Could the lines of generalised Feynman diagrams correspond to quaternion-Kähler manifolds?

In blog comments Anonymous gave a link to an article about construction of 4-D quaternion-Kähler metrics with an isometry: they are determined by so called SU(∞) Toda equation. I tried to see whether quaternion-Kähler manifolds could be relevant for TGD.

From Wikipedia one can learn that QK is characterized by its holonomy, which is a subgroup of Sp(n)×Sp(1): Sp(n) acts as linear symplectic transformations of 2n-dimensional space (now real). In 4-D case tangent space contains 3-D sub-manifold identifiable as imaginary quaternions. CP₂ is one example of QK manifold for which the subgroup in question is SU(2)× U(1) and which has non-vanishing constant curvature: the components of Weyl tensor represent the quaternionic imaginary units. QKs are Einstein manifolds: Einstein tensor is proportional to metric.

What is really interesting from TGD point of view is that twistorial considerations show that one can assign to QK a special kind of twistor space (twistor space in the mildest sense requires only orientability). Wiki tells that if Ricci curvature is positive, this (6-D) twistor space is what is known as projective Fano manifold with a holomorphic contact structure. Fano variety has the nice property that as (complex) line bundle it has enough sections to define the imbedding of its base space to a projective variety. Fano variety is also complete: this is algebraic geometric analogy of topological property known as compactness.

QK manifolds and twistorial formulation of TGD

How the QKs could relate to the twistorial formulation of TGD?

1. In the twistor formulation of TGD the space-time surfaces are 4-D base spaces of 6-D twistor spaces in the Cartesian product of 6-D twistor spaces of M⁴ and CP₂ - the only twistor spaces with Kähler structure. In TGD framework space-time regions can have either Euclidian or Minkowskian signature of induced metric. The lines of generalized Feynman diagrams have Euclidian signature.
   
   
2. Could the twistor spaces associated with the lines of generalized Feynman diagrams be projective Fano manifolds? Could QK structure characterize Euclidian regions of preferred extremals of Kähler action. Could a generalization to Minkowskian regions exist. I have proposed that so called Hamilton-Jacobi structure characterizes preferred extremals in Minkowskian regions. It could be the natural Minkowskian counterpart for the quaternion Kähler structure, which involves only imaginary quaternions and could make sense also in Minkowski signature. Note that unit sphere of imaginary quaternions defines the sphere serving as fiber of the twistor bundle.
   
   
3. Why it would be natural to have QK that is corresponding twistor space which is projective contact Fano manifold?
   
   
   1. Fano property implies that the 4-D Euclidian space-time region representing line of the Feynman diagram can be imbedded as a sub-manifold to complex projective space CP_n. This would allow to use the powerful machinery of projective geometry in TGD framework. This could also be a space-time correlate for the fact that CP_ns emerge in twistor Grassmann approach expected to generalize to TGD framework.
      
      
   2. CP₂ allows both projective (trivially) and contact (even symplectic) structures. δ M⁴₊ × CP₂ allows contact structure - I call it loosely symplectic structure. Also 3-D light-like orbits of partonic 2-surfaces allow contact structure. Therefore holomorphic contact structure for the twistor space is natural.
      
      
   3. Both the holomorphic contact structure and projectivity of CP₂ would be inherited if QK property is true. Contact structures at orbits of partonic 2-surfaces would extend to holomorphic contact structures in the Euclidian regions of space-time surface representing lines of generalized Feynman diagrams. Projectivity of Fano space would be also inherited from CP₂ or its twistor space SU(3)/U(1)× U(1) (flag manifold identifiable as the space of choices for quantization axes of color isospin and hypercharge).
      
      
4. Could the isometry (or possibly isometries) for QK be seen as a remnant of color symmetry or rotational symmetries of M⁴ factor of imbedding space? The only remnant of color symmetry at the level of imbedding space spinors is anomalous color hyper charge (color is like orbital angular momentum and associated with spinor harmonic in CP₂ center of mass degrees of freedom). Could the isometry correspond to anomalous hypercharge?
   
   

How to choose the quaternionic imaginary units for the space-time surface?

Parallellizability is a very special property of 3-manifolds allowing to choose quaternionic imaginary units: global choice of one of them gives rise to twistor structure.

1. The selection of time coordinate defines a slicing of space-time surface by 3-surfaces. GCI would suggest that a generic slicing gives rise to 3 quaternionic units at each point each 3-surface? The parallelizability of 3-manifolds - a unique property of 3-manifolds - means the possibility to select global coordinate frame as section of the frame bundle: one has 3 sections of tangent bundle whose inner products give rose to the components of the metric (now induced metric) guarantees this. The tri-bein or its dual defined by two-forms obtained by contracting tri-bein vectors with permutation tensor gives the quanternionic imaginary units. The construction depends on 3-metric only and could be carried out also in GRT context. Note however that topology change for 3-manifold might cause some non-trivialities. The metric 2-dimensionality at the light-like orbits of partonic 2-surfaces should not be a problem for a slicing by space-like 3-surfaces. The construction makes sense also for the regions of Minkowskian signature.
   
   
2. In zero energy ontology (ZEO)- a purely TGD based feature - there are very natural special slicings. The first one is by linear time-like Minkowski coordinate defined by the direction of the line connecting the tips of the causal diamond (CD). Second one is defined by the light-cone proper time associated with either light-cone in the intersection of future and past directed light-cones defining CD. Neither slicing is global as it is easy to see.
   
   

The relationship to quaternionicity conjecture and M⁸-H duality

One of the basic conjectures of TGD is that preferred extremals consist of quaternionic/ co-quaternionic (associative/co-associative) regions (see this). Second closely related conjecture is M⁸-H duality allowing to map quaternionic/co-quaternionic surfaces of M⁸ to those of M⁴× CP₂. Are these conjectures consistent with QK in Euclidian regions and Hamilton-Jacobi property in Minkowskian regions? Consider first the definition of quaternionic and co-quaternionic space-time regions.

1. Quaternionic/associative space-time region (with Minkowskian signature) is defined in terms of induced octonion structure obtained by projecting octonion units defined by vielbein of H= M⁴× CP₂ to space-time surface and demanding that the 4 projections generate quaternionic sub-algebra at each point of space-time.
   

   If there is also unique complex sub-algebra associated with each point of space-time, one obtains one can assign to the tangent space-of space-time surface a point of CP₂. This allows to realize M⁸-H duality (see this) as the number theoretic analog of spontaneous compactification (but involving no compactification) by assigning to a point of M⁴=M⁴× CP₂ a point of M⁴× CP₂. If the image surface is also quaternionic, this assignment makes sense also for space-time surfaces in H so that M⁸-H duality generalizes to H-H duality allowing to assign to given preferred extremal a hierarchy of extremals by iterating this assignment. One obtains a category with morphisms identifiable as these duality maps.
   
   

2. Co-quaternionic/co-associative structure is conjectured for space-time regions of Euclidian signature and 4-D CP₂ projection. In this case normal space of space-time surface is quaternionic/associative. A multiplication of the basis by preferred unit of basis gives rise to a quaternionic tangent space basis so that one can speak of quaternionic structure also in this case.
   
   
3. Quaternionicity in this sense requires unique identification of a preferred time coordinate as imbedding space coordinate and corresponding slicing by 3-surfaces and is possible only in TGD context. The preferred time direction would correspond to real quaternionic unit. Preferred time coordinate implies that quaternionic structure in TGD sense is more specific than the QK structure in Euclidian regions.
   
   
4. The basis of induced octonionic imaginary unit allows to identify quaternionic imaginary units linearly related to the corresponding units defined by tri-bein vectors. Note that the multiplication of octonionic units is replaced with multiplication of antisymetric tensors representing them when one assigns to the quaternionic structure potential QK structure. Quaternionic structure does not require Kähler structure and makes sense for both signatures of the induced metric. Hence a consistency with QK and its possible analog in Minkowskian regions is possible.
   
   
5. The selection of the preferred imaginary quaternion unit is necessary for M⁸-H correspondence. This selection would also define the twistor structure. For quaternion-Kähler manifold this unit would be covariantly constant and define Kähler form - maybe as the induced Kähler form.
   
   
6. Also in Minkowskian regions twistor structure requires a selection of a preferred imaginary quaternion unit. Could the induced Kähler form define the preferred imaginary unit also now? Is the Hamilton-Jacobi structure consistent with this?
   

   Hamilton-Jacobi structure involves a selection of 2-D complex plane at each point of space-time surface. Could induced Kähler magnetic form for each 3-slice define this plane? It is not necessary to require that 3-D Kähler form is covariantly constant for Minkowskian regions. Indeed, massless extremals representing analogs of photons are characterized by local polarization and momentum direction and carry time-dependent Kähler-electric and -magnetic fields. One can however ask whether monopole flux tubes carry covariantly constant Kähler magnetic field: they are indeed deformations of what I call cosmic strings (see this) for which this condition holds true?
   

Could quaternion analyticity make sense for the preferred extremals?

The 4-D generalization of conformal invariance suggests strongly that the notion of analytic function generalizes somehow. The obvious ideas coming in mind are appropriately defined quaternionic and octonion analyticity. I have used a considerable amount of time to consider these possibilities but had to give up the idea about octonion analyticity could somehow allow to preferred extemals.

One can argue that quaternion analyticity is the more natural option in the sense that the local octonionic imbedding space coordinate (or at least M⁸ or E⁸ coordinate, which is enough if M⁸-H duality holds true) would for preferred extremals be expressible in the form

o(q)= u(q) + v(q)× I .

Here q is quaternion serving as a coordinate of a quaternionic sub-space of octonions, and I is octonion unit belonging to the complement of the quaternionic sub-space, and multiplies v(q) from right so that quaternions and qiaternionic differential operators acting from left do not notice these coefficients at all. A stronger condition would be that the coefficients are real. u(q) and v(q) would be quaternionic Taylor- of even Laurent series with coefficients multiplying powers of q from right for the same reason.

I ended up to this idea after finding two very interesting articles discussing the generalization of Cauchy-Riemann equations. The first article was about so called triholomorphic maps between 4-D almost quaternionic manifolds. The article gave as a reference an article about quaternionic analogs of Cauchy-Riemann conditions discussed by Fueter long ago (somehow I managed to miss Fueter's work), and also a new linear variant of these conditions, which seems especially interesting from TGD point of view as will be found.

The so called Cauhy-Riemann-Fueter conditions generalize Cauchy-Riemann conditions. These conditions are however not unique.

1. The translationally invariant form of CRF conditions is ∂_q*
   f=0 or explicitly
   

   (∂_o+ ∂_x I+∂_yJ y+∂_zK)f=0 .
   

   This form does not allow quaternionic Taylor series although it allows functions depending on complex coordinate z of some complex-plane only.
   

   Note that the Taylor coefficients multiplying powers of the coordinate from right are arbitrary quaternions. What looks pathological is that even linear functions of q fail be solve this condition. What is however interesting that in flat space the equation is equivalent with Dirac equation for a pair of Majorana spinors.
   

2. Second form of CRF conditions is
   

   (∂₀+ (xI+yJ+zK)/r²) r∂_r)f=0 .
   

   Here r denotes the imaginary part of q. This form allows both the desired quaternionic Taylor series and ordinary holomorphic functions of complex variable in one of the 3 complex coordinate planes as general solutions.
   

   This form of CRF is neither Lorentz invariant nor translationally invariant but remains
   invariant under simultaneous scalings of t and r and under time translations. Under rotations of either coordinates or of imaginary units the spatial part transforms like vector so that quaternionic automorphism group SO(3) serves as a moduli space for these operators.
   
   

3. The interpretation of the latter solutions inspired by ZEO would be that in Minkowskian regions r corresponds to the light-like radial coordinate of the either boundary of CD, which is part of δ M⁴_+/-. The radial scaling operator is that assigned with the light-like radial coordinate of the light-cone boundary. A slicing of CD by surfaces parallel to the δ M⁴_+/- is assumed and implies that the line r=0 connecting the tips of CD is in a special role. The line connecting the tips of CDa defines coordinate line of time coordinate. The breaking of rotational invariance corresponds to the selection of a preferred quaternion unit defining the twistor structure and preferred complex sub-space.
   

   In regions of Euclidian signature r could corresponds to the the radial Eguchi-Hanson coordinates and r=0 corresponds to a fixed point of U(2) subgroup under which CP₂ complex coordinates transform linearly.
   
   

4. In TGD framework one must have also ordinary holomorphic functions mapping 4-D quaterionic space to 2-D complex space associated with real and/or imaginary part of octonion: they would be associated with extremals for which M⁴ and/or CP₂ projection is 2-dimensional (cosmic strings and massless extremals).
   

   Since the conditions are linear one can also have superpositions of different types of solutions and they could describe perturbations of say cosmic strings and massless extremals.
   
   

5. It is of course possible that an entire moduli space of tri-holomorphic operators exists and would be interesting to know the most general form of tri-holomorphic operator. For instance, if one performs a quaternion analytic map of the space-time surface the form of the operator defining C-R-F conditions changes and becomes rather complex. This suggests that the operator as it is defined above indeed refers to geometrically preferred coordinates as already suggested. One must be however very cautious. It might well be that quaternion conformal transforms of this operator are possible but that they are equivalent.
   
   
6. Both generalizations of the C-R-F conditions generalize to the octonionic situation and right multiplication of powers of octonion by Taylor coefficients plus linearity imply that there are no problems with associativity. This inspires several questions.
   

   Could octonion analytic maps of imbedding space allow to construct new solutions from the existing ones? Could quaternion analytic maps applied at space-time level act as analogs of holomorphic maps and generalize conformal invariance to 4-D context?
   

To sum up, connections between different conjectures related to the preferred extremals - M⁸-H duality, Hamilton-Jacobi structure, induced twistor space structure, quaternion-Kähler property and its Minkowskian counterpart, and even quaternion analyticity, are clearly emerging. The underlying reason is strong form of GCI forced by the construction of WCW geometry and implying strong from of holography posing extremely powerful quantization conditions on the extremals of Kähler action in ZEO. Without the conformal gauge conditions the mutual inconsistency of these conjectures looks rather infeasible.

See the chapter Classical part of the twistor story or the article Classical part of the twistor story.

Does the flow of subjective time correspond to the increase of the average value of effective Planck constant?

I like answering questions. It gives a lot of meaning to the life of a theoretician who is not allowed to enjoy the pleasures of academic existence. Career builder would of course argue that writing again and again similar answers is a waste of time: I should be building social networks to important people instead. This activity however allows to make important observations and little discoveries. This time I answered to the questions relating to non-determinism of Kähler action. How this non-determinism relates to quantum non-determinism? How the non-determinism in elementary particle scales relates to that in biology?

The unexpected fruit was a little discovery: the mechanism generating the arrow of geometric time in zero energy ontology might rely in crucial manner to a sequence of phase transitions increasing the value of Planck constant h_eff/h=n and hence the size of the causal diamond (CD) characterized by quantum average temporal distance. Since the second boundary of CD is fixed, the second one moves to future in average sense: hence the flow of experienced time and its arrow. Conscious entities become more intelligent as they age! It became also clear that large h_eff/h characterizes macroscopically quantum coherent many-particle system rather than single particle. This leads to view in which intelligent consciousness involving the experienced about the flow of time emerges as the complexity of the systems measured by the number of fundamental particles increases.

1. The non-determinism of Kähler action and quantum non-determinism

The first question was about the relationship between non-determinism of preferred extremals and quantum non-determinism. As a matter of fact, I like to use the phrase "partial failure of determinism for Kähler action" rather than "non-determinism of Kähler action".

A possible interpretation could be as a correlate for quantum non-determinism. Second interpretation would be in terms of quantum criticality implying non-determinism. I do not know whether the interpretations are actually equivalent.

I certainly do not believe that one could get rid of quantum non-determinism and there is no need for it. The generalisation of quantum-classical correspondence is however natural in ZEO, where basic objects are 4-D surfaces- classical time evolutions serving as space-time correlates for quantal evolutions.

The origin of non-determinism is following. Kähler action has a huge vacuum degeneracy. For instance, for space-time surfaces, which are maps from M⁴ to at most 2-D Lagrangian manifold of CP₂ having by definition vanishing induced Kähler form (configuration space and momentum space are Lagrangian manifolds in the context of classical mechanics) induced Kähler form of course vanishes. These vacuum extremals define an analog of gauge degeneracy of Maxwell action for vacuum extremals. For non-vacuum externals it is expected to be lifted at least partially. Hence 4-dimensional spin glass degeneracy is more appropriate analogy. One could say that classical gravitation breaks the analog of gauge invariance for non-vacuum extremals.

For CP₂ type vacuum externals one has also non-determinism, which corresponds directly to Virasoro conditions expressing the light-likeness of 1-D M⁴ projection of the CP₂ type vacuum extremal. Now induced Kähler form does not vanish.

Zero energy ontology (ZEO) and causal diamond (CD) are essential notions concerning the interpretation but I will not try to explain it here but leave it as an exercise for the reader. The ends of vacuum extremal at light-like boundaries of CD are connected by infinite number of vacuum externals. One expects that some vacuum degeneracy is present also non-vacuum externals. Part of this degeneracy must be analogous to gauge degeneracy since by strong form of general coordinate invariance (GCI) implying strong form of holography, only the partonic 2-surfaces and their 4-D tangent space data fix the physics since WCW metric depends only on this data. Hence the interiors of 3-surfaces carry very little information about quantum states.

2. Identification of gauge degeneracy as hierarchy of broken conformal gauge invariances

The conjecture is that conformal symmetries acting as partially broken gauge symmetries realize this vision. TGD allows several kinds of conformal symmetries, and a huge generalisation of string model conformal symmetries (including Kac-Moody) but I will not go to this here. Suffice it to say that the generalization of conformal symmetries means replacement of AdS/CFT correspondence with a correspondence which looks intuitively much more realistic (see this).

Classical conformal charges would vanish for sub-algebra for which the conformal weights are multiples of some integer n, n=1,2,…. These conditions would give the long-sought-for precise content to the notion of preferred extremal. These conditions would be the classical counterparts of corresponding quantum conditions and define a Bohr orbitology. This hierarchy would correspond to the hierarchy of Planck constants h_eff= n× h and to the hierarchy of dark matters. There would be infinite number of hierarchies (1, n₁, n₂, . .., n_i,...) such that n_i would divide n_i+1 . They would correspond to the hierarchies of inclusions of hyper-finite factors of type II₁ (HFFs). Included algebra defines measurement resolution, which would thus realized as conformal gauge symmetries. Evolution would correspond to a sequence of symmetry breakings: this is not a new idea but emerges naturally if $n$ serves as a quantum "IQ".

The proposal is that that there is a finite number n=h_eff/h of conformal equivalence classes of four-surfaces with fixed 3-D ends at the opposite boundaries of CD so that the non-determinism with gauge fixing would be finite and would correspond to the hierarchy of Planck constants and hierarchy of conformal symmetry breaking defined by the hierarchy of sub-algebras of various conformal algebras with weights comings as integer multiples of integer n=1,2,,…. These n surfaces would be analogous to Gribov copies for gauge conditions in non-Abelian gauge theories.

3. The non-determinisms of particle physics and biology

There was also a question about the non-determinism of partcle physics contra that of biology, where it manifests itself as partially free will.

3.1. NMP

Before continuing it is good make clear that a new principle is involved: Negentropy Maximization Principle (NMP). Also a new kind of entanglement entropy based p-padic norm is involved. This entanglement entropy is negative unlike ordinary entanglement entropy and characterizes two-particle system rather than single particle system. By consistency with quantum measurement theory it corresponds to identical entanglement probabilities p_i=1/n. This entanglement is assumed to be associated with the n-sheeted coverings (at least these) defined by the space-time surfaces in n conformal equivalence classes associated with n=h_eff/h and connecting same 3-surfaces at the ends of space-time surface. Two systems of this kind can entangle negentropically. Unitary entanglement matrix associated with quantum computation gives rise to negentropic entanglement. Also n-partite negentropic entanglement makes sense.

3.2. What could be common for particle physics and biology?

Basically the non-determinism of particle physics and of biology could be essentially the same thing but for living matter whose behave is dictated by dark matter the value of h_eff/h=n would be large and make possible macroscopic quantum coherence in spatio-temporal scales, which are longer by factor n. Note that n could characterize macroscopic quantum phase rather than single particle system: this distinction is important as will be found.

The hierarchy of CDs brings additional spatio-temporal scale identified as secondary p-adic scale characterising the minimal size of CD (that for n=1). This size scales like h_eff/h=n and one can think of a superposition of CDs with different values of n and that the average value of n measuring the age of self increases during the sequence of quantum jumps. Since n is kind of IQ, NMP says that conscious entities should become wiser as they get older: maybe this is too optimistic hypothesis in the case of human kind but maybe electrons are different!;-) I swear that this interpretation is not due to the fact that I have passed the magic threshold of 60 years when one begins to feel that the ageing means growing wisdom;-). I must confess that the interpretation of experience time flow in terms of increasing h_eff/h charactering CD scaling has not come into my mind earlier. One could even consider the possibility that there is no superposition - just a sequence of h_eff/h increasing (in average sense) phase transitions, kind of spiritual growth even at the level of elementary particles.

For instance, for electron characterised by Mersenne prime M₁₂₇=2¹²⁷-1 the minimal CD time scale is .1 seconds (note that it defines a fundamental biorhythm of 10 Hz) and thus macrotemporal. Corresponding size scale is of the order of Earth circumference. This size scale could characterize quite generally the magnetic body of the elementary particle or the magnetic body at which macroscopic quantum phase of particles resides. In both cases there would be a direct connection between elementary particle physics and macroscopic physics becoming manifest in living matter via alpha rhythm for instance. Only the interpretation in terms of
macroscopic quantum phase seems to make sense.

3.3. What distinguishes between particle physics and biology?

There are essential differences between elementary particle physics and biology. The
first differences comes from quantum measurement theory in ZEO.

1. The repeated state function reduction does nothing for the state in standard ontology. In TGD the state is invariant only at the second boundary at which the reduction occurs. For second boundary of CD the average value if n increases. This gives rise to the experienced flow of geometric time and the arrow of time. Self exists as long as reductions take place on same boundary of CD and dies as the first reduction to opposite boundary is forced by NMP.
   
   
2. In particle physics context one expects that the duration of self identified as a sequence of state function reductions at the same boundary of CD is much shorter than in living matter. Otherwise one would have too strong breaking of reversibility in elementary particle time scales.
   

Objections usually help to make formulations more precise. Now the objection is that the increase of average h_eff/h so that particles darken gradually, should have been observed long time ago since reaction rates are independent of Planck constant only the lowest order in h_eff that is in classical approximation. The attempt to circumvent this objection leads to two crucial questions?

1. Does h_eff characterize elementary particle (or fundamental fermion) or a magnetic/field body of physical system which could be also many-particle system.
   

   If h_eff/h=n corresponds to n-sheeted covering which becomes singular at the ends of space-time surface so that sheets co-incide at partonic 2-surfaces representing particles, it seems that large h_eff is a phenomenon assignable to the field/magnetic body inside CD rather than particle identified as partonic 2-surface or 3-surface at the end of CD. If so large h_eff effects would relate to the dynamics associated with the magnetic/field bodies carrying dark matter.
   
   

2. Is darkness single particle phenomenon or many-particle phenomenon? For the latter option elementary particle physics would not be any challenge so that it looks the reasonable option. Note that negentropic entanglement requires at least one pair of (say) electrons and suggests macroscopic quantum phase - say high-T_c super-conductivity or super-fluidity.
   

   The idea about evolution of many-electron systems at dark magnetic body generating increasing value of h_eff makes sense, and would conform with the observation that electrons secondary p-adic time scale defines fundamental bio-rhythm. Dark magnetic bodies carrying dark particles are indeed in key role TGD inspired quantum biology. Bose-Einstein condensates and spontaneously magnetized dark phases at magnetic bodies would conform with the idea that dark matter is many-particle phenomenon.
   

   Large h_eff would not be seen in elementary particle physics. This challenges the idea that sparticles in TGD SUSY might have same p-adic mass scale as particles but be more stable in dark phase (this would be due to the scaling up of the size of CD) (see this). Note however that in TGD already elementary particles are many-fermion systems
   so that it might be possible to circumvent this objection.
   
   

3. The original formulation for darkness was at single particle level so that h_eff characterizes elementary particles rather than many-particle systems. In elementary particle reactions the particles in the same vertex would always have the same value of h_eff/h. It was assumed that h_eff can change only in 2-vertex analogous to mass insertion vertex.
   

   The previous arguments suggest that darkness makes sense only for many-particle systems so that mass insertion vertex becomes phase transition. These phase transitions would occur routinely in living matter but as phase transitions involving large number of particles. For instance, bio-photons would result from dark photons in this manner. This picture seems to make sense at least at the level of many-particle systems but not necessary for Feynman graphs.
   

   This many-particle aspect would explain at very general level why the search for dark particles has been
   fruitless.
   

The average lifetime of elementary particle as a conscious entity cannot be longer than the life-time of particle in the sense of particle physics. In the case of electron having infinite lifetime as elementary particle the "biological" lifetime must be finite since otherwise the irreversibility would manifest itself as a breaking of time reversal invariance in electron scale. The temporal time scale of CD characterising the dimensions of the magnetic body of elementary particle is the first order of magnitude estimate for the lifetime of elementary particle self. The "biological death" of electron only means state function reduction in the sense of ordinary quantum measurement theory implying for instance localization of electron or giving eigenstate of spin in given quantization direction and these quantum jumps meaning that re-incarnations of electron certainly occur.

This time scale could give an idea about the geometric duration of elementary particle self (the growth of the temporal distance between tips of CD during the sequence of reductions or equivalently the increase of n). If this picture really makes sense, elementary particles would get more and more intelligent in TGD Universe and stable elementary particle like electron would be real sages! Could this relate to the fact that the minimal CD size for electron defines the fundamental biorhythm of 10 Hz? Strangely, I find is easier to regarded electron as intelligent creature than my working desk or a typical academic decision maker. For holographists it should be also relatively easy to think that electrons could serve as conscious holograms.

3.4. Could one regard elementary particle as a conscious entity?

The previous considerations support the view that it is macroscopic quantum phases of particles at magnetic flux tubes which can be seen as conscious and intelligent evolving entities experience the flow of time. In the case of single elementary particle previous arguments would suggest that only single state function reduction occurs at given boundary of CD so that the lifetime of elementary particle self would have zero duration! This in accordance with the absence of the arrow of time at elementary particle level. Strictly speaking this does not exclude consciousness but excludes intelligence and experience of time flow.

Could already systems with small particle number, be conscious entities and develop - not necessarily large - h_eff/h>1. Hadrons consist of quarks and I have considered the possibility that valence quarks and gluons at the color magnetic body are dark. Also nuclei as many-nucleon systems could be dark. In TGD even elementary particles consist of fundamental fermions so that one can ask whether elementary particles possess some elementary aspects of consciousness identified as the possibility of non-vanishing "biological" life-time. This kind of picture would conform with the idea about consciousness as something emerging as the complexity of the system increases.

The average lifetime of elementary particle as a conscious entity cannot be longer than the life-time of particle in the sense of particle physics. In the case of electron having infinite lifetime as elementary particle the "biological" lifetime must be finite since otherwise the irreversibility would manifest itself as a breaking of time reversal invariance in electron scale. The temporal time scale of CD characterising the dimensions of the magnetic body of the elementary particle is the first order of magnitude estimate for the lifetime of elementary particle self. The "biological death" of electron means state function reduction in the sense of ordinary quantum measurement theory implying for instance localization of electron or giving eigenstate of spin in given quantization direction and these quantum jumps meaning re-incarnations of electron certainly occur.

This time scale could give an idea about the geometric duration of elementary particle self (the growth of the temporal distance between tips of CD during the sequence of reductions or equivalently the increase of n). One expects that Δ n is by NMP rather small for single particle systems.

3.5. Could thermodynamical breaking of T symmetry relate to the CP/T breaking in particles physics?

Could the "thermodynamical" breaking of time reflection symmetry (T) correspond to the breaking of T as it is observed for elementary particles such as neutral kaon? I think that most colleagues tend to be skeptic about this kind of identification, and so do I.

The point is that particle physicist's T breaking could be purely geometric whereas thermodynamical breaking of T involves the notion of subjective time, state function reduction, and consciousness. One could however ask whether the particle physicist's T could serve as space-time correlate for thermodynamicist's T and whether systems exhibiting CP breaking could be seen as conscious entities in very primitive sense of the word (n_f/n_i>1 but small). An important point is that the time evolution for CDs corresponds to scaling so that usually exponential decay laws are replaced with their hyperbolic variants. Hyperbolic decay laws become an important signature of consciousness. For instance, bio-photon intensity decays in hyperbolic manner.

The mean lifetimes are of long-lived and short lived neutral kaon are τ_L= 1.2 × 10^-8 seconds and τ_S= 8.9× 10^-11 seconds: the ratio of the time scales is roughly 2⁷. This does not conform with the naivest guess that the size of CD gives estimate for the duration of elementary particle self (increase of the temporal distance between tips of CD): the estimate would be τ_L= 10^-7 seconds from the fact that the mass of neutral kaon is roughly 10³ times electron mass. This is not too far from the lifetime of K⁰_L but is about 2⁷ times longer than the life-time of short-lived kaon. Why K_S would be so short-lived? Could the lifetime be dictated by quark level: The longer time scale could be assigned as secondary p-adic time scale with the p-adic prime p≈ 2^k, k=104, characterising b quark. Could the short life-time be understood in terms of loops involving heavier quarks with shorter lifetimes as conscious entities: they indeed appear in the description of CP/T breaking?

For background see the chapter About nature of time. See also the article Does the flow of time correspond to the increase of the effective Planck constant?.

I am worried about Sheldon and Leonard

I am worried about Sheldon and Leonard, the theoretical physics students in The Big Bang Theory. It is not long time ago when Sheldon lost his faith on super string theory. Lubos told now that even more worrying things have happened now. Sheldon and Leonard got an idea and went to write a paper about space-time as a surface of superfluid.

Space-time as a surface: really crazy! Even worse: sounds very much like TGD! They even have the idea that surface tension could provide the negative "pressure" needed to explain the accelerated expansion. This brings to my mind another stupid idea generated by TGD. String-like magnetic monopole flux tubes and their magnetic tension explain microscopically the negative "pressure" and primordial magnetic fields. Inflationary period is replaced with a phase transition from a gas of cosmic strings to the phase in which one can speak about macroscopic space-time. Theoretical physics would lose all these nice new Higgs like inflaton fields, the intricacies of which have kept colleagues busy for so many years.

Cannot anyone help Sheldon and Leonard? If these fellows continue in this manner, they will soon be the first advocates of TGD! Horrible fate for young students.

When I chose the wrong track, my colleagues reacted immediately to turn me back to the right rail. The solution of the problem was very innovative and final: I was kicked out from Helsinki University, and after than I have been kept outside the University most of these 37 years.

I cannot blame my benevolent colleagues for being irrresponsible: they did their best to help me to adopt the correct scientific behaviors using the well-tested methods that Pavlov applied first to his dogs. It is totally my fault that I am still doing this crackpot science TGD. Yes, I got really scared - just as Pavlov's pets when they got electric shocks - but could not avoid continuing with TGD. Something was wrong with my brain circuitry.

But what to do with Sheldon and Leonard? If they are kicked out from The Big Bang Theory we will lose The Big Bang Theory?

Surface area as geometric representation of entanglement entropy?

In Thinking Allowed Original there was a link to a talk by James Sully and having the title Geometry of Compression. I must admit that I understood very little about the talk. My not so educated guess is however that information is compressed: UV or IR cutoff eliminating entanglement in short length scales and describing its presence in terms of density matrix - that is thermodynamically - is another manner to say it. The TGD inspired proposal for the interpretation of the inclusions of hyper-finite factors of type II₁ (HFFs) is in spirit with this.

The space-time counterpart for the compression would be in TGD framework discretization. Discretizations using rational points (or points in algebraic extensions of rationals) make sense also p-adically and thus satisfy number theoretic universality. Discretization would be defined in terms of intersection (rational or in algebraic extension of rationals) of real and p-adic surfaces. At the level of "world of classical worlds" the discretization would correspond to - say - surfaces defined in terms of
polynomials, whose coefficients are rational or in some algebraic extension of rationals. Pinary UV and IR cutoffs are involved too. The notion of p-adic manifold allows to interpret rthe p-adic variants of space-time surfaces as cognitive representations of real space-time surfaces.

Finite measurement resolution does not allow state function reduction reducing entanglement totally. In TGD framework also negentropic entanglement stable under Negentropy Maximixation Principle (NMP) is possible. For HFFs the projection into single ray of Hilbert space is indeed impossible: the reduction takes always to infinite-D sub-space.

The visit to the URL was however not in vain. There was a link to an article discussing the geometrization of entanglement entropy inspired by the AdS/CFT hypothesis.

Quantum classical correspondence is basic guiding principle of TGD and suggests that entanglement entropy should indeed have space-time correlate, which would be the analog of Hawking-Bekenstein entropy.

Generalization of AdS/CFT to TGD context

AdS/CFT generalizes to TGD context in non-trivial manner. There are two alternative interpretations, which both could make sense. These interpretations are not mutually exclusive. The first interpretation makes sense at the level of "world of classical worlds" (WCW) with symplectic algebra and extended conformal algebra associated with δ M⁴_+/- replacing ordinary conformal and Kac-Moody algebras. Second interpretation at the level of space-time surface with the extended conformal algebras of the light-likes orbits of partonic 2-surfaces replacing the conformal algebra of boundary of AdSⁿ.

1. First interpretation

For the first interpretation 2-D conformal invariance is generalised to 4-D conformal invariance relying crucially on the 4-dimensionality of space-time surfaces and Minkowski space.

1. One has an extension of the conformal invariance provided by the symplectic transformations of δ CD× CP₂ for which Lie algebra has the structure of conformal algebra with radial light-like coordinate of δ M⁴₊ replacing complex coordinate z.
   
   
2. One could see the counterpart of AdS_n as imbedding space H=M⁴ × CP₂ completely unique by twistorial considerations and from the condition that standard model symmetries are obtained and its causal diamonds defined as sub-sets CD×
   CP₂, where CD is an intersection of future and past directed light-cones. I will use the shorthand CD for CD× CP₂. Strings in AdS⁵× S⁵ are replaced with space-time surfaces inside 8-D CD.
   
   
3. For this interpretation 8-D CD replaces the 10-D space-time AdS⁵× S⁵. 7-D light-like boundaries of CD correspond to the boundary of say AdS⁵, which is 4-D Minkowski space so that zero energy ontology (ZEO) allows rather natural formulation of the generalization of AdS/CFT correspondence since the positive and negative energy parts of zero energy states are localized at the boundaries of CD.
   
   

2. Second interpretation

For the second interpretation relies on the observation that string world sheets as carriers of induced spinor fields emerge in TGD framework from the condition that electromagnetic charge is well-defined for the modes of induced spinor field.

1. One could see the 4-D space-time surfaces X⁴ as counterparts of AdS⁴. The boundary of AdS⁴ is replaced in this picture with 3-surfaces at the ends of space-time surface at opposite boundaries of CD and by strong form of holography the
   union of partonic 2-surfaces defining the intersections of the 3-D boundaries between Euclidian and Minkowskian regions of space-time surface with the boundaries of CD. Strong form of holography in TGD is very much like ordinary holography.
   
   
2. Note that one has a dimensional hierarchy: the ends of the boundaries of string world sheets at boundaries of CD as pointlike partices, boundaries as fermion number carrying lines, string world sheets, light-like orbits of partonic 2-surfaces, 4-surfaces, imbedding space M⁴× CP₂. Clearly the situation is more complex than for AdS/CFT correspondence.
   
   
3. One can restrict the consideration to 3-D sub-manifolds X³ at either boundary of causal diamond (CD): the ends of space-time surface. In fact, the position of the other boundary is not well-defined since one has superposition of CDs with only one boundary fixed to be piece of light-cone boundary. The delocalization of the other boundary is essential for the understanding of the arrow of time. The state function reductions at fixed boundary leave positive energy part (say) of the zero energy state at that boundary invariant (in positive energy ontology entire state would remain
   unchanged) but affect the states associated with opposite boundaries forming a superposition which also changes between reduction: this is analog for unitary time evolution. The average for the distance between tips of CDs in the superposition increases and gives rise to the flow of time.
   
   
4. One wants an expression for the entanglement entropy between X³ and its partner. Bekenstein area law allows to guess the general expression for the entanglement entropy: for the proposal discussed in the article the entropy would be the area of the boundary of X³ divided by gravitational constant: S= A/4G. In TGD framework gravitational constant might be replaced by the square of CP₂ radius apart from numerical constant. How gravitational constant emerges in TGD framework is not completely understood although one can deduce for it an estimate using dimensional analyses. In any case, gravitational constant is a parameter which characterizes GRT limit of TGD in which many-sheeted space-time is in long scales replaced with a piece of Minkowski space such that the classical gravitational fields and gauge potentials for
   sheets are summed. The physics behind this relies on the generalization of linear superposition of fields: the effects of different space-time sheets particle touching them sum up rather than fields.
   
   
5. The counterpart for the boundary of X³ appearing in the proposal for the geometrization of the entanglement entropy naturally corresponds to partonic 2-surface or a collection of them if strong form of holography holds true.
   
   

With what kind of systems 3-surfaces can entangle?

With what system X³ is entangled/can entangle? There are several options to consider and they could correspond to the two TGD variants for the AdS/CFT correspondence.

1. X³ could correspond to a wormhole contact with Euclidian signature of induced metric. The entanglement would be between it and the exterior region with Minkowskian signature of the induced metric.
   
   
2. X³ could correspond to single sheet of space-time surface connected by wormhole contacts to a larger space-time sheet defining its environment. More precisely, X³ and its complement would be obtained by throwing away the wormhole contacts with Euclidian signature of induce metric. Entanglement would be between these regions. In the generalization of the formula
   

   S= A/4hbar G
   

   area A would be replaced by the total area of partonic 2-surfaces and G perhaps with CP₂ length scale squared.
   
   

3. In ZEO the entanglement could also correspond to time-like entanglement between the 3-D ends of the space-time surface at opposite light-like boundaries of CD. M-matrix, which can be seen as the analog of thermal S-matrix, decomposes to a product of hermitian square root of density matrix and unitary S-matrix and this hermitian matrix could also define p-adic thermodynamics. Note that in ZEO quantum theory can be regarded as square root of thermodynamics.
   
   

Minimal surface property is not favored in TGD framework

Minimal surface property for the 3-surfaces X³ at the ends of space-time surface looks at first glance strange but a proper generalization of this condition makes sense if one assumes strong form of holography. Strong form of holography realizes General Coordinate Invariance (GCI) in strong sense meaning that light-like parton orbits and space-like 3-surfaces at the ends of space-time surfaces are equivalent physically. As a consequence, partonic 2-surfaces and their 4-D tangent space data must code for the quantum dynamics.

The mathematical realization is in terms of conformal symmetries accompanying the symplectic symmetries of δ M⁴_+/-× CP₂ and conformal transformations of the light-like partonic orbit. The generalizations of ordinary conformal algebras correspond to conformal algebra, Kac-Moody algebra at the light-like parton orbits and to symplectic transformations δ M⁴× CP₂ acting as isometries of WCW and having conformal structure with respect to the light-like radial coordinate plus conformal transformations of δ M⁴_+/-, which is metrically 2-dimensional and allows extended conformal symmetries.

1. If the conformal realization of the strong form of holography works, conformal transformations act at quantum level as gauge symmetries in the sense that generators with no-vanishing conformal weight are zero or generate zero norm states. Conformal degeneracy can be eliminated by fixing the gauge somehow. Classical conformal gauge conditions analogous to Virasoro and Kac-Moody conditions satisfied by the 3-surfaces at the ends of CD are natural in this respect. Similar conditions would hold true for the light-like partonic orbits at which the signature of the induced metric changes.
   
   
2. What is also completely new is the hierarchy of conformal symmetry breakings associated with the hierarchy of Planck constants h_eff/h=n. The deformations of the 3-surfaces which correspond to non-vanishing conformal weight in algebra or any sub-algebra with conformal weights vanishing modulo n give rise to vanishing classical charges and thus do not affect the value of the Kähler action.
   

   The inclusion hierarchies of conformal sub-algebras are assumed to correspond to those for hyper-finite factors. There is obviously a precise analogy with quantal conformal invariance conditions for Virasoro algebra and Kac-Moody algebra. There is also hierarchy of inclusions which corresponds to hierarchy of measurement resolutions. An attractive interpretation is that singular conformal transformations relate to each other the states for broken conformal symmetry. Infinitesimal transformations for symmetry broken phase would carry fractional conformal weights coming as multiples of 1/n.
   
   

3. Conformal gauge conditions need not reduce to minimal surface conditions holding true for all variations.
   
   
4. Note that Kähler action reduces to Chern-Simons term at the ends of CD if weak form of electric magnetic duality holds true. The conformal charges at the ends of CD cannot however reduce to Chern-Simons charges by this condition since only the charges associated with CP₂ degrees of freedom would be non-trivial.
   
   

Technicalities

The generalisation of the conjecture about surface area proportionality of entropy to TGD context looks rather straightforward but is physically highly non-trivial. There are however some technicalities involved.

1. In TGD framework it is not quite clear whether
   
   
   1. G still appears in the formula or
      
      
   2. whether G should be replaced with the square R² of CP₂ radius to give
      

      S= A/4π R²
      

      apart from numerical constant.
      
      

   For option a) one must include Planck constant explicitly to the formula to give S= A/4h_effG: the entropy would decrease as h_eff=n× h increases. The condition h_eff=h_gr= GM²/v₀ would give S= v₀/c<1. The entropy using b) would be by a factor of order 10^-5 smaller and would not depend on the value of h_eff at all. It will be found that p-adic mass calculations lead to entropy allowing to circumvent these problems.
2. There is also the question about the identification of the area A. For blackhole A would be determined by Schwartschild radius r_S= 2GM depending on mass only. In TGD framework one has several candidates.
   
   
   1. The area of partonic 2-surface is an obvious first guess. One cannot however expect that the area of partonic 2-surface is constant. Could conformal gauge fixing fixes the 3-surfaces highly uniquely. Ordinary conformal invariance for partonic 2-surface does not however seem to be consistent with the fixing of the area of partonic 2-surface since conformal transformations do not preserve area.
      
      
   2. Could the area of partonic 2-surface be replaced with the area of the boundary of space-time sheet at which particle is topologically condensed and has size scale of order Compton length? This option looks the most feasible one on basis of p-adic mass calculations as will be found.
      
      

p-Adic variant of Bekenstein-Hawking law

When the 3-surface corresponds to elementary particle, a direct connection with p-adic thermodynamics suggests itself and allows to answer the questions above. p-Adic thermodynamics could be interpreted as a description of the entanglement with environment. In ZEO the entanglement could also correspond to time-like entanglement between the 3-D ends of the space-time surface at opposite light-like boundaries of CD. M-matrix, which can be seen as the analog of thermal S-matrix, decomposes to a product of hermitian square root of density matrix and unitary S-matrix and this hermitian matrix could also define p-adic thermodynamics.

1. p-Adic thermodynamics would not be for energy but for mass squared (or scaling generator L₀) would describe the entanglement of the particle with environment defined by the larger space-time sheet. Conformal weights would comes as positive powers of integers (p^L₀ would replace exp(-H/T) to guarantee the number theoretical existence and convergence of the Boltzmann weight: note that conformal invariance that is integer spectrum of L₀ is also essential).
   
   
2. The interactions with environment would excite very massive CP₂ mass scale excitations (mass scale is about 10^-4 times Planck mass) of the particle and give it thermal mass squared identifiable as the observed mass squared. The Boltzmann weights would be extremely small having p-adic norm about 1/pⁿ, p the p-adic prime: M₁₂₇=2^{127-1 for electron.}
   
   
3. I have proposed earlier p-adic entropy as a p-adic counterpart of Bekenstein-Hawking entropy. S= (R²/hbar²)× M² holds true identically apart from numerical constant. Note that one could interpret R²M/hbar as the counterpart of Schwartschild radius. Note that this radius is proportional to 1/p^1/2 so that the area A would correspond to the area defined by Compton length. This is in accordance with the third option.
   
   

What is the space-time correlate for negentropic entanglement?

The new element brought in by TGD framework is that number theoretic entanglement entropy is negative for negentropic entanglement assignable to unitary entanglement and NMP states that this negentropy increases. Since entropy is essentially number of energy degenerate states, a good guess is that the number n=h_eff/h of space-time sheets associated with h_eff defines the negentropy. An attractive space-time correlate for the negentropic entanglement is braiding. Braiding defines unitary S-matrix between the states at the ends of braid and this entanglement is negentropic. This entanglement gives also rise to topological quantum computation.

See the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article Surface area as geometric representation of entanglement entropy?.

Planck 2013 estimates for string tension of various strings

Planck 2013 gives bounds on the string tension of cosmic strings too. The bounds depend on the type of string considered: sone can consider Nambu-Goto strings, cosmic strings of gauge theories, string like objects of field theories, etc… The upper bounds for TG are in the range 10^-6-10^-7 .

One cannot of course directly compare these bounds to cosmic strings in TGD sense (not gauge theory strings but primordial 4-D string like objects). In TGD framework the string tension characterizes the density of Kähler magnetic energy of 4-D string like object with 2-D string world sheet as Minkowski space projection.

Cosmic string tension is inversely proportional to the square of CP₂ length scale R and to Kähler coupling strength α_K for which the most recent estimate is as equal to fine structure constant: α_K≈ 1/137. The value of R is fixed by p-adic mass calculations from the conditions that electron mass comes out correctly. The velocity spectrum of distance stars in galaxy gives the same estimate if the gravitational field created by long cosmic string along which galaxies are located like pearls in string, gives an estimate consistent with this value. The estimate of cosmic string tension is TG= 6.9× 10^-7 and is therefore in the interval 10^-6-10^-7 , where the upper bounds for other string tensions reside.

A comparison with string theory is in order. For Nambu-Goto strings the estimated upper bound for string tension is GT<1.5× 10^-7 - not a good news since the Nambu-Goto string tension should satisfy GT=1 in the original approach. The same holds true also for superstrings in the original sense of the word. Therefore the situation is not very promising for superstrings. In fact, it turned out very difficult to find anything concrete about the string tension of superstrings. I however found from web a ten year old estimate estimate TG= 1/3000 for superstring tension involving experimental input. Presumably the Planck 2013 results would lower this estimate by few orders of magnitude.

For background see the chapter Cosmic Strings of "Physics in Many-Sheeted Space-time".

More detailed view about scattering amplitudes

The following represents an update view about construction of scattering amplitudes at the level of "world of classical worlds" (WCW).

Basic principles

In order to facilitate the challenge of the reader I summarize basic ideas behind the construction of scattering amplitudes.

1. Zero energy ontology

In Zero Energy Ontology (ZEO) quantum theory as hermitian square root of thermodynamics, which leads to a generalization of the notion of S-matrix to a unitary U-matrix between zero energy states having as rows M-matrices which are products of hermitian square roots of density matrices with a common unitary matrix S for given CD. Number theoretical considerations suggests that CD size comes as integer multiples of CP₂ size so that one obtain a hierarchy U-matrices having interpretation in terms of length scale evolution. For given CD also sub-CDs contribute down to some minimal scale defining UV scale. The largest CD defines the IR cutoff.

Scattering amplitudes would characterize the modes of WCW spinor field as time-like entanglement coefficients between positive and negative energy states associated with zero energy states.

The construction of scattering amplitudes - or M-matrix elements in ZEO - reduces at basic level to the construction of the Feynman diagram like entities for fundamental fermions, which serve basic building bricks of elementary particles.

2. Construction of scattering amplitudes as functional integrals in WCW

The decomposition of space-time surface to Minkowskian and Eucldian regions is the basic distinction from ordinary quantum field theories since it replaces path integral with mathematically well-defined functional integral over WCW.

1. Space-time surface decomposes to regions with Minkowskian or Euclidian signature of the induced metric. The regions with Euclidian metric are identified as lines of generalized Feynman diagrams. The boundaries between two kinds of regions - to be called parton orbits - can be regarded as carriers of elementary particle quantum numbers such as fermion number assignable to the boundaries of string world sheets at them. Induced spinor fields are localized at them from the well-definedness of electromagnetic charge requiring that induced W boson fields vanish. Hence strings emerge from TGD. Note that at boundary between Euclidian and Minkowskian regions the metric determinant vanishes.
   
   
2. Weak form of electric magnetic duality together with the assumption that the term j^αA_α in Kähler action vanishes imply that Kähler action reduces to 3-D Chern-Simons term. This hypothesis is inspired by TGD as almost topological quantum field theory conjecture. In Minkowskian regions this conjecture is very natural. In the Euclidian region the contribution to Kähler action need not reduce to a mere Chern-Simons term associated with its boundary. This would be due to the non-triviality of the U(1) bundle defined by Kähler form giving also Chern-Simons terms inside the CP₂ type vacuum extremal.
   
   
3. Scattering amplitude is a functional integral over space-time surfaces: the data about these space-time surfaces are coded by their ends about the opposite light-like boundaries of causal diamond (CD) of given scale. The weight function in the functional integral is exponential of Kähler function of "world of classical worlds" coming from Euclidian regions of the space-time surface representing lines of generalized Feynman diagram and being deformation of CP₂ type vacuum extremals representing wormhole contacts connecting two space-time sheets with Minkowskian signature of induced metric. Kähler function is the exponent of Kähler action from Euclidian regions. The real exponent takes care that the functional integral is obtained instead of path integral so that the outcome is mathematically well-defined.
   
   
4. Euclidian region would give only the analog of thermodynamics but there is also an imaginary exponential coming from the exponential of the imaginary Kähler action from Minkowskian regions. Space-time surfaces are extremals of Kähler action and for very general ansatz Minkowskian contribution to Kähler action reduces to imaginary Chern-Simons term at the light-like 3-D boundary between regions at which the 4-D metric is degenerate. This term makes possible interference of different contributions to the functional integral which is absolutely essential in quantum field theory.
   
   

3. Why it might work?

There are many reasons encouraging the hopes about calculable theory.

1. The theory has huge super-conformal symmetries dramatically reducing the dynamical degrees of freedom by the choice of conformal gauge. This implies that both the space-like 3-surfaces at the ends of space-time surface and partonic orbits satisfy classical Super conformal conditions for generalizations of ordinary super-conformal algebras perhaps extending to multilocal Yangian with locus identified as single partonic 2-surface at the light-like boundary of CD.
   

   Yangian symmetry in turn gives excellent hopes about twistorialization: in fact, M⁴× CP₂ is completely unique choice for the imbedding space by twistorial considerations and the product of the twistor spaces of M⁴ and CP₂ allows to construct the twistor spaces of space-time surfaces as liftings of the extremals of Kähler action to 6-D sphere bundles over space-time surface.
   
   

2. The integrand in the functional integral represents the analog of ordinary Feynman diagrams involving only fermions and 1-D lines. Indeed, by bosonic emergence all bosons (in fact all elementary particles) can be regarded as composites of fundamental fermions. The only fermionic vertices are 2-fermion vertices since 3-vertices correspond to space-time surfaces meeting along common 3-surface and are thus purely topological. This is of course excellent news from the point of view of finiteness. The fermionic vertices are represented by the discontinuity of the modified Dirac operator associated with the string boundary line at partonic 2-surface so that there are no coupling constants involved. The only fundamental coupling parameter is Kähler strength whose value is dictated by quantum criticality as the analog of critical temperature.
   
   

One must have a view about what elementary particles - as opposed to fundamental fermions - are, how the ordinary view about scattering based on exchanges of elementary particles emerges from this picture and how say BFF vertex reduces to a diagram at for fundamental fermions involving only 2-fermion vertices.

Elementary particles in TGD framework

The notion of elementary particle involves two aspects: elementary particles as space-time surfaces and elementary particles as many-fermion states with fundamental fermions localized at the wormhole throats and defining elementary particles as their bound states (including physical fermions).

1. Elementary particles as space-time surfaces

Let us first summarize what kind of picture ZEO suggests about elementary particles.

1. Kähler magnetically charged wormhole throats are the basic building bricks of elementary particles. The lines of generalized Feynman diagrams are identified as the Euclidian regions of space-time surface. The weak form of electric magnetic duality forces magnetic monopoles and gives classical quantization of the Kähler electric charge. Wormhole throat is a carrier of many-fermion state with parallel momenta and the fermionic oscillator algebra gives rise to a badly broken large N SUSY.
   
   
2. The first guess would be that elementary fermions correspond to wormhole throats with unit fermion number and bosons to wormhole contacts carrying fermion and anti-fermion at opposite throats. The magnetic charges of wormhole throats do not however allow this option. The reason is that the field lines of Kähler magnetic monopole field must close. Both in the case of fermions and bosons one must have a pair of wormhole contacts (see figure ) connected by flux tubes. The most general option is that net quantum numbers are distributed amongst the four wormhole throats. A simpler option is that quantum numbers are carried by the second wormhole: fermion quantum numbers would be carried by its second throat and bosonic quantum numbers by fermion and anti-fermion at the opposite throats. All elementary particles would therefore be accompanied by parallel flux tubes and string world sheets.
   
   
3. A cautious proposal in its original form was that the throats of the other wormhole contact could carry weak isospin represented in terms of neutrinos and neutralizing the weak isospin of the fermion at second end. This would imply weak neutrality and weak confinement above length scales longer than the length of the flux tube. This condition might be un-necessarily strong.
   

   The realization of the weak neutrality using pair of left handed neutrino and right handed antineutrino or a conjugate of this state is possible if one allows right-handed neutrino to have also unphysical helicity. The weak screening of a fermion at wormhole throat is possible if ν_R is a constant spinor since in this case Dirac equation trivializes and allows both helicities as solutions. The new element from the solution of the modified Dirac equation is that ν_R would be interior mode de-localized either to the other wormhole contact or to the Minkowskian flux tube. The state at the other end of the flux tube is sparticle of left-handed neutrino.
   

   It must be emphasized that weak confinement is just a proposal and looks somewhat complex: Nature is perhaps not so complex at the basic level. To understand this better, one can think about how M₈₉ mesons having quark and antiquark at the ends of long flux tube returning back along second space-time sheet could decay to ordinary quark and antiquark.

Localization of the induced spinor fields at string world sheets and fermionic propagators

The localization of induced spinors to string world sheets emerges from the condition that electromagnetic charge is well-defined for the modes of induced spinor fields. There is however an exception: covariantly constant right handed neutrino spinor ν_R: it can be de-localized along entire space-time surface. Right handed neutrino has no couplings to electroweak fields. It couples however to left handed neutrino by induced gamma matrices except when it is covariantly constant. Note that standard model does not predict ν_R but its existence is necessary if neutrinos develop Dirac mass. ν_R is indeed something which must be considered carefully in any generalization of standard model.

It has turned out that covariantly constant right-handed neutrino very probably corresponds to a pure gauge degree of freedom. Non-covariantly constant right-handed neutrino however mixes with left handed neutrino since the modified gamma matrices involve both M⁴ and CP₂ gamma matrices and latter mix M⁴ chiralities. These right-handed neutrinos localized to partonic 2-surfaces would generate broken SUSY. There are however good reasons to expect that the mass scale for SUSY breaking corresponds to that for the mixing of right and left handed neutrinos inducing neutrino massivation and that the p-adic mass scale of particle and sparticle are same. The only manner to avoid conflict with experimental facts is that sparticles are dark in TGD sense that is having Planck constant h_eff=n× h. This would conform with the idea that the hierarchy of Planck constants corresponds to a hierarchy of breakings of conformal invariance, which is indeed behind the massivation.

The localization has powerful consequences since it gives in fermionic degrees of freedom what looks like ordinary Feynman diagrams but with only 2-fermion vertex. Space-time topology describes the vertices, say BFF vertex.

Fermion lines correspond to boundaries of string world sheets at the parton orbits. At these lines one must pose a boundary condition and the boundary condition is that the action of the modified Dirac operator associated with the boundary equals to the action of massless Dirac operator in momentum space representation. This reduces the fermionic propagator to massless Dirac propagator and simplify the construction decisively. If residue integral applied in twistor approach makes sense for the fermionic virtual momenta, this in turn gives inverse of Dirac propagator contracted between massless spinors with non-physical helicities.

What is the correct choice for the modified Dirac operator?

1. Chern-Simons-Dirac operator modified gamma matrix consists of CP₂ gamma matrices only and has square which does not vanish identically. Hence the covariant derivative of the induced spinor field along the string boundary must vanish if one wants that that fermion four-momentum is light-like and it must annihilate the spinors at its ends. The helicity of fermion would be physical and one would have incoming or outgoing on mass-shell fermion. This is certainly highly undesirable.
   

   Covariant constancy of the spinor mode has however the nice implication that it gives the familiar non-integrable phase factor for the dependence of the induced spinor at the fermion line behaving like Wilson line. Wilson lines are known to lead in string model picture to twistor diagrams so that it seems that we are in correct track.
   

   This forces to ask whether fermion propagator are needed at all in the construction of fermionic Feynman diagrams. Dimensional arguments force them - at least if the scattering is just fermion scattering. The exchanged particles consist however of wormhole contacts: it is wormhole contact with propagates. Boson exchange corresponds to the exchange of wormhole contact with fermion and antifermion at throats. The four-momenta of fermion and anti-fermion making the virtual boson are tightly correlated reducing the pair of fermionic propagators to bosonic propagator and eliminating one virtual momentum integration. This conforms with the fact that in twistor approach the integration over bosonic virtual momenta over boson cancels the bosonic propagator.
   

2. What about Kähler-Dirac operator, which is indeed the most natural first guess. For Kähler-Dirac operator one could have light-like K-D gamma matrix at the string boundary and formally one can have modes, which are not covariantly constant. Unfortunately, the definition of the projection of Kähler-Dirac operator to the line is problematic since the determinant of the induced metric vanishes at partonic orbit and the component of canonical momentum density along the string boundary can diverge.
   

   One could of course consider the possibility of defining the condition as a limit. The infinitesimal value of the covariant derivative would compensate for the divergence of Kähler-Dirac gamma matrix Γ^t at the limit and one would obtain light-like momenta and non-physical fermion helicities of TGD based variant of twistor approach. For Kähler-Dirac term Γ^t can be infinite at the limit. If g_ti=0 holds true then simple matrix algebra shows that g^tt becomes infinite whereas g^ti are limiting values of form 0/0. If J_ti=0 holds true for Kähler form, manifest infinity transforms to 0/0 type limit and one has hope of finite limiting value. Weak form of electric magnetic duality could in factor guarantee J_ti=0 by forcing J_tn and J_xy to be the only non-vanishing components of induced Kähler form.
   

   Remarkably, also now one would obtain the non-integrable phase factor characterizing Wilson line. The graph would depend on space-time surface only through this phase factor appearing at 2-fermion vertices and by the discontinuity of the modified Dirac operator at partonic 2-surface defining the most natural candidate for the 2-fermion vertex.
   
   

3. If the modified Dirac operator is defined by the induced metric at the string boundary and if its is light-like. This is trivial manner to solve the problem but now one loses the non-integrable phase factor and Wilson line picture.
   
   

Vertices

Vertices can be considered at both space-time level and fermionic level.

1. At space-time level vertices correspond to the fusion of space-time surfaces representing particles along common 3-surface defining the vertex. At the parton level 3-light-like parton orbits fuse together along partonic 2-surface. In these vertices particle number changes this change correspond the change of particle number for elementary particles.
   
   
2. At fermion level vertices are localized at the partonic 2-surfaces and vertex corresponds to the discontinuity of the Kähler Dirac operator at the corner of the line representing the boundary of string world sheet. The creation of fermion pair from vacuum corresponds to an corner of string boundary at which the boundaries of string world sheets associated with two outgoing or incoming sheets meet. The creation/annihilation of a fermion pair is essential for the realization of say tree diagrams describing fermion scattering by virtual boson exchange.
   
   

The discontinuity of the modified Dirac operator is the key quantity. If one assumes continuity of the time derivative the discontinuity reduces to the discontinuity of D=Γ^tD_t acting on say incoming spinor. ∂t is continuous as operator. The continuity of Γ^t could be posed as a condition and would state that the canonical momentum density at the corner of string is continuous. This might well be achieved with the proposed conditions for J_tα=0 and g_tα=0. The outcome is non-trivial since D(out/in) need not annihilate Ψ(in/out) so that with the assumptions just listed one obtains Γ^tΔ A_t, where Δ A_t is the discontinuity of the induced spinor connection and is gauge invariant quantity. Also the non-integrable phase factor associated with the line entering to the vertex appears in the vertex and comes from covariant constancy of the induced spinor field along the line.

As already noticed, the definition of Γ_t might be problematic. For Chern-Simons term Γ_t is finite but in this case one looses the justification for M⁴ propagator of fermion as coming from the boundary condition. For Kähler-Dirac action there are hopes of obtaining a finite limiting value for Γ^t and therefore of Γ^tD_t if the induced Kähler form satisfies condition J_ti=0 in the case that one has g_tα=0. Furthermore, the continuity of Γ^t could be posed as a condition for vertices. Note that one obtains also the non-integrable phase factor allowing Wilson line interpretation.

Nothing has been said about the discontinuity of the modified Dirac operator through the wormhole contact as one traverses from Euclidian to Minkowskian side. This discontinuity might be also relevant.One could consider also the difference of the above discussed discontinuity between Euclidian and Minkowskian regions.

What one should obtain at QFT limit?

After functional integration over WCW of one should obtain a scattering amplitude in which the fermionic 2- vertices defined as discontinuities of the modified Dirac operator at partonic 2-surfaces should boil down to a contraction of an M⁸ vector with gamma matrices of M⁸. This vector has dimension of mass. This basic parameter should characterize many different physical situations. Consider only the description of massivation of elementary particles regarded as bound states of fundamental massless fermions and the mixing of left and right-handed fermions. Also CKM mixing should involve this parameter. These vectors should also appear in Higgs couplings, which in QFT description contain Higgs vacuum expectation as a factor.

In twistor approach virtual particles have complex light-like momenta. Fundamental fermions have most naturally real and light-like momenta. N=4 SUSY describes gauge bosons which correspond to bound states of fundamental fermions in TGD. This suggests that the four-momenta of bound states of massless fermions - be they hadrons, leptons, or gauge bosons - can be taken to be complex.

There is an intriguing connection with TGD based notion of space-time. In TGD one obtains at space-time level complexified four-momenta since the four-momentum from Minkowskian/ Euclidian region is real/imaginary. In the case of physical particle necessary involving two wormhole contacts and two flux tubes connecting them the total complexifies four momentum would be sum of two real and two imaginary contributions. Every elementary particle should have also imaginary part in its four-momentum and would be massless in complexified sense allowing mass in real sense given by the length of the imaginary four-momentum.

TGD predicts Higgs field although Higgs expectation does not have any role in quantum TGD proper. Higgs vacuum expectation is however a necessary part of QFT limit (Higgs decays to WW pairs require that vacuum expectation is non-vanishing). Higgs vacuum expectation must correspond in TGD framework to a quantity with dimensions of mass. In TGD Higgs cannot be scalar but a vector in CP₂ degrees of freedom. The problem is that CP₂ does not allow covariantly constant vectors. The imaginary part of classical four-momentum gives a parameter which has interpretation as a vector in the tangent space of which is same as that of M⁴× CP₂. Could M⁸-H duality be realized at the level of tangent space and for relate four-momentum and color quantum numbers to 8-momentum?

Elementary particles of course need not be eigenstates of the imaginary part of four-momentum. For a fixed mass one can have wave functions in the space of imaginary four-momentum analogous to S³ spherical harmonics at the sphere of E⁴ with radius defined by the length of imaginary four-momentum (mass). These harmonics are characterized by SO(4) quantum numbers. Could one interpret this complexification in terms of M⁸-H duality and say that SO(4) defines the symmetries for the low energy dual of WCW defining high energy description of QCD based on SU(3) symmetry. SO(4) would correspond to the symmetry group assigned to hadrons in the approach based on conserved vector currents and partially conserved axial currents. SO(4) would be much more general and associated also with leptons.

The anomalous color hyper-charge of leptonic spinors would imply that one can have also in the case of leptons a wave function in S³. Higher harmonics would correspond to color excitations of leptons and quarks. If one considers gamma matrices, complexification of M⁴ means introduction of gamma matrix algebra of complexified M⁴ requiring 8 gamma matrices. This suggests a connection with M⁸-H duality. All elementary particles have also imaginary part of four-momentum and the 8-momentum can be interpreted as M⁸-momentum combining the four-momentum and color quantum numbers together.

See the chapter A more detailed view about the construction of scattering amplitudes of "Towards M-matrix" or the article A more detailed view about the construction of scattering amplitudes.

Maintenance problem for the Earth's magnetic field

In Science Daily there was an interesting popular article about what might be called maintenance problem for Earth's magnetic field, which has very important functions such as serving as a shield against cosmic rays which is very dangerous for life.

The understanding of the Earth's magnetic field, call it B_E for shortly, is indeed still far from complete. One problem is to understand why it can exist stable at all. The idea is that the convective heat flow from the core of Earth provides the needed energy to compensate for the dissipation. Also the understanding of the orientation reversals of B_E is poor I remember that numerical simulations can reproduce them.

The popular article explained work by Zhang, Cohen and Haule in which the problem related to the maintenance of convective flow is claimed to be solved. If the conductivity and thus heat conductivity in core is low enough, heat conduction is replaced with convection and this creates the flow of charge too and one obtains convective roll pattern which gives rise to the current taking care that B_E is preserved. The problem is that the conductivity is too high in the metal core. The proposal is that an improved model for the conductivity taking care of electron-electron scattering cures the problem. Knowing how hypish science communications are nowadays, I would not take this claim as final truth.

The problem requires study of Maxwell's equations (for explicit equations see this).

1. The first basic equation for B_E is Faraday' equation stating that time derivative of B_E is the rotor of electric field. This is true in TGD too as is also the equation stating that there are no magnetic monopoles. In TGD CP₂ topology however allows monopole fluxes, which can exist without any generating currents. Second basic equation is Ohm's law saying that current is proportional to electric field: the proportionality constant is conductivity σ.
   

   Together these equations give a partial differential equation for B_E containing diffusion term proportional to the Laplacian of B_E with a coefficient inversely proportional to conductivity. Since finite conductivity means dissipation of energy, one can expect that in absence of energy feed, the current and magnetic field gradually disappear. According to a naive estimate this would take few thousands of years. This does not of course happen. Note however that the polarity of B_E can change in time scale varying from .1 My to 50 My.
   
   

2. Energy is needed to maintain B_E and the current generating it. The energy source would be the heat flowing from Earth's core to the surface. If conductivity and thus thermal conductivity (electron current carries also energy and thus heat) in the interior is not too high, the diffusion of heat proportional to conductivity is not high enough to carry out the heat and convection sets on and the stuff begins to boil. The boiling together with Earth's rotation causes rolling flow of the current above core around the earth, and this convective current generates B_E and keeps it alive.
   
   
3. The problem has been that the electron conductivity in the metal core is too high in the core to allow convective currents. The authors of paper claim that the existing model for the electronic conductivity in metals involving only electron atom scattering contains an additional contribution from electron-electron scattering and that this term cures the situation. It improves the situation but one can still remain skeptic whether this term is really enough.
   

   What worries me too is that the direction of magnetic axis can differ a lot from that of the rotation axis of Earth. How to understand this difference. Also the apparent randomness of the orientation reversals looks strange.
   

What this has to do with TGD? The basic problem is still to understand the orientation reversals and it is not quite clear that even stability problem has been solved completely.

1. TGD differs from Maxwell's theory in that monopole fluxes are possible and realized as flux tubes for which cross section is closed 2-surface carrying magnetic charge. Also flux quant are possible carrying monopole flux are possible. These flux quanta would carry also dark matter.
   

   Could the Earth's magnetic field contains also a dark contribution, call it B_D from monopole fluxes? Could the interaction of visible matter with dark matter be essential for maintaining B_E and for its orientation reversals? Could Magnetic Mother Gaia do the orientation reversals "intentionally" in order to not lose the magnetic shield against cosmic radiation?
   
   

2. The essental point is that monopole fluxes require no currents to generate them. This contribution would give approximately topologically quantized variant of dipole field with flux quanta which could be either flux tubes or sheets and return flux would be along magnetic axis. Suppose that sheets are in question. There would be also corresponding electric field E= v× B_D at the flux sheets according to the dynamo mechanism. v would be the rotation velocity of the dark particles.
   

   There would be radial Lorentz force F=q v× B_D driving charged dark particles radially outwards if the sign of the velocity is current. Could the presence of the Lorentz force causing dark current help to initiate radial convective current of ordinary charged particles bringing hot matter to the surface and cooled matter downwards and in this manner give rise to the convective heat transfer?
   
   

3. What about changes of the polarity of the Earth's magnetic flux quanta? Could they be induced by the changes of the direction of the dark magnetic field at dark flux sheets (say). If the flux sheets of monopole field carry a current of rotating charged dark charged particles, the rotation direction changes as the flux quantum changes its orientation. This guarantees a minimal convective flux is radial and towards the surface. Angular momentum would be changed to its negative in full orientation reversal and the increment of the angular momentum would go by angular momentum conservation to the ordinary matter, perhaps mostly to the ordinary electrons and generate electron currents having twice the original dark angular momentum. After this ordinary electrons would start to dissipate the just inherited newly angular momentum again.
   
   
4. What could induce the changes of the orientation of B_D? Could the two contributions to the total magnetic field be regarded as two magnetic dipoles having dipole interaction realized as a torque proportional to the cross product of the dipole moments. This would cause orientation changes of the dipoles and thus of the fields. The torque vanishes if the dipoles are parallel and dipoles gradually become parallel by dissipation for B_E. This does not seem to be the mechanism.
   

   Could it be that the dynamics of the dark magnetic flux quanta is purely quantal and induces the dynamics of B_E by angular momentum conservation? As the strength of the B_E becomes too weak to shield Earth from cosmic radiation, Magnetic Mother Gaia takes the lead and turns its magnetic body to to a new orientation, which by angular momentum conservation forces the ordinary electrons to a rotation around new magnetic axis and much brisker B_E is regenerated in new direction. Magnetic Mother Gaia takes would take good care of his prodigal son!
   
   

See the chapter About Strange Effects Related to Rotating Magnetic Systems of "TGD and Fringe Physics" or the article The maintenance problem for Earth's magnetic field .

Biochemical communications as a prerequisite for dark photon communications?

In Quantum Biology, coherence and decoherence there have been innumerable links to various hot topics in biology about which I know virtually nothing. I have managed to catch only some key notions like behavior, nutrients, nutrinogenomics, nutrinogenetic signaling, pheromone, hormones, ecology. For instance, nutrients are found to have epigenetic effect on gene expression: they affect behaviour! Pheromones have effects on behavior. I understand that molecular biologist try to reduce these effects to chemical communications and biochemical pathways.

I cannot of course say anything interesting about this horribly complex molecular biology except that I believe that something immensely important could be missing: the notion of magnetic body carrying dark matter and controlling also biochemistry. My attempts to understand rely on the conceptual framework, which is the triple (magnetic body (MB), biological body (BB), environment) replacing the pair (BB, environment) in the usual approach. MBs are the intentional agents affecting other MBs or BBs and being affected by them. So I must try to understand these concepts in terms of these notions which I dare to regard as physical.

The first thing I can do is to assign my pet notions to words like behaviour, nutrient, pheromone, hormones, etc.. and try to understand whether these horrible biochemical complexities could reflect something very simple at the deeper level.

1. TGD Universe obeys Zero Energy Ontology (ZEO) predicting that basic objects can be regarded as 4-dimensional surfaces associated with pairs of 3-surfaces at opposite boundaries of causal diamonds (CDs), which by strong form holography can be reduced to correspond pairs of collections partonic 2-surfaces and their 4-D tangent space data. Time-locality however remains and behavior is assigned to time evolution of magnetic body (MB).
   
   Quantum self organisation replaces this 4-D magnetic body with new one in each quantum jump so that our geometric future and past are not fixed but evolve to an asymptotic self-organization pattern. In ZEO quantum jumps define sequences of state function reductions on fixed boundary of CD (and are analogous to repeated measurement which however affect only the part of zero energy state associated with this CD boundary fixed whereas in ordinary ontology the entire state would remain unaffected). The acts of free will have as quantum counterparts the quantum jumps in which state function reduction agains to the opposite boundary of CD and the arrow of geometric time changes at that particular level in the hierarchy of CDs. Self corresponds to a sequences of state function reductions at fixed boundary and dies when the boundary changes. There is entire hierarchy of selves corresponding to hierarchy of CDs and sub-selves correspond to mental images of self.
   
   
2. Behavior pattern having as correlate 4-D magnetic body inside given CD is a very general notion: already DNA replication, transcription, translation, biochemical pathways, associations in nervous system, our behaviors,.. are all induced by behaviour patterns represented by 4-D magnetic bodies in appropriate time and length scales.
   
   
3. If MB wants to affect behaviour it must affect MB at the lower level or BB directly. Reconnection by using U-shaped magnetic flux tubes to generate double flux tube connection is the basic mechanism for this and identifiable as correlate for directed attention. Stable reconnection requires that cyclotron frequencies of dark charge particles at flux tubes and therefore also the strengths of magnetic fields and thicknesses are same for the reconnected U-shaped flux tubes. The signalling is thus based on dark photons which can transform to ordinary photons identified as bio-photons. Dark photons can in this manner affect biochemistry since their energy spectrum is in the energy range of excitations of biomolecules (ranging from .5 eV (metabolic energy quantum) to visible and UV).
   

   In the general situation several frequencies are involved and serve as kind of passwords. The model for musical harmony and genetic code in terms of bio-harmony relying on icosahedral Hamilton's cycles predicts that DNA codons and amino-acids correspond to 3-chords defining what might be called bio-harmony: in fact, 256 different bio-harmonies are predicted. The corresponding frequencies can be in the range of audible frequencies and it is quite possible that music of dark photons is realized in biology. Each molecule could correspond and produce its own collection of chords, maybe even melody somewhat like the characters in the operas of Wagner!
   

4. This process would operate at all levels. Basic biomolecules are scanning their environment using these U shaped flux tubes and reconnecting. Immune system tries to detect invader molecules using reconnection followed by resonant exchange of dark photons, followed by the reduction of h_eff shortening the length of the flux tube and bringing the unlucky invader near the immune soldier to be mercilessly destroyed. This scanning can be done also with positive intentions: DNA replication, transcription to mRNA, translation, etc.. are examples
   of this.
   
   
5. This picture leads to an interpretation what happens , when biomolecule attaches to a receptor. Biomolecule - say neural transmitter or hormone - is the end of (a potential) communication line - plug formed by the U-shaped flux tube. When it attaches to the receptor, the owner of the receptor is plugged to the web and can send and receive dark photons resonantly to the receiver. Therefore bio-chemical communications are at deeper level not yet communications but only sending of plugs making possible real communication by dark photons.
   
   

In the following some more or less random comments inspired by this proposal.

1. States have both armies and diplomats. Immune system is the counterpart of army trying to detect invaders and destroy them. There must be also a system trying to find potential friends and collaborators. We indeed co-operate with bacteria and the significance of this aspect seem to be increasingly realized. For instance, in Quantum Biology, coherence and decoherence there is a link to an article about collaboration some exotic sea animal with micro-organisms: the animal actively builds connections to the micro-organisms and here also reconnection mechanism is highly suggestive.
   
   
2. Hormones are usually regarded as purely chemical means of communication. I have proposed that they make possible communication using dark photon signals propagating along communication lines defined by flux tubes. The attachment of biomolecule to a receptor in cell membrane is for plugging in: the biomolecule in receptor can be connected by flux tube pair to quite specific biomolecules or magnetic bodies.
   

3. The effects of psychedelics and entheogens such as naturally occurring psylocybin (see also this) could involve even flux tubes connections to distant civilizations or higher level conscious entities! This sounds of course totally outlandish but is not my original proposal;-). In ZEO finite light velocity is not a problem since signals can travel also backwards in time). In this case the lengths of flux tubes would be very long and extend to distant galaxies.
   
   
4. Pheromones (see this link as example) are like hormones except that they affect the behaviour of another member of the same species (or more generally?) by inducing epigenetic influences. Female butterfly emits pheromones and male receives them and connection is established by magnetic flux tube to the female's BB or MB. After than male flies to the direction in which the connection becomes stronger. This mechanism is same as used by birds in TGD Universe to find from Africa to the same place in Norther Finland every year;-). Also food odors have epigenetic influences. The same plugin model
   applies also to these effects: chemical signals are actually plugs connecting to the web and making possible signalling
   by dark photons.
   
   
5. Also nutrients could fit nicely to this picture if also they are plugs connecting BB to some MB rather than just source of metabolic energy. To stay alive means is to stay connected to the web;-). Magnetic zombies die soon! The MB in question could be not only personal MB but also the dark MB of Mother Gaia as I have suggested (see this . The explanation of the Pioneer and Flyby anomalies (see this ) allow to consider a concrete identification as approximately spherical flux sheet carrying dark matter and having radius of Moon's orbital radius. The density of the dark matter would be universal and about .8 kg/m². It would be approximately spherical and involve also a flux tube through the magnetic axis so that closed flux lines would result. This is of course just an innocent suggestion.
   
   
6. Even mushrooms communicate via underground network analogous to neural networks and formed with roots and mycellium in forests (see this) and magnetic flux tube networks could be naturally at the background.
   

To sum up, communications would be a crucial aspect of being intelligent living system and the proposal is that magnetic body carrying dark matter and photons plays key role in these communications.