takes place in the delocalization of the single particle states to the N branches of the cover, implies that the volume per particle increases by a factor N and single particle wave function is delocalized in a larger region of 3-space. If the particles reside at effectively one-dimensional 3-surfaces - say magnetic flux tubes - this would increase their de Broglie wave length in the direction of the flux tube and also the length of the flux tube. This seems to be enough for various applications.
   

One important notion in TGD inspired quantum biology is dark cyclotron state.

1. The scaling hbar→ k× hbar in the formula E_n= (n+1/2)hbar eB/m implies that cyclotron energies are scaled up for dark cyclotron states. What this means microscopically has not been obvious but the recent picture gives a rather clearcut answer. One would have k-particle state formed from cyclotron states in N-fold branched cover of space-time surface. Each branch would carry magnetic field B and ion or electron. This would give a total cyclotron energy equal to kE_n. These cyclotron states would be excited by k-photons with total energy E= khf and for large enough value of k the energies involved would be above thermal threshold. In the case of Ca⁺⁺ one has f=15 Hz in the field B_end=.2 Gauss. This means that the value of hbar is at least the ratio of thermal energy at room temperature to E=hf. The thermal frequency is of order 10¹² Hz so that one would have k≈ 10¹¹. The number branches would be therefore rather high.
   
   
   
2. It seems that this kinds of states which I have called cyclotron Bose-Einstein condensates could make sense also for fermions. The dark photons involved would be Bose-Einstein condensates of k photons and wall of them would be simultaneously absorbed. The biological meaning of this would be that a simultaneous excitation of large number of atoms or molecules can take place if they are localized at the branches of N-furcation. This would make possible coherent macroscopic changes. Note that also Cooper pairs of electrons could be n=2-particle states associated with N-furcation.
   

There are experimental findings suggesting that photosynthesis involves delocalized excitations of electrons and it is interesting so see whether this could be understood in this framework.

1. The TGD based model relies on the assumption that cyclotron states are involved and that dark photons with the energy of visible photons but with much longer wavelength are involved. Single electron excitations (or single particle excitations of Cooper pairs) would generate negentropic entanglement automatically.
   
   
2. If cyclotron excitations are the primary ones, it would seem that they could be induced by dark n-photons exciting all n electrons simultaneously. n-photon should have energy of a visible photon. The number of cyclotron excited electrons should be rather large if the total excitation energy is to be above thermal threshold. In this case one could not speak about cyclotron excitation however. This would require that solar photons are transformed to n-photons in N-furcation in biosphere.
   
   
3. Second - more realistic looking - possibility is that the incoming photons have energy of visible photon and are therefore n=1 dark photons delocalized to the branches of the N-furcation. They would induce delocalized single electron excitation in WCW rather than 3-space.
   

Charge fractionalization and anyons

It is easy to see how the effective value of Planck constant as an integer multiple of its standard value emerges for multi-sheeted states in second quantization. At the level of Kähler action one can assume that in the first approximation the value of Kähler action for each branch is same so that the total Kähler action is multiplied by n. This corresponds effectively to the scaling α_K→ α_K/n induced by the scaling hbar₀→ nhbar₀.

Also effective charge fractionalization and anyons emerge naturally in this framework.

1. In the ordinary charge fractionalization the wave function decomposes into sharply localized pieces around different points of 3-space carrying fractional charges summing up to integer charge. Now the same happens at at the level of WCW ("world of classical worlds") rather than 3-space meaning that wave functions in E³ are replaced with wave functions in the space-time of 3-surfaces (4-surfaces by holography implied by General Coordinate Invariance) replacing point-like particles. Single particle wave function in WCW is a sum of N sharply localized contributions: localization takes place around one particular branch of the multi-sheeted space time surface. Each branch carries a fractional charge q/N for teh analogs of plane waves.
   

   Therefore all quantum numbers are additive and fractionalization is only effective and observable in a localization of wave function to single branch occurring with probability p=1/N from which one can deduce that charge is q/N.
   
   

2. The is consistent with the proposed interpretation of dark photons/gravitons since they could carry large spin and this kind of situation could decay to bunches of ordinary photons/gravitons. It is also consistent with electromagnetic charge fractionalization and fractionalization of spin.
   
   
3. The original - and it seems wrong - argument suggested what might be interpreted as a genuine fractionalization for orbital angular momentum and also of color quantum numbers, which are analogous to orbital angular momentum in TGD framework. The observation was that a rotation through 2π at space-time level moving the point along space-time surface leads to a new branch of multi-furcation and N+1:th branch corresponds to the original one. This suggests that angular momentum fractionalization should take place for M⁴ angle coordinate φ because for it 2π rotation could lead to a different sheet of the effective covering.
   

   The orbital angular momentum eigenstates would correspond to waves exp(iφ m/N ), m= 0,2,...,N-1 and the maximum orbital angular momentum would correspond the sum ∑_m=0^N-1m/N = (N-1)/2. The sum of spin and orbital angular momentum be therefore fractional.
   

   The different prediction is due to the fact that rotations are now interpreted as flows rotating the points of 3-surface along 3-surface rather than rotations of the entire partonic surface in imbedding space. In the latter interpretation the rotation by 2π does nothing for the 3-surface. Hence fractionalization for the total charge of the single particle states does not take place unless one adopts the flow interpretation. This view about fractionalization however leads to problems with fractionalization of electromagnetic charge and spin for which there is evidence from fractional quantum Hall effect.
   

Negentropic entanglement between branches of multi-furcations

The application of negentropic entanglement and effective hierarchy of Planck constants to photosynthesis and metabolism suggests that these two notions might be closely related. Negentropic entanglement is possible for rational (and even algebraic) entanglement probabilities. If one allows number theoretic variant of Shannon entropy based on the p-adic norm for the probability appearing as argument of logarithm, it is quite possible to have negative entanglement entropy and the interpretation is as genuine information carried by entanglement. The superposition of state pairs a_i⊗ b_i in entangled state would represent instances of a rule. In the case of Schrödinger cat the rule states that it is better to not open the bottle: understanding the rule consciously however requires that cat is somewhat dead! Entanglement provides information about the relationship between two systems. Shannon entropy represents lack of information about single particle state.

Negentropic entanglement would replace metabolic energy as the basic quantity making life possible. Metabolic energy could generate negentropic entanglement by exciting biomolecules to negentropically entangled states. ATP providing the energy for generating the metabolic entanglement could also itself carry negentropic entanglement, and transfer it to the target by the emission of large hbar photons.

How the large hbar photons could carry negentropic entanglement?

1. In zero energy ontology large hbar photons could carry the negentropic entanglement as entanglement between positive and negative energy parts of the photon state.
   
   
2. The negentropic entanglement of large hbar photon could be also associated with its positive or energy part or both. Large hbar_eff=nhbar photon with n-fold energy E= n× hf is n-sheeted structure consisting of n-photons with energy E=hf delocalized in the discrete space formed by the N space-time sheets. The n single photon states can entangle and since the branches effectively form a discrete space, rational and algebraic entanglement is very natural. There are many options for how this could happen. For instance, for N-fold branching the superposition of all N!/(N-n)!n! states obtained by selecting n branches are possible and the resulting state is entangled state. If this interpretation is correct, the vacuum degeneracy and multi-furcations implied by it would the quintessence of life.
   
   
3. The identification of negentropic entanglement as entanglement between branches of a multi-furcation is not the only possible option. The proposal is that non-localized single particle excitations of cyclotron condensate at magnetic flux tubes give rise to negentropic entanglement relevant to living matter. Dark photons could transfer the negentropic entanglement possibly assignable to electron pairs of ATP molecule.
   
   
4. The negentropic entanglement associated with cyclotron condensate could be associated with the branches of the large hbar variant of the condensate. In this case single particle excitation would not be sum of single particle excitations at various positions of 3-space but at various sheet of covering representing points of WCW. If each of the n branches carries 1/n:th part of electron one would have an anyonic state in WCW.
   
   
5. One can also make a really crazy question. Could it be that ATP and various bio-molecules form n-particle states at the n-sheet of N-furcations and that the bio-chemistry involves simultaneous reactions of large numbers of biomolecules at these sheets? If so, the chemical reactions would take place as large number of copies.
   

Note that in this picture the breaking of time reversal symmetry in the presence of metabolic energy feed would be accompanied by evolution involving repeated multi-furcations leading to increased complexity. TGD based view about the arrow of time implies that for a given CD this evolution has definite direction of time. At the level of ensemble it implies second law but at the level of individual system means increasing complexity.

Dark variants of nuclear and atomic physics

During years I have in rather speculative spirit considered the possibility of dark variants of nuclear and atomic - and perhaps even molecular physics. Also the notion of dark cyclotron state is central in the quantum model of living matter. One such notion is the idea that dark nucleons could realize vertebrate genetic code.

Before the real understanding what charge fractionalization means it was possible to imagine several variants of say dark atoms depending on whether both nuclei and electrons are dark or whether only electrons are dark and genuinely fractionally charged. The recent picture however fixes these notions completely. Basic building bricks are just ordinary nuclei and atoms and they form n-particle states associated with n-branches of N-furcation with n=1,...,N. The fractionalization for a single particle state delocalized completely to the discrete space of N branches as the analog of plane wave means that single branch carriers charge 1/N.

The new element is the possibility of n-particle states populating n branches of the N-furcation: note that there is superposition over the states corresponding to different selections of these n branches. N-k and k-nuclei/atoms are in sense conjugates of each other and they can fuse to form N-nuclei/N-atoms which in fermionic case are analogous to Fermi sea with all states filled.

Bio-molecules seem to obey symbolic dynamics which does not depend much on the chemical properties: this has motivated various linguistic metaphors applied in bio-chemistry to describe the interactions between DNA and related molecules. This motivated the wild speculation was that N-atoms and even N-molecules could make possible the emergence of symbolic representations with n≤ N serving as a name of atom/molecule and that k- and N-k atom/molecule would be analogous to opposite sexes in that there would be strong tendency for them to fuse together to form N-atom/-molecule. For instance, in bio-catalysis k- and N-k-atoms/molecules would be paired. The recent picture about n and N-k atoms seems to be consistent with these speculations which I had already given up as too crazy. It is difficult to avoid even the speculation that bio-chemistry could replace chemical reactions with their n-multiples. Synchronized quantum jumps would allow to avoid the distastrous effects of state function reductions on quantum coherence. The second manner to say the same thing is that the effective value of Planck constant is large.

Summary

The hierarchy of Planck constants reduces to second quantization of multi-furcations in TGD framework and the hierarchy is only effective. Anyonic physics and effective charge fractionalization are consequences of second quantized multi-furcations. This framework also provides quantum version for the transition to chaos via quantum multi-furcations and living matter represents the basic application. The key element of dynamics of TGD is vacuum degeneracy of Kähler action making possible quantum criticality having the hierarchy of multi-furcations as basic aspect. The potential problems relate to the question whether the effective scaling of Planck constant involves scaling of ordinary wavelength or not. For particles confined inside linear structures such as magnetic flux tubes this seems to be the case.

There is also an intriguing connection with the vision about physics as generalized number theory. The conjecture that the preferred extremals of Kähler action consist of quaternionic or co-quaternionic regions led to a construction of them using iteration and also led to the hierarchy of multi-furcations. Therefore it seems that the dynamics of preferred extremals might indeed reduce to associativity/co-associativity condition at space-time level , to commutativity/co-commutativity condition at the level of string world sheets and partonic 2-surfaces, and to reality at the level of stringy curves (conformal invariance makes stringy curves causal determinants so that conformal dynamics represents conformal evolution).

For more details see the chapter Does TGD Predict Spectrum of Planck Constants? of "Towards M-Matrix".

Dark haze

Doug Finkbeiner tells about the discovery microwave haze haze in Cosmic Variance. See also the article Milky Way 'Haze' May Be Dark Matter Signature. There is radio-wave and micro-wave haze in galactic center. There is also nearly monochromatic gamma ray glow in the center. Also so called Fermi bubbles forming a 3-D figure eight have been discovered.

There is no standard physics mechanism explaining these findings. It has been proposed that dark matter annihilation creates electron-positron pairs and electrons. The annihilation of single dark particle or their annihilation in collision could create gamma ray pairs and give rise to nearly monochromatic gamma ray glow in the galactic center. Electrons and positrons created in the annihilations would accelerate in the strong galactic and magnetic fields and create radio and microwave haze as brehmstrahlung. It has also become clear that the haze is associated with the so called Fermi bubbles, which also represent a new discovery.

My own proposal has been that the observed mono-energetic gamma rays - with energy equal to electron's rest energy rather precisely - come from the decays of electro-pions which are bound states of color excited electrons: these would represent dark matter in TGD sense and thus have non-standard value of Planck constant. Electro-pions decay to gamma ray pairs and create a gamma ray glow in the center of galaxy. Electro-pions would have been found also at Earth in heavy ion collisions: the model for them is discussed here.

Also muo-pions and tau-pions are predicted and there is evidence also for them but neglected because their existence is not consistent with the decay widths of weak bosons: the assumption that they correspond to dark matter in the sense that their Planck constant has non-standard value prevents the decay of weak bosons to them. Quite generally, darkness means that they have non-standard value of Planck constant and cannot appear in the same vertex with ordinary elementary particles.

Electro-pions can also decay to pairs of electron and positron. These would accelerate in the galactic magnetic field and in this manner create radio wave and microwave haze, and affect also the microwave background by inverse Compton scattering of microwave photon from electron transforming it to gamma ray.

Fermi bubbles form a kind of 3-D figure eight at galactic center with halves of the eight at the opposite sides of the galactic plane. In TGD Universe could represent magnetic flux sheets (they could decompose to flux tubes) associated with what resembles magnetic dipole field with dipole in the direction of galactic "bar" in the galactic plane. The dark matter would reside inside the flux tube defining dipole (the "bar"). Electrons and positrons resulting in the decays would rotate around the flux lines and travel along the magnetic flux tube and gain energy by accelerating along them. This brings in mind a similar situation in van Allen belt around Earth, where electrons go forth and back between poles along flux lines (flux tubes).

No standard physics mechanism explaining all these findings is known and in TGD Universe the hierarchy of phases with non-standard values of Planck constant and the confinement of magnetic field and electrons and positrons at flux tubes and sheets could represent the needed new physics. The explanation of these findings would provide one additional piece of support for the notion of many-sheeted space-time. It seems that the notion of magnetic flux quanta realized as space-time sheets gains new experimental support almost every week.

Still one attempt understand generalized Feynman diagrams

The only manner to develop the understanding about generalized Feynman diagrams is to articulate the basic questions again and again in the hope that something new might emerge. There are many questions to be answered.

What Grasmannian twistorialization means when imbedding space spinor fields are the fundamental objects. How does ZEO make twistorialization possible? How twistorialization emerges from the functional integral in WCW from the proposed stringy construction of spinor modes.

One must also understand in detail the realization of super-conformal symmetries and how n-point functions of conformal field theory are associated with scattering amplitudes, and how cm degrees of freedom described using imbedding space spinor harmonics are treated in the scattering amplitudes. Also the braiding and knotting should be understood. The challenge is to find a universal form for the vertices and to identify the propagators. Also the modular degrees of freedom of partonic 2-surfaces explaining family replication phenomenon should be taken into account.

Zero energy ontology, twistors, and Grassmannian description?

In ZEO also virtual wormhole throats are massless particles and four-momentum conservation at vertices identifiable as partonic 2-surfaces at which wormhole throats meet expressed in terms of twistors leads to Grassmannian formulation automatically. This feature is thus not specific to N=4 SYM.

Momentum conservation and massless on mass-shell conditions at vertices defined as partonic 2-surfaces at which the orbits of wormhole contacts meet, are extremely restrictive, and one has good hopes that huge reduction in the number of twistorial diagrams takes place and could even lead to finite number of diagrams (number theoretic arguments favor this).

Realization of super-conformal algebra

Thanks to the advances in the construction of preferred extremals and solutions of the modified Dirac equation there has been considerable progress in the understanding of super-conformal invariance and its 4-D generalization (see this).

1. In ordinary SYM ground states correspond to both maximal helicites or only second maximal helicity of super multiplet ( N=4 case). Now these ground states are replaced by the modes of imbedding space spinor fields assignable to center of mass degrees of freedom for partonic 2-surfaces. The light-like four-momenta of these modes can be expressed in terms of twistor variables. Spin-statistics connection seems to require that the total number of fermions and antifermions associated with given wormhole throat is always odd.
   
   
2. Super-algebra consists of oscillator operators with non-vanishing quark or lepton number. By conformal invariance fermionic oscillator operators obey 1-D anti-commutation relations. The integral over CD boundary defines a bi-linear form analogous to inner product. If a reduction to single particle level takes place, the vertex is expressible as a matrix element between two fermion-anti-fermion states: the first one assignable to the incoming and outgoing wormhole throats one and second to the virtual boson identified as wormhole contact on one hand. The exchange boson entangled fermion-anti-fermion state represented by a bi-local generalization of the gauge current. This picture applying to gauge boson exchanges generalizes in rather obvious manner.
   
   
3. Unitary demands correlation between fermionic oscillator operators and spinor harmonics of imbedding space
   as following argument suggets. The bilocal generalization of gauge current defines a "norm" for spinor modes as generalization of what in QFT regarded as charge. On basis of experience with Dirac spinors one expect that this norm is not positive definite. This "norm" must be consistent with the unitarity of the scattering amplitude and the experience with QFT suggests a correlation between creation/annihilation operator character of fermionic oscillator operators and the sign of the "norm" in imbedding space degrees of freedom.
   
   
4. The modes with negative norm should correspond to negative energy fermions and annihilation operators and modes with positive norm to positive energy fermions and creation operators. Therefore the anti-commutators of fermionic oscillator operators must be linear in four-momentum or its longitudinal projection and thus proportional to p^kγ_k or p^k_Lγ_k.
   

   On the other hand, the primary anti-commutators for the induced spinor fields are proportional to the modified gamma matrix in a direction normal to the 1-D quantization curve at the boundary of string world sheet or at the partonic 2-surface. These two anti-commutators should be consistent.
   
   

   1. Does the functional integral somehow lead from the primordial anti-commutators to the anti-commutators involving longitudinal momentum and perhaps 1-D delta function in the intersection of M² with CD boundary (light-like line)?
      
      
   2. Or does the connection between the two quantizations emerge as boundary conditions stating that the normal component of modified gamma matrix at the boundary and along string world sheet equals to p_L^kγ_k? This would also realize quantum classical correspondence in the sense that the longitudinal momentum is reflected in the geometry of the space-time sheet. Quaternionic space-time surfaces indeed contain integrable distribution of M²(x) subset M⁴ at their tangent spaces. The restriction to braid strands would mean that the condition indeed makes sense. Note that braid strands should correspond to same M²(x).
      

How conformal time evolution corresponds to physical time evolution?

The only internally consistent option is conformally invariant meaning that induced spinor fields anti-commute only along as set of 1-D curves belonging to partonic 2-surfaces. This means that one can speak about conformal time evolution at partonic 2-surface.

This suggests a huge simplification of the conformal dynamics.

1. Conformal time evolution can be translated to time evolution along light-like orbit of wormhole throat by projecting the intersections of this surface with shifted light-cone boundary to the upper or lower light-like boundary of CD: whether it is upper or lower boundary of CD depends on the arrow of imbedding space time associated with the zero energy state. All partonic 2-surfaces would be mapped to same light-cone boundary. The orbits of braid strands at wormhole throat project to orbits at light-cone boundary in question and can be further projected to the sphere r_M=constant at light-boundary. 3-D dynamics would project to simplest possible stringy 2-D dynamics (spherical string orbit) and dictated by conformal invariance.
   
   
2. The conformal field theory in question is for conformal fermionic fields realized in terms of fermionic oscillator operators and n-point functions correspond to fermionic n-point functions. The non-triviality of dynamics in these degrees of freedom follows from the non-triviality of the conformal field theory. The entire collection of partonic 2-surfaces at the ends of CD would reduce to its projection to S².
   
   
3. One can try to build a geometric view about the situation using as a guideline conformal Hamiltonian quantum evolution. Time=constant slices would correspond to 1-D curve or collection of them. At these slices fermionic oscillator operators would satisfy the conformal anti-commutation relations. This kind of slice would be associated with both ends of CD. Braid strands would connect these 1-D slices as kind of hairs. One can however ask whether there is any need to restrict the end points of braid strands to line on a curve at which fermionic oscillator operators satisfy stringy anti-commutation relations.
   

What happens in 3-vertices?

The vision is that only 3-vertices are needed. Idealize particles as wormhole contacts (in reality pair of wormhole contacts connected by a flux tube would describe elementary particles). A very convenient visualization of wormhole contact is as a very short string like object with throats at its ends so that stringy diagrammatics allows to identify the vertices as the analogs of open string vertices. One can even consider the possibility that string theory amplitudes define the vertices. This would conform with the p-adic mass calculations applying conformal invariance in CP₂ scale. Note also that partonic 2-surfaces are effectively replaced by braids so that very stringy picture results.

1. Consider a three vertex representing the emission of boson by incoming fermion (FFB) or by incoming boson (BBB) described as wormhole contact such that throats carry fermion and anti-fermion number in the bosonic case. In the fermionic the first throat carries fermion and second one represents vacuum state. The exchanged boson can be regarded as fermion anti-fermion pair such that second fermion travels to future and second one to the past in the vertex. 3-vertex would reduce to two 2-vertices representing the transformation of fermion line from incoming line to exchanged line or from latter to outgoing line.
   
   
2. The minimal option is that the same vertex describes the situation if both cases. Essentially a combination of incoming free fermions to boson like state is in question and corresponds in string picture an exchange of open string between open strings. If so, second wormhole throat is passive and suffers forward scattering in the vertex. The fermion and anti-fermion of the exchanged virtual boson (the light-like momenta of wormhole throats need not be collinear for virtual bosons and also the sign of energy can be different form them) would suffer scattering before the transformation to fermions belonging to incoming and outgoing wormhole contact.
   

One expects the vertices to factorize into products of two kinds of factors: the inner products of fermionic Fock states defined by conformal n-point functions at sphere of light-cone boundary, and the bi-linear forms for the modes of imbedding space spinor fields involving integral over cm degrees of freedom and allowing twistorialization by previous arguments. Let us continue with the simple example in which wormhole throats carry fermion number 0 or 1.

1. If second wormhole throat is passive, it is enough to construct only FFB vertex, with B identified as a wormhole contact carrying fermion and anti-fermion. One has 4 fermions altogether, and one expects that in cm degrees of freedom incoming and outgoing fermion are un-correlated whereas the fermions of the boson exchange are correlated and the correlation is expressible as the analog of gauge current.
   
   
2. This suggests a sum over bi-local counterparts of electro-weak and color gauge currents at opposite ends of the exchanged line. Bi-local gauge currents would contain a spinor mode from both wormhole throats, and the strict locality of M⁴ gauge currents would be replaced with a bi-locality in CP₂ scale.
   
   
3. The current assignable to a particular boson exchange must involve the matrix element of corresponding charge matrix between spinor modes besides the quantity. Is it possible to find a general expression for the sum over current - current interaction terms? If this is the case, there would be no need to perform the summation over bosonic exchanges explicitly. One would have the analog for the ∑_n |n ><| in propagator line but summation allowing the momenta of fermion and antifermion to be arbitrary massless momenta rather than summing up to the on mass shell momentum of boson. The counterpart of gauge coupling should be universal and naturally given by Kähler coupling.
   
   
4. The TGD counterparts of scalar and pseudo-scalar bosons would be vector bosons with polarization in CP₂ direction and they could be also seen both as Higgs like states and Euclidian pions assignable to wormhole contacts. Genuine H-scalars are excluded implied by 8-D chiral symmetry implying also separate conservation of B and L.
   

In the general case the wormhole throats carry arbitrary odd fermion number but for fermion numbers n>1 at any wormhole throat exotic super-partner with propagator decaying faster than 1/p² is in question. Furthermore, wormhole contact is accompanied by second wormhole contact since the flux lines of monopole flux must closed. Therefore one has a pair of "long" string like flux tubes connected by short flux tubes at their ends. Its length is given by weak length scale quite generally or possibly by Compton length. The other end of the long flux tube can also contain fermions at both flux tubes.

The identification of propagators

A natural guess is that the propagator for single fermion state is just the longitudinal Dirac propagator D_pL for a massless fermion in M⁴⊃ M². For states, which by statistics constraint always contain an odd number M=2N+1 of fermions and antifermions, the propagator would be M:th power of fermionic longitudinal propagator so that it would reduce to p_L^-2ND_pL meaning that only the single fermion states would be behave like ordinary elementary particles. States with higher fermion number would represent radiative corrections reflecting the non-point-like nature of partons. Longitudinal mass squared would be equal to the sum of the contribution from CP₂ degrees of freedom and the integer valued conformal contribution from spinor harmonics. The M⁴ momenta associated with wormhole throats would be light-like. In the prescription using fermionic longitudinal propagators assigned to the braid strands, braid strands are analogous to the edges of polygons appearing in twistor Grassmannian approach.

Some open questions

A long list of open questions remains without a final answer. Consider first twistor Grassmannian approach.

1. Does this prescription follow from quantum criticality? Recall that quantum criticality formulated in terms of preferred extremals and modified Dirac equation leads to a stringy perturbation theory involving fermionic propagator defined by the modified Dirac operator and functional integral over WCW for the deformations of space-time surface preserving the preferred extremal property (see this). This propagator could be called space-time propagator to distinguish it from the imbedding space propagator associated with the longitudinal momentum.
   
   
2. One expects that one still has topological Feynman diagrammatic expansion (besides that defined by functional integral over small deformations of space-time surface with given topology) involving in principle an arbitrary number of vertices defined by the intermediate partonic 2-surfaces. Momentum conservation and massless on mass-shell conditions however pose powerful restrictions on the allowed diagrams, and one might hope that the simplicity of the outcome is comparable to Grassmannian twistor approach for N=4 SYM. One can even hope that the number of contributing diagrams is finite. The important point would be that Grassmannian diagrams would give the outcome of the functional integral over 3-surfaces. Twistorial Grassmann representation is the first guess hitherto for the explicit outcome of the functional integral over WCW.
   
   
3. The lines of Feynman graph are replaced with braids. A new element is that braid strands are braided as curves inside light-like 3-surfaces defined by the orbit of the wormhole throat. Twistorial construction applies only to the planar amplitudes of N=4 SYM. Can one imagine TGD counterparts for non-planar amplitudes in TGD framework or does the stringy picture imply that they are completely absent?
   

   A possible answer to the question is based on the M² projection of the lines of braid strands (or on the projection to the 2-surface defined by an integrable distribution of tangent planes M²(x)). For non-planar diagrams the projections intersect and the intersection cannot be eliminated by a small deformation. It does not make sense to say that line goes over or below the second line. One can speak only about crossings. In the theory or algebraic knots (see this) algebraic knots with crossings are possible. Could algebraic knot theory allow to reduce non-planar diagrams to sums of planar diagrams?
   
   

4. Does one obtain Yangian symmetry using longitudinal propagators and by integrating over the moduli labeling among other things the choices of the preferred plane M²⊂ M⁴ or integrable distribution of preferred planes M²(x)⊂ M⁴? The integral over the choices M²⊂ M⁴ gives formally a Lorentz invariant outcome. Does it also give rise to physically acceptable scattering amplitudes? Are the gauge conditions for the incoming gauge boson states formulated in terms of longitudinal momentum and thus allowing also the third polarization physical? Can one apply this gauge condition also to the virtual boson like exchanges?
   
   
5. It is still somewhat unclear whether one should assume single global choice of M² or an integrable distribution of M²(x).
   
   1. The choice of M²(x) must be same for all braid strands of given partonic 2-surface and remain constant along braid strand and therefore be same also at second end of the strand. Otherwise the fermionic propagator would vary along braid strand. A possible additional condition on braids is that braid strands correspond to the same choice of M²(x). In quantum measurement theory this corresponds to the choice of same spin quantization axes for all fermions inside parton and is physically extremely natural condition. The implication is that one can indeed assign a fixed M² with CD and choice of braid strands via boundary conditions. The simplest boundary conditions would require M²(x) to be constant at light-like 3-surfaces and at the ends of space-time surface at boundaries of CD. This is in spirit with holography stating that quantum measurements can be carried out only at these 3-surfaces (or at least those at the ends of CD).
      
      
   2. One cannot exclude the possibility that M²(x) does not depend on x for a particular space-time sheet and even entire CD although this looks rather strong a restriction. On the other hand, one can ask whether the preferred M² assigned with CD should be generalized to an integrable distribution M²(x) assigned with CD such that M²(x) is contained in the tangent space of preferred Minkowskian extremal.
      
      
   3. Is the functional integral over integrable distributions M²(x) needed? It would be analogous to a functional integral over string world sheets. It is enough to integrate over Lorentz transforms of a given intebrable distribution M²(x) to achieve Lorentz invariance. This because the choice of the integrable distribution reduces effectively to the choice of M² for the disconnected pieces of generalized Feynman diagram. Physical intuition suggests that a particular choice of M²(x) corresponds to fixing of zero modes of WCW and is essentially fixing of classical variables needed to fix quantization axes. The fixing of value distributions of induced Kähler fields n 4-D sense at partonic 2-surfaces would be similar fixation of zero modes.
      
      
   4. If only M² momentum makes it visible in anti-commutators, how the other components of four-momentum can make themselves visible in dynamics? This is possible via momentum conservation at vertices making possible twistor Grassmannian approach. The dynamics in transversal momenta would be dictated completely by the conservation laws.
      

There are also other challenges.

1. Family replication phenomenon has TGD based explanation in terms of the conformal moduli of partonic 2-surfaces. How conformal moduli should be taken into account in the Feynman diagrammatics? Phenomena like topological mixing inducing in turn the mixing of partonic 2-topologies responsible for CKM mixing in TGD Universe should be understood in this description.
   
   
2. Number theoretical universality requires that also the p-adic variants of the amplitudes should make sense. One could even require that the amplitudes decompose to products of parts belonging to different number fields (see this). If one were able to formulate this vision precisely, it would provide powerful constraints on the amplitudes. For instance, a reduction of the amplitudes to a sum over finite number of generalized Feynman diagrams is plausible since this would guarantee that individual contributions which must give rise to algebraic numbers for algebraic 4-momenta, would sum up to an algebraic number.
   

New evidence for anomalies of radio-active decay rates

Lubos Motl told about new evidence for periodic variations of nuclear deay rates reported by Sturrock et al in their article
Analysis of Gamma Radiation from a Radon Source: Indications of a Solar Influence . The abstract of the article summarizes the results.

This article presents an analysis of about 29,000 measurements of gamma radiation associated with the decay of radon in a sealed container at the Geological Survey of Israel (GSI) Laboratory in Jerusalem between 28 January 2007 and 10 May 2010. These measurements exhibit strong variations in time of year and time of day, which may be due in part to environmental influences. However, time-series analysis reveals a number of periodicities, including two at approximately 11.2 year^-1 and 12.5 year^-1. We have previously found these oscillations in nuclear-decay data acquired at the Brookhaven National Laboratory (BNL) and at the Physikalisch-Technische Bundesanstalt (PTB), and we have suggested that these oscillations are attributable to some form of solar radiation that has its origin in the deep solar interior. A curious property of the GSI data is that the annual oscillation is much stronger in daytime data than in nighttime data, but the opposite is true for all other oscillations. This may be a systematic effect but, if it is not, this property should help narrow the theoretical options for the mechanism responsible for decay-rate variability.

Quantitative summary of findings

The following gives a brief quantitative summary of the findings. Radioactive decays of nuclei have been analyzed in three earlier studies and also in the recent study.

1. BNL data are about ³⁶Cl and ³²Si nuclei. Strong day-time variation in month time scale was observed. Twofrequency bands ranging from 11.0 to 11.2 year^-1 and from 12.6 to 12.9 year^-1 were observed.
   
2. PTB data are about ²²⁶Ra nuclei. Also now strong day-time variation was observed with frequency bands ranging from 11.0 to 11.3 year^-1 and from 12.3 to 12.5 year^-1 .
   
3. GIS data are about ²²²Ra nuclei. Instead of strong day-time variation a strong night-time variation was observed. Annual oscillation was centered on mid-day. 2 year^-1 is the next strongest feature. Also a night time feature with a peak at 17 hours was observed. There are also features at 12.5 year^-1 and 11.2 year^-1 and 11.9 year^-1. All these three data sets lead to oscillations in frequency bands ranging from 11.0 to 11.4 year^-1 and from 12.1 to 12.9 year^-1.
   
4. Bellotti et al studied ¹³⁷Cl nuclei deep underground in Gran Sasso. No variations were detected.
   

Could exotic nuclear states explain the findings?

The TGD based new physics involved with the effect could relate to the excitations of exotic nuclear states induced by em radiation arriving from Sun. This would change the portions of various excited nuclei with nearly the same ground state energy and affect the average radio-active decay rates.

1. The exotic nuclei emerge in the model of nucleus as a nuclear string with nucleons connected by color flux tubes having quark and antiquark at ends (see this). The excitations could be also involved with cold fusion. For the normal nuclei color flux tubes would be neutral but one can consider also excitations for which quark pair carries a net change +/- e. This would give rise to a large number of nuclei with same em charge and mass number but having actually abnormal proton and neutron numbers. If the energy differences for these excitations are in keV range they might represent a fine structure of nuclear levels not detected earlier.
   

   Could these exchanges take place also between different nuclei? For instance, could it be that in the collision of deuterium nuclei the second nucleus can be neutralized by the exchange of scaled down W boson leading to neutralization of second deuterium nucleus so that Coulomb wall could disappear and make possible cold nuclear reaction. It seems that the range of this scaled variant of weak interaction is quite too short. M₁₂₇ variant of weak interactions with W boson mass very near to electron mass could make possible this mechanism.
   
   

2. The exchange of weak bosons could be responsible for generating these excitations: in this case two neutral color bonds would become charged with opposite charges. If one takes seriously the indications for 38 MeV new particle (see this), one can even consider a scaled variant of weak interaction physics with weak interaction length scale given by a length scale near hadronic length scale (see this). E(38) could be scaled down Z boson with mass of about 38 MeV.
   

Em radiation from Sun inducing transitions of ordinary nuclei to their exotic counterparts could be responsible for the variation of the radio-active decay rates. If course, exotic nuclei in the above sense are only one option and the following argument below applies quite generally.

Kinetic model for the evolution for the number of excited nuclei

A simple model for the evolution of the number of excited nuclei is as follows:

dN/dt= kJ-k₁N for t∈ [t₀,t₁] ,

dN/dt= -k₁N for t∈ [t₁,t₀+T] .

J denotes the flux of incoming radiation and N the number of excited nuclei. t₀ corresponds to the time of sunrise and t₁ to the time of sunset and T is 24 hours in the approximation that sun rises at the same time every morning. The time evolution of N(t) is given by

N(t) = k/k₁ J+(N(t₀)-kJ/k₁ )exp[-k₁(t-t₀] for t∈ [t₀,t₁] ,

N(t)= N(t₁)exp[-k₁(t-t₁)] for t∈ [t₁,t₀+T] .

Explanation for the basic features of the data

The model can explain the qualitative features of the data rather naturally.

1. The period of 1 year obviously correlates with the distance from Sun. .5 year period correlates with the fact that the distance from Sun is minimal twice during a year. Day-time night-time difference can be explained with the fact that em radiation at night-time does not penetrate Earth. This explains also why Gran Sasso in deep underground observes nothing.
   
   
2. The large long time scale variation for the day-time data for BNL and PTB seems to be in apparent constrast with that for the night-dime data at GIS. It is however possible to understand the difference.
   
   
   1. If the rate parameter k₁ is large, one can understand why variations are strong at day-time in BNL and BTB. For large value of k₁ N(t) increases rapidly to its asymptotic value N_max= kJ/k₁ and stays in it during day so that day-time variations due to solar distance are large. At night-time N(t) rapidly decreases to zero so that night-time variation due to the variation of the solar distance is small.
      
      
   2. For GIS the strong variation is associated with the night-dime data. This can be understood in terms of small value of k₁ which can be indeed smaller for ²²⁶Ra than for the nuclei used in the other studies. During daytime N(t) slowly increases to its maximum at N(t₁) and decreases slowly during night-time. Since N(t₁) depends on the time of the year, the night-time variation is large.
      
      
   3. The variations in time scales of roughly the time scale of month should be due to the variations in the intensity of the incoming radiation. The explanation suggested in the article is that the dynamics of solar core has these periodicities manifested also as the periodicities of the emission of radiation at the frequencies involved. These photons would naturally correspond to the photons emitted in the transitions between excited states of nuclei in the solar core or possibly in solar corona having temperature of about 300 eV. One could in fact think that the mysterious heating of solar corona to a temperature of 3 million K could be due to the exotic excitations of the nuclei by radiation coming from Sun. At this temperature the maximum of black body distribution with respect to frequency corresponds to energy of .85 keV consistent with the proposal that the energy scale for excitations is keV.
      
      
   4. The difference of frequencies 12.49 year^-1 and 11.39 year^-1 is in good approximation 1 year^-1, which suggests modulation of the average frequency with a period of year being due to the rotation of Earth around Sun. The average frequency is 11.89 year^-1 that is 1/month. The explanation proposed in the article explanation proposed in the article is in terms of rotation velocity of the inner core which would be smaller but same order of magnitude as that of the outer core (frequency range from 13.7 to 14.7 year^-1). It is however not plausible that the keV photons could propagate from the iinner core of Sun unless they are dark in TGD sense. In TGD framework it would be natural to assign the frequency band to solor Corona.
      

Can one assign the observed frequency band to the rotation of solar corona?

The rotation frequency band assignable to photosphere is too high by about Δ f=3 year^-1 as compared to that appearing in decay rate variation. Could one understand this discrepancy?

1. One must distinguish between the synodic rotation frequency f_S measured in the rest system of Sun and the rotation frequency observed in Earth rotating with frequency f=1 year^-1 around Sun: these frequencies relate by f_E= f_S-f giving frequency range 12.7 to 13.7 year^-1. This is still too high by about Δ f=2 year^-1.
   
   
2. Could corona rotate slower than photosphere? The measurements by Mehta give the value range 22 - 26.5 days meaning that the the coronal synodic frequency f_C would be in the range 14.0-16.6 year^-1. The range of frequences observed at Earth would be 13-15.6 year^-1 and too high by about Δ =2 year^-1.
   

   If I have understood correctly, the coronal rotational velocity is determined by using solar spots as markers and therefore refers to the magnetic field rather than the gas in the corona. Could the rotation frequency of the gas in corona be about Δ f=2 year^-1 lower than that for the magnetic spots?
   

One can develop a theoretical argument in order to understand the rotational periods of photosphere and corona and why they could differ by about Δ f=2 year^-1.

1. Suppose that one can distinguish between the rotation frequencies of magnetic fields (magnetic body in many-sheeted space-time) and gas. Suppose that photosphere (briefly 'P') and corona (briefly 'C') can be treated in the first approximation as rigid spherical shells having thus moment of inertia I= (2/3)mR² around the rotational axis. The angular momentum per unit mass is dL/dm= (2/3)R²ω. Suppose that the value of dL/dm is same for the photosphere and Corona. If the rotation velocity magnetic fields determined from magnetic spots is same as the rotation velocity of gas in corona, this implies f_C/f_P= (R_S/R_C)², where R_S is solar radius identifiable as the radius of photosphere. The scaling of 13 year^-1 down to 11 year^-1 would require R_C/R_S≈ 1.09. This radius should correspond to the hottest part of the corona at temperature about 1-2 million K.
   

   The inner solar corona extends up to (4/3)R_S (see this). This would give average radius of the inner coronal shell about 1.15R_S. The constancy of dL/dm(R) would give a differential rotation with frequency varying as 1/R². If the frequency band reflects the presence of differential rotation, one has R_max/R_min≈ (f_max/f_min)^1/2 ≈ (15/13)^1/2≈ 1.07.
   
   

2. One can understand why angular momentum density per mass is constant if one accepts a generalization of the Bohr quantization of planetary orbits originally proposed by Nottale and based on the notion of gravitational Planck constant hbar_gr. One has hbar_gr= GMm/v₀ and is assigned with the flux sheets mediating gravitational interaction between Sun and the planet or some other astrophysical object near Sun. The dependence on solar mass and planetary mass is is fixed by Equivalence Principle. v₀ has dimensions of velocity and therefore naturally satisfies v₀<c. For the three inner planets one has v₀/c≈ 2^-11. Angular momentum quantization gives mR²ω= n× hbar_gr giving R²ω= nGM/v₀ so that the angular momentum per mass is integer valued. For the inner planets n has values 3,4,5.
   
   
3. One could argue that for the photosphere and corona regarded as rigid bodies a similar quantization holds true but with the same value of n since the radii are so near to each other. Also v₀ should be larger. Consider first photosphere. One can apply the angular momentum quantization condition to photosphere approximate as a spherical shell and rigid body. Iω_P= nGmM/v_0P for n=1 gives (2/3)R²ω= GM/v_0P. For v_0P=c one would obtain ω_P/ω_E= (3/2) (R_E/R)²(v₀/v_0P). For R_P= .0046491 R_E (solar radius) this gives ω_P/ω_E ≈ 12.466 for the v₀/c= 4.6× 10^-4 used by Nottale (see this): I have often used the approximate nominal value v₀/c= 2^-11 but now it this approximation is too rough. Taking into account the frequency shift due to Earth's orbital motion one obtains ω_P/ω_E ≈ 11.466 which is consistent with the lower bound of the observed frequency band and would correspond to R_max. The value v_0P=v_0C=c looks unrealistic if interpreted as a physical velocity of some kind the increase of R_C allows however to reduce the value of v_0C so that it seems possible to understand the situation quantitatively.
   

   If one wants to generalize this argument to differential rotation, one must decompose the system spherical shells or more general elements rotating at different velocities and having different value of hbar_gr assignable to the flux tubes connecting them to Sun and mediating gravitational interaction. This decomposition must be physical.
   

For the background see the chapter Nuclear string hypothesis of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy". See also the article.

Do galaxies have preferred handedness?

New Scientist tells that spiral galaxies which seem to have tendency to be left handed along two lines which have angle of 85 degrees with respect to each other. Galaxies would be therefore like biomolecules which also have preferred handedness in living matter.

Handedness in geometric sense requires that the mirror image of the galaxy is not identical with galaxy itself. In good approximation galaxies are however rotationally symmetric around the spin axis. In dynamical sense handedness results if the total angular momentum of galaxy is non-vanishing. Spiral galaxies indeed have spin.

What has been observed that along these two lines of sight there are more left- than right handed galaxies. The length for the light of sight was 1.2 billion ly in the survey of Michael Longo and 3.4 billion ly in the survey of Lior Shamir. The scale of our large void is about .1 billion light years so that cosmic length scales are in question. The findings could of course be statistical flukes. Future surveys will resolve this issue.

The existence of a preferred axes of symmetry in cosmic scales does not fit well with isotropy and homogenuity assumptions of the standard cosmology. The TGD based proposal for the formation of galaxies and other astrophysical structures relies on a fractal network of string like objects defined by Kähler magnetic flux tubes. These magnetic flux tubes were present in primordial cosmology and had 1-D M⁴ projection at that time: they indeed defined string world sheets in M⁴. During the cosmic expansion the thickness of their M⁴ projections has increased gradually. These string like objects carry dark energy as magnetic energy and also the magnetic fields have become weaker during expansion. These flux tubes could also correspond to a gigantic value of Planck constant. Various astrophysical structures consisting of ordinary and dark matter would have formed via the decay of the magnetic energy of the flux tubes to ordinary and dark particles. The basic difference with respect to the inflationary scenario is that the energy of inflaton field is replaced with Kähler magnetic field and identified as dark energy.

Galaxies would be like pearls in a necklace. The dark matter and energy along the galactic necklaces causes a logarithmic 2-D gravitational potential producing constant velocity spectrum for distant stars. The basic prediction is that the galaxies can move freely along the flux tubes: this could explain the observed systematic motions in cosmic scale also challenging the basic assumptions of standard cosmology. Galaxies moving along different flux tubes can also collide if the flux tubes go near each other: this could be caused by their gravitational attraction already during the primordial period. One can imagine a cosmic highway network consisting of flux tubes intersecting at nodes and formed during the primordial period. Galaxies not obeying cosmic traffic rules could collide at crossings;-).

Since the necklace would have been much shorter during the primordial period, the proto galaxies possibly existing already at that time would have very near to each other and dynamically strongly coupled. Therefore the correlation of the directions of the angular momenta of proto galaxies - roughly in the direction of the long string like flux tube - could be a remnant from this time. This remnant manifesting itself as a definite handedness would be stabilized by the conservation of angular momentum after the decoupling of the galaxies from each other. The large value of Planck constant could also make possible quantum coherence in astrophysical scales for dark matter and energy and in this manner explain the correlations.

That there are two axes of this kind would suggest that our galaxy resides at junction of cosmic highways as a victim of cosmic traffic accident: that is in the node at which to cosmic necklaces touch. This is what I suggested in the earlier posting inspired by one particular finding challenging the assumption that galactic dark matter forms a spherical halo.

The finding was that near galactic center there is a distribution of satellite galaxies and star clusters, which rotate around Milky Way in a plane orthogonal to the plane of Milky Way. The observation could be interpreted by assuming that two orthogonal magnetic flux tubes (90 degrees is not far form 85 degrees) containing galaxies along them intersect at our galaxy. The newly found distribution of matter would correspond to a matter rotating around the flux tube - call it B - in the same way as the matter of our own galaxy rotates around the second flux tube - call it A. These flux tubes could correspond to the lines of sight found in the two surveys.