Are space-time surfaces minimal surfaces everywhere except at 2-D interaction vertices?

The action S determining space-time surfaces as preferred extremals follows from twistor lift and equals to the sum of volume term Vol and Kähler action S_K. The field equation is a geometric generalization of d'Alembert (Laplace) equation in Minkowskian (Eucidian) regions of space-time surface coupled with induced Kähler form analogous to Maxwell field. Generalization of equations of motion for particle by replacing it with 3-D surface is in question and the orbit of particle defines a region of space-time surface.

1. Zero energy ontology (ZEO) suggests that the external particles arriving to the boundaries of given causal diamond (CD) are like free massless particles and correspond to minimal surfaces as a generalization of light-like geodesic. This dynamic reduces to mere algebraic conditions and there is no dependence on the coupling parameters of S. In contrast to this, in the interaction regions inside CDs there could be a coupling between Vol and S_K due to the non-vanishing divergences of energy momentum currents associated with the two terms in action cancelling each other.
   
   
2. Similar algebraic picture emerges from M⁸-H duality at the level of M⁸ and from what is known about preferred extremals of S assumed to satisfy infinite number of super-symplectic gauge conditions at the 3-surfaces defining the ends of space-time surface at the opposite boundaries of CD.
   
   
3. At M⁸ side of M⁸-H duality associativity is realized as quaternionicity of either tangent or normal space of the space-time surface. The condition that there is 2-D integral distribution of sub-spaces of tangent spaces defining a distribution of complex planes as subspaces of octonionic tangent space implies the map of the space-time surface in M⁸ to that of H. Given point m₈ of M⁸ is mapped to a point of M⁴× CP₂ as a pair of points (m₄,s) formed by M⁴ ⊂ M⁸ projection m₄ of m₈ point and by CP₂ point s parameterizing the tangent space or the normal space of X⁴⊂ M⁸.
   
   
4. If associativity or even the condition about the existence of the integrable distribution of 2-planes fails, the map to M⁴× CP₂ is lost. One could cope with the situation since the gauge conditions at the boundaries of CD would allow to construct preferred extremal connecting the 3-surfaces at the boundaries of CD if this kind of surface exists at all. One can however wonder whether giving up the map M⁸→ H is necessary.
   
   
5. Number theoretic dynamics in M⁸ involves no action principle and no coupling constants, just the associativity and the integrable distribution of complex planes M²(x) of complexified octonions. This suggests that also the dynamics at the level of H involves coupling constants only via boundary conditions. This is the case for the minimal surface solutions suggesting that M⁸-H duality maps the surfaces satisfying the above mentioned conditions to minimal surfaces. The universal dynamics conforms also with quantum criticality.
   
   
6. One can argue that the dependence of field equations on coupling parameters in interactions leading to a perturbative series in coupling parameters in the interior of the space-time surface spoils the extremely beautiful purely algebraic picture about the construction of solutions of field equations using conformal invariance assignable to quantum criticality. Classical perturbation series is also in conflict with the vision that the TGD counterparts twistorial Grassmannian amplitudes do not involve any loop contributions coming as powers of coupling constant parameters.
   
   

Thus both M⁸-H duality, number theoretic vision, quantum criticality, twistor lift of TGD reducing dynamics to the condition about the existence of induced twistor structure, and the proposal for the construction of twistor scattering amplitudes suggest an extremely simple picture about the situation. The divergences of the energy momentum currents of Vol and S_K would be non-vanishing only at discrete points at partonic 2-surfaces defining generalized vertices so that minimal surface equations would hold almost everywhere as the original proposal indeed stated.

1. The fact that all the known extremals of field equations for S are minimal surfaces conforms with the idea. This might be due to the fact that these extremals are especially easy to construct but could be also true quite generally apart from singular points. The divergences of the energy momentum currents associated with S_K and Vol vanish separately: this follows from the analog of holomorphy reducing the field equations to purely algebraic conditions.
   

   It is essential that Kähler current j_K vanishes or is light-like so that its contraction with the gradients of the imbedding space coordinates vanishes. Second condition is that in transversal degrees of freedom energy momentum tensor is tensor of form (1,1) in the complex sense and second fundamental form consists of parts of type (1,1) and (-1-1). In longitudinal degrees of freedom the trace H^k of the second fundamental form H^k_αβ= D_β∂_αh^k vanishes.
   
   

2. Minimal surface equations are an analog of massless field equation but one would like to have also the analog of massless particle. The 3-D light-like boundaries between Minkowskian and Euclidian space-time regions are indeed analogs of massless particles as are also the string like word sheets, whose exact identification is not yet fully settled. In any case, they are crucial for the construction of scattering amplitudes in TGD based generalization of twistor Grassmannian approach. At M⁸ side these points could correspond to singularities at which Galois group of the extension of rationals has a subgroup leaving the point invariant. The points at which roots of polynomial as function of parameters co-incide would serve as an analog.
   

   The intersections of string world sheets with the orbits of partonic 2-surface are 1-D light-like curves X¹_L defining fermion lines. The twistor Grassmannian proposal is that the ends of the fermion lines at partonic 2-surfaces defining vertices provide the information needed to construct scattering amplitudes so that information theoretically the construction of scattering amplitudes would reduce to an analog of quantum field theory for point-like particles.
   
   

3. Number theoretic vision reduces coupling constant evolution to a discrete evolution. This implies that twistor scattering amplitudes for given values of discretized coupling constants involve no radiative corrections. The cuts for the scattering amplitudes would be replaced by sequences of poles. This is unavoidable also because there is number theoretical discretization of momenta from the condition that their components belong to an extension of rationals defining the adele.
   
   

What could the reduction of cuts to poles for twistorial scattering amplitudes at the level of momentum space mean at space-time level?

1. Poles of an analytic function are co-dimension 2 objects. d'Alembert/Laplace equations holding true in Minkowskian/Euclidian signatures express the analogs of analyticity in 4-D case. Co-dimension 2 rule forces to ask whether partonic 2-surfaces defining the vertices and string world sheets could serve analogs of poles at space-time level? In fact, the light-like orbits X³_L of partonic 2-surfaces allow a generalization of 2-D conformal invariance since they are metrically 2-D so that X³_L and string world sheets could serve in the role of poles.
   

   X³_L could be seen as analogs of orbits of bubbles in hydrodynamical flow in accordance with the hydrodynamical interpretations. Particle reactions would correspond to fusions and decays of these bubbles. Strings would connect these bubbles and give rise to tensor networks and serve as space-time correlates for entanglement. Reaction vertices would correspond to common ends for the incoming and outgoing bubbles. They would be analogous to the lines of Feynman diagram meeting at vertex: now vertex would be however 2-D partonic 2-surface.
   
   

2. What can one say about the singularities associated with the light-like orbits of partonic 2-surfaces? The divergence of the Kähler part T_K of energy momentum current T is proportional to a sum of contractions of Kähler current j_K with gradients ∇ h^k of H coordinates. j_K need not be vanishing: it is enough that its contraction with ∇ h^k vanishes and this is true if j_K is light-like. This is the case for so called massless extremals (MEs). For the other known extremals j_K vanishes.
   

   Could the Kähler current j_K be light-like and non-vanishing and singular at X³_L and at string world sheets? This condition would provide the long sought-for precise physical identification of string world sheets. Minimal surface equations would hold true also at these surface. Even more: j_K could be non-vanishing and thus also singular only at the 1-D intersections X¹_L of string world sheets with X³_L - I have called these curves fermionic lines?
   

   What it means that j_K is singular - that is has 2-D delta function singularity at string world sheets? j_K is defined as divergence of the induced Kähler form J so that one can use the standard definition of derivative to define j_K at string world sheet as the limiting value j_K^α= (Div_+- J)^α = lim_{Δ xⁿ→ 0} (J₊^{α n}- J_-^{α n})/Δ xⁿ, where xⁿ is a coordinate normal to the string world sheet. If J is not light-like, it gives rise to isometry currents with non-vanishing divergence at string world sheet. This current should be light like to guarantee that energy momentum currents are divergenceless. This is guaranteed if the isometry currents T^{&n; A} are continuous through the string world sheet.
   
   

3. If the light-like j_K at partonic orbits is localized at fermionic lines X¹_L, the divergences of energy momentum currents could be non-vanishing and singular only at the vertices defined at partonic 2-surfaces at which fermionic lines X¹_L meet. The divergences of energy momentum tensors T_K of S_K and T_Vol of Vol would be non-vanishing only at these vertices. They should of course cancel each other: Div T_K=-Div T_Vol.
   
   
4. Div T_K should be non-vanishing and singular only at the intersections of string world sheets and partonic 2-surfaces defining the vertices as the ends of fermion lines. How to translate this statement to a more precise mathematical form? How to precisely define the notions of divergence at the singularity?
   

   The physical picture is that there is a sharing of conserved isometry charges of the incoming partonic orbit i=1 determined T_K between 2 outgoing partonic orbits labelled by j=2,3 . This implies charge transfer from i=1 to the partonic orbits j=2,3 such that the sum of transfers sum up to to the total incoming charge. This must correspond to a non-vanishing divergence proportional to delta function. The transfer of the isometry charge for given pair i,j of partonic orbits that is Div_{i→ j} T_K must be determined as the limiting value of the quantity Δ_{i→ j} T_K^α,A/Δ x^α as Δ x^α approaches zero. Here Δ_{i→ j} T_K^α,A is the difference of the components of the isometry currents between partonic orbits i and j at the vertex. The outcome is proportional delta function.
   
   

5. Similar description applies also to the volume term. Now the trace of the second fundamental form would have delta function singularity coming from Div T_K. The condition Div T_K= -Div T_Vol would bring in the dependence of the boundary conditions on coupling parameters so that space-time surface would depend on the coupling constants in accordance with quantum-classical correspondence. The manner how the coupling constants make themselves visible in the properties of space-time surface would be extremely delicate.
   
   

This picture conforms with the vision about scattering amplitudes at both M⁸ and H sides of M⁸-H duality.

1. M⁸ dynamics based on algebraic equations for space-time surfaces leads to the proposal that scattering amplitudes can be constructed using the data only at the points of space-time surface with M⁸ coordinates in the extension of the rationals defining the adele. I call this discrete set of points cognitive representation.
   
   
2. At H side the information theoretic interpretation would be that all information needed to construct scattering amplitudes would come from points at which the divergences of the energy momentum tensors of S_K and Vol are non-vanishing and singular.
   
   

Both pictures would realize extremely strong form of holography, much stronger than the strong form of holography that stated that only partonic 2-surfaces and string world sheets are needed.

See the article The Recent View about Twistorialization in TGD Framework or the shorter article Further comments about classical field equations in TGD framework, or the chapter chapterwith the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Three dualities of the field equations of TGD

The basic field equations of TGD allow several dualities. There are 3 of them at the level of basic field equations (and several other dualities such as M⁸-M⁴× CP₂ duality).

1. The first duality is the analog of particle-field duality. The spacetime surface describing the particle (3-surface of M⁴× CP₂ instead of point-like particle) corresponds to the particle aspect and the fields inside it geometrized in terms of sub-manifold geometry in terms of quantities characterizing geometry of M⁴× CP₂ to the field aspect. Particle orbit serves as wave guide for field, one might say.
   
   
2. Second duality is particle-spacetime duality. Particle identified as 3-D surface means that particle orbit is space-time surface glued to a larger space-time surface by topological sum contacts. It depends on the scale used, whether it is more appropriate to talk about particle or of space-time.
   
   
3. The third duality is hydrodynamics-massless field theory duality Hydrodynamical equations state local conservation of Noether currents. Field equations indeed reduce to local conservation conditions of Noether currents associated with isometries of M⁴× CP₂. One the other hand, these equations have interpretation as non-linear geometrization of massless wave equation with coupling to Maxwell fields. This realizes the ultimate dream of theoretician: symmetries dictate the dynamics completely. This is expected to be realized also at the level of scattering amplitudes and the generalization of twistor Grassmannian amplitudes could realize this in terms of Yangian symmetry.
   

   Hydrodynamics-wave equations duality generalizes to the fermionic sector and involves superconformal symmetry.
   
   

4. What I call modified gamma matrices are obtained as contractions of the partial derivatives of the action defining space-time surface with respect to the gradients of imbedding space coordinate with imbedding space gamma matrices. Their divergences vanish by field equations for the space-time surface and this is necessary for the internal consistency the Dirac equation. The modified gamma matrices reduces to ordinary ones if space-time surface is M⁴ and one obtains ordinary massless Dirac equation.
   
   
5. Modified Dirac equation expresses conservation of super current and actually infinite number of super currents obtained by contracting second quantized induced spinor field with the solutions of modified Dirac. This corresponds to the super-hydrodynamic aspect. On the other hand, modified Dirac equation corresponds to fermionic analog of massless wave equation as super-counterpart of the non-linear massless field equation determining space-time surface.
   
   

See the article The Recent View about Twistorialization in TGD Framework or the shorter article Further comments about classical field equations in TGD framework, or the chapter chapterwith the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

About the physical interpretation of the velocity parameter in the formula for the gravitational Planck constant

Nottale's formula for the gravitational Planck constant hbar_gr= GMm/v₀ involves parameter v₀ with dimensions of velocity. I have worked with the quantum interpretation of the formula but the physical origin of v₀ - or equivalently the dimensionless parameter β₀=v₀/c (to be used in the sequel) appearing in the formula has remained open hitherto. In the following a possible interpretation based on many-sheeted space-time concept, many-sheeted cosmology, and zero energy ontology (ZEO) is discussed.

A generalization of the Hubble formula β=L/L_H for the cosmic recession velocity, where L_H= c/H is Hubble length and L is radial distance to the object, is suggestive. This interpretation would suggest that some kind of expansion is present. The fact however is that stars, planetary systems, and planets do not seem to participate cosmic expansion. In TGD framework this is interpreted in terms of quantal jerk-wise expansion taking place as relative rapid expansions analogous to atomic transitions or quantum phase transitions. The TGD based variant of Expanding Earth model assumes that during Cambrian explosion the radius of Earth expanded by factor 2.

There are two measures for the size of the system. The M⁴ size L_M⁴ is identifiable as the maximum of the radial M⁴ distance from the tip of CD associated with the center of mass of the system along the light-like geodesic at the boundary of CD. System has also size L_ind defined defined in terms of the induced metric of the space-time surface, which is space-like at the boundary of CD. One has L_ind<L_M⁴. The identification β₀= L_M⁴/L_H<1 does not allow the identification L_H=L_M⁴. L_H would however naturally corresponds to the size of the magnetic body of the system in turn identifiable as the size of CD.

One can deduce an estimate for β0 by approximating the space-time surface near the light-cone boundary as Robertson-Walker cosmology, and expressing the mass density ρ defined as ρ=M/VM4, where VM4=(4π/3) LM43 is the M4 volume of the system. ρ can be expressed as a fraction ε2 of the critical mass density ρcr= 3H2/8π G. This leads to the formula β0= [rS/LM4]1/2 × (1/ε), where rS is Schwartschild radius.

This formula is tested for planetary system and Earth. The dark matter assignable to Earth can be identified as the innermost part of inner core with volume, which is .01 per cent of the volume of Earth. Also the consistency of the Bohr quantization for dark and ordinary matter is discussed and leads to a number theoretical condition on the ratio of the ordinary and dark masses.

See the article About the physical interpretation of the velocity parameter in the formula for the gravitational Planck constant or the new chapter About the Nottale's formula for h_gr and the possibility that Planck length l_P and CP₂ length R are identical giving G= R²/ℏ_eff of "Physics in many-sheeted space-time"..

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

An island at which body size shrinks

I encountered in Facebook an article claiming that the bodies of animals shrink at the island of Flores belonging to Indonesia. This news is not Dog's days news (Dog's days news is a direct translation from the finnish synonym for fake news).

Both animals and humans really are claimed to have shrinked in size. The bodies of both hominins (predecessors of humans, humans, ane even elephants) have shrinked at Flores.

1. In 2003, researchers discovered in a mountain cave in the island of Flores fossils of tiny, humanlike individual. It had chimp sized brain and was 90 cm tall. Several villages at the area are inhabited by people with average body height about 1.45 meters.
   
   
2. Could the small size of the recent humans at Flores be due to interbreeding between modern humans with Homo Florensiensis (HF) occurred long time ago? The hypothesis could be tested by studying the DNA of HF. Since the estimate age of fossils of HF was 10,000 years, researchers hoped that they could find some DNA to HF. DNA was not found but researchers realized that if HF as interbreeded with humans, this DNA could show itself in DNA of modern humans at Flores. It was found that this DNA can be identified but differs insignificantly from that of modern humans. It was also found that the age of the fossils was about 60,000 years.
   
   
3. Therefore it seems that the interbreeding did not cause the reduction in size. The study also showed that at least twice in the ancient history of humans and their relatives arrived as Flores and then grew shorter. This happened also for elephants that arrived to Flores at twice.
   
   

This looks really weird! Weirdness in this proportion allows some totally irresponsible speculation.

1. The hierarchy of Planck constants h_eff=nh₀ (h=6h₀ is a good guess ) assigned with dark matter as phases of ordinary matter and responsible for macroscopic quantum coherence is central in TGD inspired biology . Quantum scales are proportional to or its power (h_eff² for atoms, h_eff for Compton length, and h_eff^{1/2 for cyclotron states).}
   
   
2. The value of gravitational Planck constant h_gr (=h_eff) at the flux tubes mediating gravitational interaction could determine the size scale of the animals. Could one consider a local anomaly in which the value of h_gr is reduced and leads to a shrinkage of also body size?
   
   
3. h_gr is of form h_gr=GM_Dm/v₀, where v₀ a velocity parameter (see this, this, and this). M_D is a large dark mass of order 10^-4 times the mass of Earth. Gravitational Compton length Λ_gr= h_gr/m=GM_D/v₀ for a particle with mass m. Λ_gr= h_gr/m does not depend on the mass of the particle - this conforms with Equivalence Principle.
   

   The estimate of this article gives Λ_gr= 2πM _D/v₀= 2.9× r_S(E)$, where the Schwartshild radius of Earth is $r_S(E)=2GM_E=.9$ mm. This gives Λ_gr= 2.6 mm, which corresponds to p-adic length scale L(k=187). Brain contains neuron blobs with this size scale. The size scale of organism is expected to be some not too large multiple of this scale.
   

   Could one think that v₀ at Flores is larger than normally and reduces the value of Λ_gr so that the size for the gravitational part of the magnetic body of any organism shrinks, and that this gradually leads to a reduction of the size of the biological body. Second possibility is that the value of dark mass M_D is at Flores smaller than elsewhere: one would have a dark analogy of ordinary local gravitational anomaly. The reduction of h_gr should be rather large so that the first option looks more plausible.
   
   

See the article An island at which body size shrinks or the chapter Quantum Criticality and dark matter of "Hyper-finite factors, p-adic length scale hypothesis, and dark matter hierarchy".

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

What perfectoid and its tilt introduced by Fields medalist Peter Scholze are?

Quanta Magazine tells that 30-year old Peter Scholze is now the youngest Fields medalist due to the revolution that he launched in arithmetic geometry (see this).

Scholze's work might be interesting also from the point of view physics, at least the physics according to TGD. I have already made a attempt to understand Scholze's basic idea and to relate it to physics. About the theorems that he has proved I cannot say anything with my miserable math skills.

The notion of perfectoid

Scholze introduces first the notion of perfectoid.

1. This requires some background notions. The characteristic p for field is defined as the integer p for which px=0 for all elements x. Frobenius homomorphism (Frob familiarly) is defined as x→ x^p. For a field of characteristic p Frob: x-->x^p is an algebra homomorphism mapping product to product and sum to sum: this is very nice and relatively easy to show even by a layman like me.
   
   
2. Perfectoid is a field having either characteristic p=0 (reals, p-adics for instance) or for which Frob is a surjection meaning that Frob maps at least one number to a given number x.
   

   For finite fields Frob is identity: x^p=x as proved already by Fermat. For reals and p-adic number fields with characteristic p=0 Frob maps all elements to unit element and is not a surjection but this is not required now.
   

The tilt of the perfectoid

What Scholze introduces besides perfectoids K also what he calls tilt of the perfectoid: K_b. K_b is something between p-adic number fields and reals and leads to theorems giving totally new insights to arithemetic geometry

1. As we learned during the first student year, real numbers can be defined as Cauchy sequences of rationals converging to a real number, which can be also algebraic number or transcendental. The numbers in the tilt K_b would be this kind of sequences.
   
   
2. Scholze starts from (say) p-adic numbers and considers infinite sequence of iterates of 1/p:th roots. At given step x→ x^1/p. This gives the sequences $(x,x^1/p,x^1/p2,x^1/p3,...)$ as elements of K_b. At the limit one obtains 1/p^∞ root of x.
   
   
   1. For finite fields each step is trivial (x^p=x) so that nothing interesting results: one has (x,x,x,x,x...).
      
      
   2. For p-adic number fields the situation is non-trivial. x^1/p exists as p-adic number for all p-adic numbers with unit norm having x= x₀+x₁p+... In the lowest order x ≈ x₀ the root is just x since x is effectively an element of finite field in this approximation. One can develop the x^1/pto a power series in p and continue the iteration. The sequence obtained defines the element of tilt K_b of field $K$, now p-adic numbers.
      
      
   3. If the p-adic number x has norm pⁿ and is therefore not unity, the root operation makes sense only if one performs an extension of p-adic numbers containing all the roots p^1/pk) . These roots define one particular kind of extension of p-adic numbers and the extension is infinite-dimensional since all roots are needed. One can approximate K_b by taking only finite number iterated roots: I call these almost perfectoids as precursors of perfectoids.
      
      
3. The tilt is said to be fractal: this is easy to understand from the presence of the iterated p:th root. Each step in the sequence is like zooming. One might say that p-adic scale p becomes p:th root of itself. In TGD p-adic length scale L_p is proportional to p^1/2: does the scaling mean that the p-adic length scale would defined a hierarchy of scales proportional to p^1/2kp approaching the CP₂ scale since the root of p approaches unity. Tilts as extensions by iterated roots would improve the length scale resolution.
   
   

One day later I got the feeling that I might have understood one more important thing about the tilt of p-adic number field: changing of the characteristic 0 of p-adic number field to characteristics p>0 of the corresponding finite field for its tilt (thanks for Ulla for the links). What could this mean?

Characteristic p (p is now the prime labelling p-adic number field) means nx=0. This property makes the mathematics of finite fields extremely simple: in the summation one need not take care of the residue as in the case of reals and p-adics. The tilt of the p-adic number field would have the same property! In the infinite sequence of the p-adic numbers coming as iterated p:th roots of starting point p-adic number one can sum each p-adic number separately. This is really cute if true!

It seems that one can formulate the arithmetics problem in the tilt where it becomes in principle as simple as in finite field with only p elements! Does the existence of solution in this case imply its existence in the case of p-adic numbers? But doesn't the situation remain the same concerning the existence of the solution in the case of rational numbers? The infinite series defining p-adic number must correspond a sequence in which binary digits repeat with some period to give a rational number: rational solution is like a periodic solution of a dynamical system whereas non-rational solution is like chaotic orbit having no periodicity? In the tilt one can also have solutions in which some iterated root of p appears: these cannot belong to rationals but to their extension by an iterated root of p.

The results of Scholze could be highly relevant for the number theoretic view about TGD in which octonionic generalization of arithematic geometry plays a key role since the points of space-time surface with coordinates in extension of rationals defining adele and also what I call cognitive representations determining the entire space-time surface if M⁸-H duality holds true (space-time surfaces would be analogous to roots of polynomials). Unfortunately, my technical skills in mathematics needed are hopelessly limited.

TGD inspires the question is whether the finite cutoffs of K_b - almost perfectoids - could be particularly interesting physically. At the limit of infinite dimension one would get an ideal situation not realizable physically if one believes that finite-dimensionality is basic property of extensions of p-adic numbers appearing in number theoretical quantum physics (they would related to cognitive representations in TGD). Adelic physics involves all extensions of rationals and the extensions of p-adic number fields induced by them and thus also extensions of type K_b. I have made some naive speculations about why just these extensions might be physically of a special signiticance.

See the articles Could the precursors of perfectoids emerge in TGD? and Does M⁸-H duality reduce classical TGD to octonionic algebraic geometry?.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

How life began?

The central question of biology is "How life began?" and dark variants of biomolecules suggest not only a solution to various paradoxes but also a concrete answer to this question.

The transcription machinery for rRNA including ribozymes and mRNA coding for the proteins associated with ribosomes is central for the translation. The DNA coding for rRNA is associated with nucleolus (see this) in the center of the nucleus.

1. After the emergence of the first ribosome the ribosomes of the already existing nucleus can take care of the translation of the ribosomal proteins. But how could the first ribosome emerge? This question leads to a paradox bringing in mind self-reference - the basic theme of Gödel-Escher-Bach of Douglas Hofstadter, perhaps the most fascinating and inspiring book I have ever read. The ribosomal proteins associated with the first ribosomes should have been translated using ribosome, which did not yet exist!
   
   
2. Could the translation of the first ribosomal proteins directly from the dark variants of these proteins solve the paradox? The idea of shadow dynamics induced by the pairing of basic biomolecules with their dark variants even allows to ask whether the replication, transcription, and translation could occur at dark level so that dark genes for ribosomes would be transcribed to dark ribosomal RNA and dark mRNA translated to dark AA associated with the ribosomes. These in turn would pair with ordinary ribosomal RNA and AA.
   
   
3. But what about dark variants of ribosomes? One can encounter the same paradox with them if they are needed for the translation. Could it be that dark variants of the ribosomes are not needed at all for the translation but would only give rise to ordinary ribosomes by the pairings basic biomolecules and their dark variants. Dark DNA would pair with dark mRNA, which pairs spontaneously with dark tRNA. Once the ordinary ribosomes are generated from the dark ribosomes by pairing, they could make the translation much faster.
   
   
4. There is however a problem. Both dark RNA and AA correspond to dark nuclear strings. Dark tRNA realized as nuclear string in the proposed manner does not have a decomposition to dark AA and dark RNA as ordinary tRNA has. The pairing of dark tRNA and dark mRNA should rise to dark AA and dark nuclear string - call it X - serving as the analog for the pairing of mRNA sequence with "RNAs" of tRNAs in the ordinary translation.
   
   
5. How to identify X? Could the translation be analogous to a reaction vertex in which dark mRNA and dark tRNA meet and give rise to dark AA and X? X cannot be completely trivial. Could X correspond to the dark DNA?! If so, the process would transcribe from dark DNA dark RNA and translate from dark RNA and dark tRNA AA and dark DNA. This would lead to an exponential growth of dark DNA and other dark variants of bio-molecules. This exponential growth would induce exponential growth of the basic bio-molecules by pairing. Life would emerge! No RNA era or lipid era might be needed. All basic biomolecules or their precursors could emerge even simultaneously - presumably in presence of lipids - but this is not the only possibility.
   
   

One can take a more precise look at the situation and try to understand the emergence of bio-molecules and their basic reactions as shadows of the dark variants of bio-molecules appearing in dark particle reactions. The basic idea is that same dark reaction can give rise to several reactions of biomolecules if varying number of the external dark particles are paired with corresponding bio-molecules. Under what conditions this pairing could occur, is left an open question. Consider now the dark 2→ 2 reactions and possible reactions obtained by pairing of some particles.

1. The reaction
   

   DmRNA+DtRNA→ DAA + DDNA
   

   gives rise to translation mRNA+tRNA → AA if DDNA-DNA pairing does not occur in the final state but other dark particles are paired with the their ordinary variants. If only DmRNA-mRNA and DDNA-DNA pairings occur, the reaction gives the reversal mRNA → DNA of transription.
   

   It should be easy to check whether this is allowed by the tensor product decomposition for the group representations associated with dark proton triplets. Same applies to other reactions considered below.
   

   If this reaction is possible then also the reversal
   

   DAA + DDNA → DmRNA+DtRNA.
   

   can occur. If only DDNA-DNA and DmRNA-mRNA pairings occur this gives rise to transcription of DNA→ mRNA.
   Also reverse translation AA → mRNA is possible.
   
   

2. One can consider also the reaction
   

   DmRNA+DtRNA → DAA + DmRNA
   

   If all pairings except DAA-AA pairing are present, the outcome is instead of translation the replication of mRNA such that the amino-acid in tRNA serves the role of catalyzer. I have considered the possibility that this process preceded the ordinary translation: in a phase transition increasing h_eff the roles of AA and RNA in tRNA would have changed.
   

   If this reaction is possible then also its reversal
   

   DAA + DmRNA → DmRNA+DtRNA
   

   is allowed. If all pairing except DmRNA-mRNA occur, this gives rise to AA +RNA → tRNA allowing to generate tRNA from AA and RNA (not quite RNA).
   
   

3. The replication of DNA strand would correspond at dark level to a formation of bound states by the reaction
   

   DDNA+DDNA→ DDNA +_bound DDNA
   

   in which all particles are paired. The opening of DNA double strand would correspond to the reverse of this bound state formation.
   
   

These dark particle reactions behind the shadow dynamics of life should be describable by S-matrices, which one might call the S-matrix of life.

1. For instance for
   

   DmRNA+DtRNA→ X,
   

   where X can be DmRNA+DtRNA (nothing happens - forward scattering) or DAA + DDNA and perhaps even DAA+DmRNA, one would have unitary S-matrix satisfying SS^{dagger=Id giving probability conservation as ∑_n p_m,n= |S_mn|2 =1 as a special case. Writing S=1+iT unitarity gives i(T-T†)+TT†=0 giving additional constraints besides probability conservation.}
   

   For
   

   DmRNA+DtRNA→ DAA + DDNA
   

   the non-vanishing elements of T are only between pairs [(DmRNA,DtRNA), (DAA,DDNA)] for which mRNA pairs with tRNA and DNA codes for AA. Unitary matrix would be coded by amplitudes t(AA,DNA_i(A)) satisfying ∑_ip_i(DAA)=p(DDNA+DAA), p_i(AA)=|t(DAA,DDNA_i(A)|². p(DDNA+DAA) equals to p(DDNA+DAA)= (1-p) Br(DDNA+DAA), where p is the probability that nothing happens (forward scattering) and Br(DDNA+DAA) is the branching ratio to DDNA+DAA channel smaller than 1 if Br(DDNA+DmRNA) is non-vanishing. The natural interpretation for p_i(AA) would be as probability that DNA_i codes for it.
   
   

2. For the reverse reaction
   

   DAA + DDNA rightarrow DmRNA+DtRNA
   

   it is natural to assume that DtRNA corresponds to any tRNA, which pairs with RNA. The AA associated with this tRNA is always the same but the counterpart of RNA can vary (wobbling). One can speak of the decomposition of dark genetic code to DmRNA→ DtRNA → DAA to a pair of codes mapping DmRNA to DtNRA and DtRNA to DAA. There is a set tRNA_i(mRNA) of tRNAs coding for given mRNA, and the probabilities p_i(DmRNA) sum up to p= ∑_i p_i(DmRNA)= (1-p) Br(DmRNA+DtRNA) , where p is the probability for forward scattering and Br(DmRNA+DtRNA) is the branching fraction. The natural identification of p_i(DmRNA) is as the probability that mRNA pairs with tRNA_i.
   
   

A possible weak point of the proposal is pairing: what are the conditions under which it occurs and are different pairing patterns possible. Possible second weak point is purely group theoretic: one should check whether which reactions are allowed by the tensor product decompositions for the states of dark proton triplets.

See the article Getting philosophical: some comments about the problems of physics, neuroscience, and biology or the chapter Quantum Mind, Magnetic Body, and Biological Body of of "TGD based view about living matter and remote mental interactions".

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

More about badly behaving photons

I wrote about two years ago about strange halving of the unit of angular momentum for photons. The article had title Badly behaving photons and space-time as 4-surface).

Now I encountered a popular article (see this) telling about this strange halving of photon angular momentum unit two years after writing the above comments. I found nothing new but my immediate reaction was that the finding could be seen as a direct proof for h_eff=nh₀ hierarchy, where h₀ is the minimal value of Planck constants, which need not be ordinary Planck constant h as I have often assumed in previous writings.

Various arguments indeed support for h=6h₀. This hypothesis would explain the strange findings about hydrogen atom having what Mills calls hydrino states having larger binding energy than normal hydrogen atom (see this): the increase of the binding energy would follow from the proportionality of the binding energy to 1/h_eff². For n₀=6→ n<6 the binding energy is scale up as (n/6)². The values of n=1,2,3 dividing n are preferred. Second argument supporting h=6h₀ comes from the model for the color vision (see this).

What is the interpretation of the ordinary photon angular momentum for n=n₀= 6? Quantization for angular momentum as multiples of hbar₀ reads as l= l₀hbar₀= (l₀/6)hbar, l₀=1,2... so that fractional angular momenta are possible. l₀=6 gives the ordinary quantization for which the wave function has same value for all 6 sheets of the covering. l₀=3 gives the claimed half-quantization.

See the article Badly behaving photons and space-time as 4-surface.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

TGD approach to the hen-or-egg problems of biology

Standard biology suffers from several hen-or-egg problems. Which came first: genes or metabolism? The problem is that genes require metabolism and metabolism requires genes! Genes-first leads to the vision about RNA world and metabolism-first to lipids world idea.

The emergence of basic biomolecules is the second problem. What selected these relatively few basic molecules from huge multitude of molecules? And again the hen-egg problems emerge. Which came first: proteins or the translation machinery producing them from RNA? Did RNA arrive before proteins or did proteins and RNAs necessary for their transcription and translation machinery emerge first. One can argue that ribozymes served as catalysts for RNA replication but how RNAs managed to emerge without replication machinery involving ribozymes? What about DNA: did it emerge before RNA or could it have emerged from RNA? It seems that something extremely important is missing from the picture.

TGD predicts the existence of dark variants of basic biomolecules DNA, RNA, tRNA, and amino-acids (AAs).
One can ask whether something very simple could be imagined by utilizing the potential provided by dark variants of bio-molecules present already from beginning and providing both genes and metabolism simultaneously.

One can start from a couple of observations which forced myself to clarify myself some aspects of TGD view and also to develop an alternative vision about prebiotic period.

1. Viruses are probable predecessors of cellular life. So called positive sense single stranded RNA (ssRNA) associated with viruses can form temporarily double strands and in this state replicate just like DNA (see this). The resulting single stranded RNA can in turn be translated to proteins by using ribosomal machinery. RNA replication takes place in so called viral replication complexes associated with internal cell membranes, and is catalyzed by proteins produced by both virus and host cell.
   

   Could ribozyme molecules have catalyzed RNA replication during RNA era? For this option AA translation would have emerged later and the storage of genetic information to DNA only after that. There is however the question about the emergence of AAs and of course, DNA and RNA. Which selected just them from enormous variety of options.
   
   

2. Lipid membranes are formed by self-organization process from lipids and emerge spontaneously without the help of genetic machinery. It would be surprising if prebiotic life would not have utilized this possibility. This idea leads to the notion of lipid life as a predecessor of RNA life. In this scenario metabolism would have preceded genes (see this and this).
   
   

Consider now the situation in TGD.

1. The dark variants of DNA, RNA, AA, and tRNA would provide the analogs of genes and all basic biomolecules present from the beginning together with lipid membranes whose existence is not a problem. They would also provide a mechanism of metabolism in which energy feed by (say) solar radiation creates so called exclusion zones (EZs) of Pollack in water bounded by a hydrophilic substance. EZs are negatively charged regions of water giving rise to a potential gradient (analog of battery) storing chemically the energy provided by sunlight and the formation of these regions gives rise to dark nuclei at magnetic flux tubes with scaled down binding energy.
   

   When the p-adic length scale of these dark nuclei is liberated binding energy is liberated as metabolic energy so that metabolic energy feed giving basically rise to states with non-standard value h_eff/h=n of Planck constant is possible. For instance, processes like protein folding and muscle contraction could correspond to this kind of reduction of h_eff liberating energy and also a transformation of dark protons to ordinary protons and disappearance of EZs.
   

   The cell interiors are negatively charged and this is presumably true for the interiors of lipid membranes in general and they would therefore correspond to EZs with part of protons at magnetic flux tubes as dark nuclei representing dark variants of basic biomolecules. Already this could have made possible metabolism, the chemical storage of metabolic energy to a potential gradient over the lipid membrane, and also the storing of the genetic information to dark variants of biomolecules at the magnetic flux tubes formed in Pollack effect.
   
   

2. Biochemistry would have gradually learned to mimic dark variants of basic processes as a kind of shadow dynamics. Lipid membranes could have formed spontaneously in water already during prebiotic phase when only dark variants of DNA, RNA, AAs and tRNA, water, and lipids and some simple bio-molecules could have been present. The dark variants of replication, transcription and translation would have been present from the beginning and would still provide the templates for these processes at the level of biochemistry.
   

   Dark-dark pairing would rely on resonant frequency pairing by dark photons and dark-ordinary pairing to resonant energy pairing involving transformation of dark photon to ordinary photon. The direct pairing of basic biomolecules with their dark variants by resonance mechanism could have led to their selection explaining the puzzle of why so few biomolecules survived.
   

   This is in contrast with the usual view in which the emergence of proteins would have required the emergence of translation machinery in turn requiring enzymes as catalyzers so that one ends up with hen-or-egg question: which came first, the translation machinery or proteins. In RNA life option similar problem emerges since RNA replication must be catalyzed by ribozymes.
   
   

3. Gradually DNA, RNA, tRNA, and AA would have emerged by pairing with their dark variants by resonance mechanism. The presence of lipid membranes could have been crucial in catalyzing this pairing. Later ribozymes could have catalyzed RNA replication by the above mentioned mechanism during RNA era: note however that the process could be only a shadow of much simpler replication for dark DNA. One can even imagine membrane RNAs as analogs of membrane proteins serving as receptors giving rise to ionic channels. Note however that in TGD framework membrane proteins could have emerged very early via their pairing with dark AA associated with the membrane. These membrane proteins and their RNA counterparts could have evolved into transcription and translation machineries.
   

   DNA molecules would have emerged through pairing with dark DNA molecules. The difference between deoxi-ribose and ribose would correspond to the difference between dark RNA and dark DNA manifesting as different cyclotron frequencies and energies making possible the resonant pairing for frequencies and energies. Proteins would have emerged as those proteins able to pair resonantly with dark variants of amino-acid sequences without any pre-existing translational machinery. It is difficult to say in which order the basic biomolecules would have emerged. They could have emerged even simultaneously by resonant pairing with their dark variants.
   
   

See either the article New results in the model of bio-harmony or the article Getting philosophical: some comments about the problems of physics, neuroscience, and biology.

For a summary of earlier postings see Latest progress in TGD.

Homeostasis and zero energy ontology

Homeostasis means that system is able to preserve its flow equilibrium under changing conditions. This involves many-layered hierarchies of pairs of control signals with opposite effects so that the system stays in equilibrium. For instance, we could not stand without this control system as one can easily check by using non-living test body! For instance, in bio-chemical homeostasis the ratios of concentrations remain constant. It is not at all clear whether ordinary chemistry can explain homeostasis.

In zero energy ontology (ZEO) one can imagine very fundamental mechanism of homeostasis.

1. Zero energy states are pairs of ordinary 3-D states with members located at opposite boundaries of causal diamond (CD). Their total quantum numbers are opposite, which is only a manner to say that conservation laws hold true. The space-time surfaces connecting the 3-surfaces are preferred extremals of the action principle.
   

   In quantum field theory this picture can be seen only as a book keeping trick and one assumes that space-time continues beyond causal diamond. There is however no need for this in TGD framework although it is natural to assume that there is some largest CD beyond which space-time surfaces do not continue. CDs form a hierarchy and sub-CDs of this CD can be connected by minimal surfaces, which are analogs of external particles. One obtains networks analogs to twistor Grassmannian diagrams.
   
   

2. Conscious entities (selves) correspond in ZEO to a sequences of state function reductions having interpretation as weak measurements, "small" state function reductions. In given weak measurement the members of the zero energy state at the passive boundary of CD are not affected: this is essentially Zeno effect associated with repeated measurements in ordinary quantum theory. The members of the state pairs at the active boundary of CD change and also the temporal distance between the tips of CD increases: this assigns a clock time to the experienced flow of time as sequence of state function reductions.
   

   Eventually it becomes impossible to find observables, whose measurement would leave the passive parts of the zero energy state invariant. First "big" state function reduction changing the roles of active and passive boundaries of CD takes place and time begins to run in opposite direction since the formerly passive boundary recedes away from the formerly active boundary which is now stationary. Self dies and re-incarnates with an opposite arrow of time. In TGD biology these two time-reversed selves are proposed to correspond to motor actions and sensory perceptions.
   

   Already Fantappie realized that two arrows of time seem to be present in living matter (consider only spontaneous assembly of bio-molecules as decay in opposite direction) and introduced the notion of syntropy as time-reversed entropy. For an observer with given arrow of time, a system with opposite arrow of time seems to break the second law. Temperature and concentration gradients develop, system self-organizes.
   
   

3. These two quantal time evolutions with opposite arrows of time look very much like competing control signals in homeostasis. The 4-D conscious entities corresponding to control signals would have finite lifetime so that in their ensemble the effects of the signals with opposite arrows of time tend to compensate. This would give rise to homeostasis.
   
   

See the article Getting philosophical: some comments about the problems of physics, neuroscience, and biology or the chapter Quantum Mind, Magnetic Body, and Biological Body.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Which came first: genes or metabolism?

The key hen-egg question in biology is which came first, genes or metabolism. Both seem to need the other one. Genes-first view has led to the notion of RNA era as prebiotic biology. Metabolism-first view assumes that lipid life preceded the life controlled by basic biomolecules. In TGD framework this hen-egg question disappears.

1. Dark variants of DNA, RNA, tRNA, and AAs are dark proton sequences representing dark nuclei with scaled down nuclear binding energy. They come in different length scales and their decay to dark nuclei with shorter scale liberates dark nuclear binding energy, which could have been used as metabolic energy during the prebiotic period and could even in recent biology. Therefore genes and metabolic genes would have emerged simultaneously.
   
   
2. In so called Pollack effect the irradiation of a water bounded by a hydrophilic material creates negatively charged exclusion zone (EZ). Positive charge must go somewhere and the TGD based proposal is that it goes to magnetic flux tubes and forms dark proton sequences provoding a representation of DNA,RNA,tRNA,AA and of vertebrate genetic code.
   

   This effect occurs for several light wavelengths but infrared light present in thermal radiation provides the most effective manner to generate the charge separation. One can say that charge separation generates a battery so that the energy of light is transformed to a usable chemical energy. EZs have presumably preceded cell membrane (cell interior has negative charge).
   

   According to Pollack protein unfolding and folding could involve Pollack effect and its reverse. Also muscle contraction liberating energy could involve the reverse of Pollack effect. This mechanism works also for humans and one can even say that photosynthesis which induces charge separation by splitting water to H⁺ and OH^- works also for animals but in somewhat more complex manner. This could explain the positive effects of sunlight on well-being. Some spiritually oriented people are even claimed to survive by using only sunlight as metabolic energy: light and water would be enough.
   
   

3. In this framework one can consider the possibility of lipid membranes living in symbiosis with dark variants DNA, RNA, tRNA, and AAs during prebiotic era. Membranes would have generated charge separation by Pollack effect in order to transform the energy of light to chemical energy usable as metabolic energy. The resulting dark nuclei at magnetic flux tubes would have given rise to a dark realization of genetic code realized later chemically perhaps in terms of some simpler molecules attached to the dark variants of DNA, RNA or AA. Replication at this period could have been simply splitting of the membrane to two pieces perhaps induced by the splitting of the magnetic body in the manner described.
   
   

See the article Getting philosophical: some comments about the problems of physics, neuroscience, and biology or the chapter Quantum Mind, Magnetic Body, and Biological Body.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Two new findings related to high Tc super-conductivity

I learned simultaneously about two findings related to high Tc super-conductivity leading to a proposal of a general mechanism of bio-control in which small signal can serve as a control knob inducing phase transition producing macroscopically quantum coherent large h_eff phases in living matter.

1. High Tc superconductivity at room temperature and pressure

Indian physicists Kumar Thapa and Anshu Pandey have found evidence for superconductivity at ambient (room) temperature and pressure in nanostructures (see this). There are also earlier claims about room temperature superconductivity that I have discussed in my writings.

1.1 The effect

Here is part of the abstract of the article of Kumar Thapa and Anshu Pandey.

We report the observation of superconductivity at ambient temperature and pressure conditions in films and pellets of a nanostructured material that is composed of silver particles embedded into a gold matrix. Specifically, we observe that upon cooling below 236 K at ambient pressures, the resistance of sample films drops below 10-4 Ohm, being limited by instrument sensitivity. Further, below the transition temperature, samples become strongly diamagnetic, with volume susceptibilities as low as -0.056. We further describe methods to tune the transition to temperatures higher than room temperature.

During years I have developed a TGD based model of high Tc superconductivity and of bio-superconductivity (see this and this).

Dark matter is identified as phases of ordinary matter with non-standard value h_eff/h=n of Planck constant (see this) (h=6h₀ is the most plausible option). Charge carriers are h_eff/h₀=n dark macroscopically quantum coherent phases of ordinary charge carriers at magnetic flux tubes along which the supra current can flow. The only source of dissipation relates to the transfer of ordinary particles to flux tubes involving also phase transition changing the value of h_eff.

This superconductivity is essential also for microtubules exhibit signatures for the generation of this kind of phase at critical frequencies of AC voltages serving as a metabolic energy feed providing for charged particles the needed energy that they have in h_eff/h₀=n phase.

Large h_eff phases with same parameters than ordinary phase have typically energies large than ordinary phase. For instance. Atomic binding energies scale like 1/h_eff² and cyclotron energies and harmonic oscillator energies quite generally like h_eff. Free particle in box is however quantum critical in the sense that the energy scale E= hbar_eff²/2mL² does not depend on the h_eff if one has L∝ h_eff. At space-time level this is true quite generally for external (free) particles identified as minimal 4-surfaces. Quantum criticality means independence on various coupling parameters.

What is interesting is that Ag and Au have single valence electron. The obvious guess would be that valence electrons become dark and form Cooper pairs in the transition to superconductivity. What is interesting that the basic claim of a layman researcher David Hudson is that ORMEs or mono-atomic elements as he calls them include also Gold. These claims are not of course taken seriously by academic researchers. In the language of quantum physics the claim is that ORMEs behave like macroscopic quantum systems. I decided to play with the thought that the claims are correct and this hypothesis served later one of the motivations for the hypothesis about dark matter as large h_eff phases: this hypothesis follows from adelic physics (see this), which is a number theoretical generalization of ordinary real number based physics.

TGD explanation of high Tc superconductivity and its biological applications strongly suggest that a feed of "metabolic" energy is a prerequisite of high Tc superconductivity quite generally. The natural question is whether experimenters might have found something suggesting that the external energy feed - usually seen as a prerequisite for self-organization - is involved with high Tc superconductivity. During same day I got FB link to another interesting finding related to high Tc superconductivity in cuprates and suggesting positive answer to this question!

1.2 The strange observation of Brian Skinner about the effect

After writing the above comments I learned from a popular article (see this) about and objection (see this) challenging the claimed discovery (see this). The claimed finding received a lot of attention and physicist Brian Skinner in MIT decided to test the claims. At first the findings look quite convincing to him. He however decided to look for the noise in the measured value of volume susceptibility χ_V. χ_V relates the magnetic field B in superconductor to the external magnetic field B_ext via the formulate B= (1+χ_V)B_ext (in units with μ₀=1 one has B_ext=H, where H is used usually).

For diamagnetic materials χ_V is negative since they tend to repel external magnetic fields. For superconductors one has χ_V=-1 in the ideal situation. The situation is not however ideal and stepwise change of χ_V from χ_V=0 to χ_V to some negative value but satisfying |μ_V| <1 serves as a signature of high Tc superconductivity. Both superconducting and ordinary phase would be present in the sample.

Figure 3a of the article of authors gives χ_V as function of temperature for some values of B_ext with the color of the curve indicating the value of B_ext. Note that μ_V depends on B_ext, whereas in strictly linear situtation it would not do so. There is indeed transition at critical temperature T_c= 225 K reducing χ_V=0 to negative value in the range χ_V ∈ [-0.05 ,-.06 ] having no visible temperature dependence but decreasing somewhat with B_ext.

The problem is that the fluctuations of χ_V for green curve (B_ext=1 Tesla) and blue curve (B_ext=0.1 Tesla) have the same shape. With blue curve only only shifted downward relative to the green one (shifting corresponds to somewhat larger dia-magnetism for lower value of B_ext). If I have understood correctly, the finding applies only to these two curves and for one sample corresponding to Tc= 256 K. The article reports superconductivity with Tc varying in the range [145,400] K.

The pessimistic interpretation is that this part of data is fabricated. Second possibility is that human error is involved. The third interpretation would be that the random looking variation with temperature is not a fluctuation but represents genuine temperature dependence: this possibility looks infeasible but can be tested by repeating the measurements or simply looking whether it is present for the other measurements.

1.3 TGD explanation of the effect found by Skinner

One should understand why the effect found by Skinner occurs only for certain pairs of magnetic fields strengths B_ext and why the shape of pseudo fluctuations is the same in these situations.

Suppose that B_ext is realized as flux tubes of fixed radius. The magnetization is due to the penetration of magnetic field to the ordinary fraction of the sample as flux tubes. Suppose that the superconducting flux tubes assignable 2-D surfaces as in high Tc superconductivity. Could the fraction of super-conducting flux tubes with non-standard value of h_eff - depends on magnetic field and temperature in predictable manner?

The pseudo fluctuation should have same shape as a function temperature for the two values of magnetic fields involved but not for other pairs of magnetic field strengths.

1. Concerning the selection of only preferred pairs of magnetic fields Haas-van Alphen effect gives a
   clue. As the intensity of magnetic field is varied, one observes so called de Haas-van Alphen effect (see this) used to deduce the shape of the Fermi sphere: magnetization and some other observables vary periodically as function of 1/B. In particular, this is true for χ_V.
   

   The value of P is
   

   P_H-A== 1/B_H-A= 2π e/hbar S_e ,
   

   where S_e is the extremum Fermi surface cross-sectional area in the plane perpendicular to the magnetic field and can be interpreted as area of electron orbit in momentum space (for illustration see this).
   

   Haas-van Alphen effect can be understood in the following manner. As B increases, cyclotron orbits contract. For certain increments of 1/B n+1:th orbit is contracted to n:th orbit so that the sets of the orbits are identical for the values of 1/B, which appear periodically. This causes the periodic oscillation of say magnetization. From this one learns that the electrons rotating at magnetic flux tubes of B_ext are responsible for magnetization.
   
   
   

2. One can get a more detailed theoretical view about de Haas-van Alphen effect from the article of Lifschitz and Mosevich (see this). In a reasonable approximation one can write
   

   P= e× ℏ/m_eE_F = [4α/3^2/3π^1/3]× [1/B_e] , B_e == e/a_e² =[x^-216 Tesla ,
   

   a_e= (V/N)^1/3= =xa , a=10^-10 m .
   

   Here N/V corresponds to valence electron density assumed to form free Fermi gas with Fermi energy E_F= ℏ²(3pi²N/V)^2/3/2m_e. a=10^-10 m corresponds to atomic length scale. α≈ 1/137 is fine structure constant. For P one obtains the approximate expression
   

   P≈ .15 x² Tesla^-1 .
   

   If the difference of Δ (1/B_ext) for B_ext=1 Tesla and B_ext=.1 Tesla correspond to a k-multiple of P, one obtains the condition
   

   kx² ≈ 60 .
   
   

3. Suppose that B_ext,1=1 Tesla and B_ext,1=.1 Tesla differ by a period P of Haas-van Alphen effect. This would predict same value of χ_V for the two field strengths, which is not true. The formula used for χ_V however holds true only inside given flux tube: call this value χ_V,H-A.
   

   The fraction f of flux tubes penetrating into the superconductor can depend on the value of B_ext and this could explain the deviation. f can depend also on temperature. The simplest guess is that that two effects separate:
   

   χ_V= χ_V,H-A(B_H-A/B_ext)× f(B_ext,T) .
   

   Here χ_V,H-A has period P_H-A as function of 1/B_ext and f characterizes the fraction of penetrated flux tubes.
   
   

4. What could one say about the function f(B_ext,T)? B_H-A=1/P_H-A has dimensions of magnetic field and depends on 1/B_ext periodically. The dimensionless ratio E_c,H-A/T of cyclotron energy E_c,H-A= hbar eB_H-A/m_e and thermal energy T and B_ext could serve as arguments of f(B_ext,T) so that one would have
   

   f(B_ext,T)=f₁(B_ext)f₂(x) ,

   x=T/E_H-A(B_ext)) .
   

   One can consider also the possibility that E_c,H-A is cyclotron energy with hbar_eff=nh₀ and larger than otherwise. For h_eff=h and B_ext= 1 Tesla one would have E_c= .8 K, which is same order of magnitude as variation length for the pseudo fluctuation. For instance, periodicity as a function of x might be considered.
   

   If B_ext,1=1 Tesla and B_ext,1=.1 Tesla differ by a period P one would have
   

   χ_V(B_ext,1,T)/χ_V(B_ext,2,T) =f₁(B_ext,1)/f₁(B_ext,2)
   

   independently of T. For arbitrary pairs of magnetic fields this does not hold true. This property and also the predicted periodicity are testable.
   
   
   

2. Transition to high Tc superconductivity involves positive feedback

The discovery of positive feedback in the transition to hight Tc superconductivity is described in the popular article " Physicists find clues to the origins of high-temperature superconductivity" (see this). Haoxian Li et al at the University of Colorado at Boulder and the Ecole Polytechnique Federale de Lausanne have published a paper on their experimental results obtained by using ARPES (Angle Resolved Photoemission Spectroscopy) in Nature Communications (see this).

The article reports the discovery of a positive feedback loop that greatly enhances the superconductivity of cupra superconductors. The abstract of the article is here.

Strong diffusive or incoherent electronic correlations are the signature of the strange-metal normal state of the cuprate superconductors, with these correlations considered to be undressed or removed in the superconducting state. A critical question is if these correlations are responsible for the high-temperature superconductivity. Here, utilizing a development in the analysis of angle-resolved photoemission data, we show that the strange-metal correlations don’t simply disappear in the superconducting state, but are instead converted into a strongly renormalized coherent state, with stronger normal state correlations leading to stronger superconducting state renormalization. This conversion begins well above Tc at the onset of superconducting fluctuations and it greatly increases the number of states that can pair. Therefore, there is positive feedback––the superconductive pairing creates the conversion that in turn strengthens the pairing. Although such positive feedback should enhance a conventional pairing mechanism, it could potentially also sustain an electronic pairing mechanism.

The explanation of the positive feedback in TGD TGD framework could be following. The formation of dark electrons requires "metabolic" energy. The combination of dark electrons to Cooper pairs however liberates energy. If the liberated energy is larger than the energy needed to transform electron to its dark variant it can transform more electrons to dark state so that one obtains a spontaneous transition to high Tc superconductivity. The condition for positive feedback could serve as a criterion in the search for materials allowing high Tc superconductivity.

The mechanism could be fundamental in TGD inspired quantum biology. The spontaneous occurrence of the transition would make possible to induce large scale phase transitions by using a very small signal acting therefore as a kind of control knob. For instance, it could apply to bio-superconductivity in TGD sense, and also in the transition of protons to dark proton sequences giving rise to dark analogs of nuclei with a scaled down nuclear binding energy at magnetic flux tubes explaining Pollack effect. This transition could be also essential in TGD based model of "cold fusion" based also on the analog of Pollack effect. It could be also involved with the TGD based model for the finding of macroscopic quantum phase of microtubules induced by AC voltage at critical frequencies (see this).

See the article Two new findings related to high Tc super-conductivity or the chapter Quantum criticality and dark matter of "Hyper-finite factors, p-adic length scale hypothesis, and dark matter hierarchy".

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

How to describe family replication phenomenon gauge theoretically?

In TGD framework family replication phenomenon is described topologically (see this). The problem is to modify the gauge theory approach of the standard model to model to describe family replication phenomenon at QFT limit.

1. Identification of elementary particles

1.1 Original picture

The original view about family replication phenomenon assumed that fermions correspond to single boundary component of the space-time surface (liquid bubble is a good analogy) and thus characterized by genus g telling the number of handles attached to the sphere to obtain the bubble topology.

1. Ordinary bosons would correspond to g=0 (spherical) topology and the absorption/emission of boson would correspond to 2-D topological sum in either time direction. This interpretation conforms with the universality of ordinary ew and color interactions.
   
   
2. The genera of particle and antiparticle would have formally opposite sign and the total genus would be conserved in the reaction vertices. This makes sense if the annihilation of fermion and anti-fermion to g=0 boson means that fermion turns backwards in time emitting boson. The vertex is essentially 2-D topological sum at criticality between two manifold topologies. In the vertex 2-surface would be therefore singular manifold. The analogy to closed string emission in string model is obvious.
   
   

1.2 The recent vision

Later the original picture was replaced with a more complex identification.

1. Fundamental particles - partons - serving as building bricks of elementary particles are partonic 2-surfaces identified as throats of wormhole contacts at which the Euclidian signature of the induced metric of the wormhole contact changes to Minkowskian one. The orbit of partonic 2-surface corresponds to a light-like 3-surface at which the Minkowskian signature of the induced metric changes to Euclidian, and carries fermion lines defining of boundaries of string world sheets. Strings connect different wormhole throats and mean generalization of the notion of point like particle leading to the notion of tensor network (see this).
   

   Elementary particles are pairs of two wormhole contacts. Both fermions and bosons are pairs of string like flux tubes at parallel space-time sheets and connected at their ends by CP₂ sized wormhole contacts having Euclidian signature of induced metric. A non-vanishing monopole flux loop runs around the extrenely flattened rectangle loop connecting wormhole throats at both space-time sheets and traverses through the contacts.
   
   

2. The throats of wormhole contacts are characterized by genus given by the number g of handles attached to sphere to get the topology. If the genera g_a,g_b of the opposite throats of given wormhole contact are same, one can assign genus to it : g=g_a=g_b. This can be defended by the fact, that the distance between the throats is given by CP₂ length scale and thus extremely short so that g_a≠ g_b implies strong gradients and by Uncertainty Principle mass of order CP₂ mass.
   

   If the genera of the two wormhole contacts are same: g₁=g₂, one one can assign genus g to the particle. This assumption is more questionable if the distance between contacts is of order of Compton length of the particle. The most general assumption is that all genera can be different.
   
   

3. There is an argument for why only 3 lowest fermion generations are observed (see this). Assume that the genus g for all 4 throats is same. For g=0,1,2 the partonic 2-surfaces are always hyper-elliptic allowing thus a global conformal Z₂ symmetry. Only these 3 2-topologies would be realized as elementary particles whereas higher generations would be either very heavy or analogous to many-particle states with a continuum mass spectrum. For the latter option g=0 and g=1 state could be seen as vacuum and single particle state whereas g=2 state could be regarded as 2-particle bound state. The absence of bound n-particle state with n>2 implies continuous mass spectrum.
   
   
4. Fundamental particles would wave function in the conformal moduli space associated with its genus (Teichmueller space). For fundametal fermions the wave function would be strongly localized to single genus. For ordinary bosons one would have maximal mixing with the same amplitude for the appearance of wormhole throat topology for all genera g=0,1,2. For the two other u(3)_g neutral bosons in octet one would have different mixing amplitudes and charge matrices would be orthogonal and universality for the couplings to ordinary fermions would be broken for them. The evidence for the breaking of the universality (see this) is indeed accumulating and exotic u(3)_g neutral gauge bosons giving effectively rise to two additional boson families could explain this.
   
   

2. Two questions related to bosons and fermions

What about gauge bosons and Higgs, whose quantum numbers are carried by fermion and anti-fermion (or actually a superposition of fermion-anti-fermion pairs). There are two options.

1. Option I: The fermion and anti-fermion for elementary boson are located at opposite throats of wormhole contact as indeed assumed hitherto. This would explain the point-likeness of elementary bosons. u(3) charged bosons having different genera at opposite throats would have vanishing couplings to ordinary fermions and bosons. Together with large mass of g_a≠ g_b wormhole contacts this could explain why g_a≠ g_b bosons and fermions are not observed and would put the Cartan algebra of u(3)_g in physically preferred position. Ordinary fermions would effectively behave as u(3)_g triplet.
   
   
2. Option II: The fermion and anti-fermion for elementary boson are located at throats of different wormhole contacts making them non-point like string like objects. For hadron like stringy objects, in particular graviton, the quantum numbers would necessarily reside at both ends of the wormhole contact if one assumes that single wormhole throats carries at most one fermion or anti-fermion. For this option also ordinary fermions could couple to (probably very massive) exotic bosons different genera at the second end of the flux tube.
   
   

There are also two options concerning the representation of u(3)_g assignal to fermions corresponding ot su(3)_g triplet 3 and 8⊕ 1.

Option I: Since only the wormhole throat carrying fermionic quantum numbers is active and since fundamental fermions naturally correspond to u(3)_g triplets, one can argue that the wormhole throat carrying fermion quantum number determines the fermionic u(3)_g representation and should be therefore 3 for fermion and 3bar anti-fermion.

At fundamental level also bosons would in the tensor products of these representations and many-sheeted description would use these representations. Also the description of graviton-like states involving fermions at all 4 wormhole throats would be natural in this framework. At gauge theory limit sheets would be identified and in the most general case one would need U(3)_g× U(3)_g× U(3)_g× U(3)_g with factors assignable to the 4 throats.

1. The description of weak massivation as weak confinement based on the neutralization of weak isospin requires a pair of left and right handed neutrino located with ν_L and νbar_R or their CP conjugates located at opposite throats of the passive wormhole contact associated with fermion. Already this in principle requires 4 throats at fundamental level. Right-handed neutrino however carries vanishing electro-weak quantum numbers so that it is effectively absent at QFT limit.
   
   
2. Why should fermions be localized and su(3)_g neutral bosons delocalized with respect to genus? If g labels for states of color triplet 3 the localization of fermions looks natural, and the mixing for bosons occurs only in the Cartan algebra in u(3)_g framework: only u(3)_g neutral states an mix.
   
   

Option II: Also elementary fermions belong to 8+1. The simplest assumption is that both fermions and boson having g₁≠ g₂ have large mass. In any case, g₁≠ g₂ fermions would couple only to u(3)_g charged bosons. Also for this option ordinary bosons with unit charge matrix for u(3)_g would couple in a universal manner.

1. The model for CKM mixing (see this) would be modified in trivial manner. The mixing of ordinary fermions would correspond to different topological mixings of the three states su(3)-neutral fermionic states for U and D type quarks and charged leptons and neutrinos. One could reduce the model to the original one by assuming that fermions do not correspond to generators Id, Y, and I₃ for su(3)_g but their linear combinations giving localization to single valued of g in good approximation: they would correspond to diagonal elements e_aa, a=1,2,3 corresponding to g=0,1,2.
   
   
2. p-Adic mass calculations (see this) assuming fixed genus for fermion predict an exponential sensitivity on the genus of fermion. In the general case this prediction would be lost since one would have weighted average over the masses of different genera with g=2 dominating exponentially. The above recipe would cure also this problem. Therefore it seems that one cannot distinguish between the two options allowing g₁≠ g₂. The differences emerge only when all 4 wormhole throats are dynamical and this is the case for graviton-like states (spin 2 requires all 4 throats to be active).
   
   

The conclusion seems to be that the two options are more or less equivalent for light fermions. In the case of exotic fermions expected to be extremely heavy the 8+1 option looks more natural. At this limit however QFT limit need not make sense anymore.

3. Reaction vertices

Consider next the reaction vertices for the option in which particles correspond to string like objects identifiable as pairs of flux tubes at opposite space-time sheets and carrying monopole magnetic fluxes and with ends connected by wormhole contacts.

1. Reaction vertex looks like a simultaneous fusing of two open strings along their ends at given space-time sheets. The string ends correspond to wormhole contacts which fuse together completely. The vertex is a generalization of a Y-shaped 3-vertex of Feynman diagram. Also 3-surfaces assignable to particles meet in the same manner in the vertex. The partonic 2-surface at the vertex would be non-singular manifold whereas the partonic orbit would be singular manifold in analogy with Y shaped portion of Feynman diagram.
   
   
2. In the most general case the genera of all four throats involved can be different. Since the reaction vertex corresponds to a fusion of wormhole contacts characterized in the general case by (g₁,g₂), one must have (g₁,g₂)=(g₃,g₄). The rule would correspond in gauge theory description to the condition that the quark and antiquark su(3)_g charges are opposite at both throats in order to guarantee charge conservation as the wormhole contact disappears.
   
   
3. One has effectively pairs of open string fusing along their and and the situation is analogous to that in open string theory and described in terms of Chan-Paton factors. This suggests that gauge theory description makes sense at QFT limit.
   
   
   1. If g is same for all 4 throats, one can characterize the particle by its genus. The intuitive idea is that fermions form a triplet representation of u(3)_g assignable to the family replication. In the bosonic sector one would have only u(3)_g neutral bosons. This approximation is expected to be excellent.
      
      
   2. One could allow g₁≠ g₂ for the wormhole contacts but assume same g for opposite throats. In this case one would have U(3)_g× U(3)_g as dynamical gauge group with U(3)_g associated with different wormhole contacts. String like bosonic objects (hadron like states) could be therefore seen as a nonet for u(3)_g. Fermions could be seen as a triplet.
      

      Apart from topological mixing inducing CKM mixing fermions correspond in good approximation to single genus so that the neutral members of u(3)_g nonet, which are superpositions over several genera must mix to produce states for which mixing of genera is small. One might perhaps say that the topological mixing of genera and mixing of u₃(g) neutral bosons are anti-dual.
      
      

   3. If all throats can have different genus one would have U(3)_g× U(3)_g× U(3)_g× U(3)_g as dynamical gauge group U(3)_g associated with different wormhole throats. This option is probably rather academic.
      Also fermions could be seen as nonets.
      

4. What would the gauge theory description of family replication phenomenon look like?

For the most plausible option bosonic states would involve a pair of fermion and anti-fermion at opposite throats of wormhole contact. Bosons would be characterized by adjoint representation of u(3)_g=su(3)_g× u(1)_g obtained as the tensor product of fermionic triplet representations 3 and 3bar.

1. u(1)_g would correspond to the ordinary gauge bosons bosons coupling to ordinary fermion generations in the same universal manner giving rise to the universality of electroweak and color interactions.
   
   
2. The remaining gauge bosons would belong to the adjoint representation of su(3)_g. One indeed expects symmetry breaking: the two neutral gauge bosons would be light whereas charged bosons would be extremely heavy so that it is not clear whether QFT limit makes sense for them.
   

   Their charge matrices Q_gⁱ would be orthogonal with each other (Tr(Q_gⁱQ_g^j)=0, i≠ j) and with the unit charge matrix u(1)_g charge matrix Q⁰∝ Id (Tr(Q_gⁱ)=0) assignable to the ordinary gauge bosons.These charge matrices act on fermions and correspond to the fundamental representation of su(3)_g. They are expressible in terms of the Gell-Mann matrices λ_i (see
   this).
   
   

How to describe family replication for gauge bosons in gauge theory framework? A minimal extension of the gauge group containing the product of standard model gauge group and U(3)_g does not look promising since it would bring in additional generators and additional exotic bosons with no physical interpretation. This extension would be analogous to the extension of the product SU(2)× SU(3) of the spin group SU(2) and Gell-Mann's SU(3) to SU(6)). Same is true about the separate extensions of U(2)_ew and SU(3)_c.

1. One could start from an algebra formed as a tensor product of standard model gauge algebra g= su(3)_c× u(2)_ew and algebraic structure formed somehow from the generators of u(3)_g. The generators would be
   

   J_i,a= T_i ⊗ T_a ,
   

   where i labels the standard model Lie-algebra generators and a labels the generators of u(3)_g.
   

   This algebra should be Lie-algebra and reduce to the same as associated with standard model gauge group with generators T^b replacing effectively complex numbers as coefficients. Mathematician would probably say, that standard model Lie algebra is extended to a module with coefficients given by u(3)_g Lie algebra generators in fermionic representation but with Lie algebra product for u(3)_g replaced with a product consistent with the standard model Lie-algebra structure, in particular with the Jacobi-identities.
   
   

2. By writing explicitly commutators and Jacobi identifies one obtains that the product must be symmetric: T_a• T_b= T_b• T_a and must satisfy the conditions T_a• (T_b• T_c)= T_b• (T_c• T_a)= T_c• (T_a• T_b) since these terms appear as coefficients of the double commutators appearing in Jacobi-identities
   

   [J_i,a,[J_j,b],J_k,c]]+[J_j,b,[J_k,c],J_i,a]] + [J_k,c,[J_i,a],J_j,b]]=0 .
   

   Commutativity reduces the conditions to associativity condition for the product •. For the sub-algebra u(1)³_g these conditions are trivially satisfied.
   
   

3. In order to understand the conditions in the fundamental representation of su(3), one can consider the product the su(3)_g product defined by the anti-commutator in the matrix representation provided by Gell-Mann matrices λ_a (see this and this):
   

   {λ_a,λ_b}= 43δ_a,b Id + 4d_abcλ^c , & Tr(λ_aλ_b) =2δ_ab , & d_abc= Tr(λ_aλ_b,λ_c)
   

   d_abc is totally symmetric under exchange of any pair of indices so that the product defined by the anti-commutator is both commutative and associative. The product extends to u(3)_g by defining the anti-commutator of Id with λ_a in terms of matrix product. The product is consistent with su(3)_g symmetries so that these dynamical charges are conserved. For complexified generators this means that generator and its conjugate have non-vanishing coefficient of Id.
   

   Remark: The direct sum u(n)⊕ u(n)_s formed by Lie-algebra u(n) and its copy u(n)_s endowed with the anti-commutator product • defines super-algebra when one interprets anti-commutator of u(n)_s elements as an element of u(n).
   
   

4. Could su(3) associated with 3 fermion families be somehow special? This is not the case. The conditions can be satisfied for all groups SU(n), n≥ 3 in the fundamental representation since they all allow completely symmetric structure constants d_abc as also higher completely symmetric higher structure constants d_abc... up to n indices. This follows from the associativity of the symmetrized tensor product: ((Adj⊗ Adj)_S⊗ Adj)_S =(Adj⊗ (Adj⊗ Adj)_S)_S for the adjoint representation.
   
   

To sum up, the QFT description of family replication phenomenon with the extension of the standard model gauge group would bring to the theory the commutative and associative algebra of u(3)_g as a new mathematical element. In the case of ordinary fermions and bosons and also in the case of u(3)_g neutral bosons the formalism would be however rather trivial modification of the intuitive picture.

See the article Topological description of family replication and evidence for higher gauge boson generations.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.