Considerations inspired by LLM summaries of TGD articles

Considerations inspired by LLM summaries of TGD articles

Tuomas Sorakivi prepared LLM summaries about some articles related to TGD, in particular the article (see this) in which the relation of the holography = holomorphy vision to elliptic surfaces and the notion of partonic orbits are considered.

The discussions and the LLM summaries inspired considerations related to the general view about the definition of the partonic orbits involving the conditions g₄^1/2=0 assuming generalized holomorphy and to the details related to the model for the pairs of space-time sheets connected by wormhole contacts.

What conjugation means for generalized complex coordinates?

Generalized complex structure involves hypercomplex coordinates and this involves non-trivial delicacies related to the counterpart of generalized complex conjugation.

1. The expression for g_uv involves conjugation of CP₂ cooordinates ξ^k. It is important to note that conjugation means that means

   ξ^k(u,w)→ ξ^k(v,w) .

   This is because v is the hypercomplex conjugate of u. In the conditions f_i=0, only the hypercomplex and therefore real u coordinate occurs in the functions f_i(u,w,ξ¹,ξ²), i=1,2.

2. What is the interpretation of the fact that the hypercomplex conjugation u→ v is involved? The presented model for a pair of spacetime sheets is that, for example, the upper sheet has an active coordinate u and the lower one has v. Conjugation would take from the "upper" spacetime sheet to the "lower" one if both are involved. This would indicate that the sheets are the relations of generalized complex conjugation. This is not a necessary assumption, but it is possible and I have suggested it.
3. This formal interpretation seems strange, but in ordinary complex conjugation it is like this. x+iy, y≥ 0 corresponds to the upper half plane and x-iy, y≥ 0 to the lower half plane. Conjugation takes from the upper half plane to the lower one. On the real axis y=0 the planes meet.

   So two 4-D Minkowski spacetime sheets would be generalizations of the half planes. The real axis would be the Euclidean 3-D CP₂ inside the extremal: it is not the same as the parto orbit: the language model had mixed them up. In the used H-J coordinates, u=t-z=v=t+z, that is z=0, would hold. This 3-surface in the direction of time would correspond to the world line of a particle at rest in M⁴.

Connection with particle massivation and ideas of Connes

The fact that this 3-surface inside the CP₂ type extremal is like a particle at rest necessarily means that there are 2 space-time sheets and they are connected by a wormhole contact. Massification has necessarily occurred.

1. If only one space-time sheet is involved, it is a half-plane equivalent of one of the two. Is this possible? Could the light-like 3-D orbit of the parton surface be a track edge in the Minkowski region? Is such a solution possible or are wormhole contacts and a pair of space-time sheets necessarily needed. In any case, the fermion lines would be on partonic 2-surfaces, so a partonic surface is needed.
2. Interestingly, a top French mathematician Connes ended up proposing that the Higgs mechanism in non-commutative geometry would correspond to the Minkowski space doubling in the same way. Also in TGD framework the massivation would occur in the same way!

   I have been in Schrödinger's cat-like state regarding this question: it would seem that the boundary conditions do not allow boundaries at all. On the other hand, I have also considered the possibility allowing light-like boundaries.

3. The fact that only the coordinate u or v appears in the generalized analytic functions f₁ and f₂ means that an analogy is made between the wave motion at the speed of light t-z or t=z and the coordinate on which the wave depends. In the string model, the terms left mover and right mover are used.

   The situation in which both space-time sheets are involved would correspond in the string model to the fact that a wave coming along one space-time sheet is reflected back on this three-surface of CP₂ type extremal and returns along the other space-time sheet.

   If a single sheet with light-like boundaries are possible, it would correspond to massless particles. Either a left-mover or a right-mover, but not both. On the other hand, p-adic thermodynamics predicts that photons and gravitons also have a small mass.

Testing whether the conditions g_uv=0 allow solutions

Tuomas had, using the language model, come up with a proposal to investigate whether there are analytical solutions to the condition g_uv=0 on a partonic surface. If there are, then we can be satisfied. On the other hand, it could happen that there are none. I thought about it at night and found out that such solutions really do exist. The task is to find such a simple situation that numerical calculations are not needed.

1. I already made a simplifying assumption earlier that f₂ is of the form f₂= ξ²-wⁿ. There would be no u-dependence at all. f₂=0 would give ξ₂= wⁿ. There would be no need to find the roots either.

   A more general solution would be f₂= P₂(ξ²,w) without u-dependence. Now the roots of the polynomial must be solved. This does not change the situation.

2. We could make a similar assumption for f₁, but assume u-dependence.

   f₁= f₁(ξ₁,w,u) = ξ₁- g(w,u) .

   We can simplify it even further by assuming

   g(w,u)= u h(w) .

   So we can solve ξ₁ as

   ξ₁= uh(w) .

3. Now we have everything we need to solve the condition g_uv=0.
   1. The CP₂ metric s_{ξⁱ ξ^j} is known. Here we must remember that conjugation means u→ v!

   2. The vanishing condition g_uv= 0 gives

      s_kl ∂_uξ^k ∂_v ξ^l =-1 .

   3. The non-vanishing partial derivatives are

      ∂_uξ¹=h(w)
      ∂_v ξ¹ =h(w) .

      This gives

      h(w)h(w) s₁₁ = -1 .

   4. The component of the CP₂ metric s₁₁ ≤ 0 appears in the formula (the CP₂ metric is Euclidean) and is known and is proportional to 1/(1+r²) (see this),

      r²= ξ¹ξ¹+ ξ²ξ²

      and depends on the uv via ξ¹ξ¹ = uvg(w)g(w) . The equation can be solved for the uv function in terms of a function k(w,w) deducible from the condition:

      uv= k(w,w).

      In the (u,v) plane, this is a hyperbola for the given values of w. So there are solutions. We can breathe a sigh of relief.

See the article Holography= holomorphy vision: analogues of elliptic curves and partonic orbits or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

About the TGD based models for Cambrian Explosion and the formation of planets and Moon

TGD based cosmology (see this) predicts that the cosmic expansion occurs as a sequence of rapid phase transitions, increasing the thickness of the monopole flux tubes and liberating energy as the string tension is reduced.

One application is Expanding Earth hypothesis (see this, this, this), which states that Cambrian Explosion about half billion years ago was induced by a relatively rapid increase of Earth radius by factor 2. The details of the energetics of this transition is still far from well-understood although I have considered several options (see this). The same problem appears in the model for the formation of the Moon (see this) as a transition in which a surface layer of the Earth was thrown out to form Moon by gravitational condensation.

Where did the energy needed to compensate for the decrease of the gravitational binding energy in these explosive transitions come from? A rough estimate for the gravitational binding energy of a proton at the surface of the Earth is about 1 eV. Gravitational energy is not the only energy involved so that the estimates involving only gravitational energy are very uncertain. What seems clear is that the needed energy cannot be electromagnetic.

I have already earlier proposed that the transformations of the so-called dark nuclei to ordinary nuclei and liberating almost all ordinary nuclear binding energy could explain "cold fusion" (see this). The TGD counterpart of cold fusion could also provide the energy needed to compensate for the reduction of the gravitational binding energy in the both processes.

This inspired an attempt to fuse the TGD views about the formation of the Moon and Expanding Earth hypothesis (EE) explaining CE to a unified narrative about geological and biological evolution is made by using various guidelines. The observation that angular momentum conservation predicts for both models the same result for the rotation velocity of the Earth before CE and also before the formation of the Moon, plays a key role in the model.

Zero energy ontology (ZEO) suggests that the two events correspond to subsequent "big" state function reductions (BSFRs) in astrophysical scales changing the arrow of geometric time on a scale of billions of years.

The original versions of both models made some un-necessarily strong assumptions and comparison with empirical data allows us to loosen them.

1. The recent Mars with radius near R_E/2 was taken as an analog model for the evolution of the Earth before CE. A more precise assumption takes into account that the formation of the moons of Mars took place about much later than for the Earth.

   Therefore the analogy applies only to the early geological evolution of the Earth after the formation of the Moon. This allows us to circumvent conflicts with the empirical data. The existence of the oceans, continents, and plate tectonics, not present in the recent Mars, does not lead to a conflict.

2. Angular momentum conservation was applied originally by assuming that angular momentum transfer from Moon to Earth was instantaneous so that the rotation velocity Ω of the Earth was 4 times the recent rotation velocity before CE. A more realistic assumption is that the transfer was gradual. One can however assume that the radius was roughly R_E/2 before the EE event. This allows to avoid conflicts with the empirical determinations of the rotation velocity Ω.
3. The so-called Great Uniformity, which looks mysterious in the standard physics framework, provides very direct evidence for the occurrence of the second BSFR leading to the doubling of the Earth radius. Also the very large increase of oceans conforms with the TGD view of the EE event. The Snowball Earth hypothesis used to explain the Great Uniformity is not needed.
4. The proposal that the formation of the Moon and EE event were induced by explosions associated with the core allows us to understand why the radius of the earth was reduced in the formation of the Moon and increased in the EE event. The identification of these events as "cold fusion" transforming dark nuclei to ordinary ones would have liberated a huge energy allowing to compensate for the reduction of the gravitational binding energy.
5. Zero energy ontology (ZEO) allows us to interpret the period before CE as a period with a reversed arrow of the geometric time. The paradoxical looking prediction is that the Moon was formed in the geometric future of the recent Earth! This forces a careful reconsideration of the empirical data obtained by various dating methods. Since the dating methods do not give information about the time associated with say systems with scale much larger than the Earth it seems that they are not sensitive to the arrow of the geometric time. If this is really true it means that ZEO not only solves the measurement problem but correctly predicts a change of the geometric arrow of time in the scale of billions of years.
6. During recent years it has become clear that the so called superionic phases in the mantle and core could be central for the understanding of geology. Some of the superionic phases could also have dark variants, which raises the question whether life in some exotic form is or could have existed also in the Earth's mantle and core.
7. The role of superionic phases already found to play a potential role in the interior physics of the Earth are discussed from the TGD point of view.

See the article About the TGD based models for Cambrian Explosion and the formation of planets and Moon or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Simulation hypothesis and TGD

The original simulation hypothesis did not make sense to me since I find it impossible to imagine how the simulation hypothesis could solve any problem of physics or of theory of consciousness. Living systems are of course mimicking each other all the time so that conscious simulation is a very real phenomenon.

The new view of the simulation hypothesis (see this) seems to be analogous to what the simulation of a second computer by computer means. Already in classical physics the coupling of two systems, in particular resonance coupling, produces what might be called a simulation. Complex enough simulating a simpler system can produce rather faithful simulations. This is not new but makes sense.

One can also speak of conscious simulations.

1. In TGD inspired theory of consciousness all perception as a sequence of quantum measurements produces representations of an external system and the slightly non-determinism internal degrees of freedom of the space-time surface representing conscious entities can produce this kind of simulation in the more complex system, a kind of cognitive model. The hierarchy of algebraic extensions of rationals defines the entire complexity hierarchy.
2. Holography = holomorphy hypothesis (see this and (see this) makes this view concrete. Consider as an example two systems described as roots of (f₁,f₂)=0 and say (g^{_º n}_º f₁,f₂)=(0,0). Here f_i are analytic functions of generalized complex coordinates of H=M⁴×CP₂ (one hypercomplex coordinate is involved). The latter system has for any n at its roots also (f₁,f₂)=0 for g(0)=0 and the latter system can simulate the first system exactly at the space-time level. The larger the value of n, the higher the simulatory capacities. One obtains simulations and simulations of simulations of ....
3. For elementary particles the p-adic length scale hypothesis stating that p-adic primes p near power of 2 are important could mean the following. Polynomials g with prime degree are of special interest since they cannot be decomposed with respect to _º. For any f₁,f₂ defining kind of ground state one can have any prime polynomial g of prime degree p and can form iterates g^_ºn (see this). For p = 2 or 3, one can solve the roots of the iterates g^_ºn exactly (Galois) (see this). This exceptional feature suggests that the p-adic length scale hypothesis is true for p=2 and 3 (see and they form cognitive hierarchies by iterations. p=2 is realized in particle physics and there is evidence also for p=3 in biology (see this).

See the article Classical non-determinism in relation to holography, memory and the realization of intentional action in the TGD Universe or the chapter Quartz crystals as a life form and ordinary computers as an interface between quartz life and ordinary life?.

See also the video Topological Geometrodynamics and Consciousness prepared by Marko Manninen and Tuomas Sorakivi using LLM as a tool.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Still about the energetics of the TGD based models for the formation of planets and Moon

TGD based cosmology (see this) predicts that the cosmic expansion occurs as a sequence of rapid phase transitions, increasing the thickness of the monopole flux tubes and liberating energy as the string tension is reduced.

One application is Expanding Earth hypothesis (see this, this, this), which states that Cambrian Explosion about half billion years ago was induced by a relatively rapid increase of Earth radius by factor 2. The details of the energetics of this transition is still far from well-understood although I have considered several options (see this). The same problem appears in the model for the formation of the Moon (see this) as a transition in which a surface layer of the Earth was thrown out to form Moon by gravitational condensation.

Where did the energy needed to compensate for the decrease of the gravitational binding energy in these explosive transitions come from? A rough estimate for the gravitational binding energy of a proton at the surface of the Earth is about 1 eV. Gravitational energy is not the only energy involved so that the estimates involving only gravitational energy are very uncertain. What seems clear is that the needed energy cannot be electromagnetic.

I have already earlier proposed that the transformations of the so-called dark nuclei to ordinary nuclei and liberating almost all ordinary nuclear binding energy could explain "cold fusion" (see this). The TGD counterpart of cold fusion could also provide the energy needed to compensate for the reduction of the gravitational binding energy in the both processes.

See the article About the TGD based models for Cambrian Explosion and the formation of planets and Moon or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Does the Lorentz invariance for p-adic mass calculations require the p-adic mass squared values to be Teichmüller elements?

p-Adic mass calculations involve canonical identification I: x= ∑_nx_npⁿ ∑ x_np^-n mapping the p-adic values of mass squared to real numbers. The momenta p_i at the p-adic side are mapped to real momenta I(p_i) at the real side. Lorenz invariance requires I(p_i· p_j)= I(p_i)· I(p_j). The predictions for mass squared values should be Lorentz invariant. The problem is that without additional assumptions the canonical identification I does not commute with arithmetics operations.

Sums are mapped to sums and products to products only at the limit of large p-adic primes p and mass squared values, which correspond to x_n≤≤ p. The p-adic primes are indeed large: for the electron one has p= M₁₂₇=2¹²⁷-1∼ 10³⁸. In this approximation, the Lorentz invariant inner products p_i· p_j for the momenta at the p-adic side are indeed mapped to the inner products of the real images: I(p_i· p_j)= I(p_i)· I(p_j). This is however not generally true.

1. Should this failure of Lorentz invariance be accepted as being due to the approximate nature of the p-adic physics or could it be possible to modify the canonical identification? It should be also noticed that in zero energy ontology (see this), the finite size of the causal diamond (CD) (see this) reduces Lorent symmetries so that they apply only to Lorenz group leaving invariant either vertex of the CD.
2. Or could one consider something more elegant and ask under what additional conditions Lorentz invariance is respected in the sense that inner products for momenta on the p-dic side are mapped to inner products of momenta on the real side.

The so called Teichmüller elements of the p-adic number field could allow to realize exact Lorentz invariance.

1. Teichmüller elements T(x) associated with the elements of a p-adic number field satisfy x^p=x, and define therefore a finite field G_p, which is not the same as that given by p-adic integers modulo p. Teichmüller element T(x) is the same for all p-adic numbers congruent modulo p and involves an infinite series in powers of p.

   The map x→ T(x) respects arithmetics. Teichmüller elements of for the product and sum of two p-adic integers are products and sums of their Teichmüller elements: T(x₁+x₂)= T(x₁)+T(x₂) and T(x₁x₂)= T(x₁)T(x₂).

2. If the thermal mass squared is Teichmüller element, it is possible to have Lorentz invariance in the sense that the p-adic mass squared m²_p= p^kp_k defined in terms of p-adic momenta p_k is mapped to m²_R=I(m²_p) satisfying I(m²_p)= I(p^k)I(p_k). Also the inner product p₁· p₂ of p-adic momenta mapped to I(p₁· p₂)=I(p₁)· I(p₂) if the momenta are Teichmüller elements.
3. Should the mass squared value coming as a series in powers of p mapped to Teichmüller element or should it be equal to Teichmüller element?
   1. If the mass squared value is mapped to the Teichmüller element, the lowest order contribution to mass squared from p-adic thermodynamics fixes the mass squared completely. Therefore the Teichmüller element does not differ much from the p-adic mass squared predicted by p-adic thermodynamics. For the large p-adic primes assignable to elementary particles this is true.

   2. The radical option is that p-adic thermodynamics and momentum spectrum is such that it predicts that thermal mass squared values are Teichmüller elements. This would fix the p-adic thermodynamics apart from the choice of p-adic number field or its extension. Mass squared spectrum would be universal and determined by number theory. Note that the p-adic mass calculations predict that mass squared is of order O(p): this is however not a problem since one can consider the m²/p.

This would have rather dramatic physical implications.

1. If the allowed p-adic momenta are Teichmüller elements and therefore elements of G_p then also the mass squared values are Teichmüller elements. This would mean theoretical momentum quantization. This would imply Teichmüller property also for the thermal mass squared since p-adic thermodynamics in the approximation that very higher powers of p give a negligible contribution give a finite sum over Teichm\"muller elements. Number theory would predict both momentum and mass spectra and also thermal mass squared spectrum.

   What does it mean that the product of Teichmüller elements is Teichmüller element? The product xy can be written as ∑_k (xy)_k p^k, (xy)_k=∑_l x_k-ly_l. For Teichmüller elements (xy)_k has no overflow digits. This is true also for I(xy) so that I(xy)= I(x)I(y). Similar argument applies to the sum.

2. The number of possible mass squared values in p-adic thermodynamics would be equal to the p-adic prime p and the mass squared values would be determined purely number theoretically as Teichmüller representatives defining the elements of finite field G_p. The p-adic temperature (see this), which is quantized as 1/T_p=n, can have only p values 0,1,...p-1 and 1/T_p=0 corresponds to high temperature limit for which p-adic Boltzman weights are equal to 1 and the p-adic mass squared is proportional to m²= ∑ g(m) m/∑g(m), where g(m) is the degeneracy of the state with conformal weight h=m. T_p=1/(p-1) corresponds to the low temperature limit for which Boltzman weights approach rapidly zero.

See the article Could the precursors of perfectoids emerge in TGD? or the chapter Does M⁸-H duality reduce classical TGD to octonionic algebraic geometry?: Part III

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

3 kielimallin avulla tehtyä videota TGD:stä ja TGD:n inspiroimasta tietoisuuden teoriasta.

Marko Manninen ja Tuomas Sorakivi tekivät kielimallia käyttänen videon TGD inspired theory of consciousness TGD:n insproimasta tietoisuuden teoriasta.

Tuomas Sorakivi teki kielimallin avulla videon Topologinen Geometrodynamiikka ja Tietoisuus liittyen TGD:n inspiroimaan tietoisuuden teoriaan ja videon Todellisuus yhtälönä liittyen holografia = holomorfia periaatteeseen.

Vaikka mukana on hypeä niin videot antavat mielestäni hyvän kokonaiskuvan siitä mistä on kyse.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Periodic table of topological insulators and topological superconductors and discrete symmetries at the space-time level

At the level of the embedding space H=M⁴× CP₂, the TGD view about discrete symmetries T, P, and C is essentially the same as in the standard model except that C has a geometric interpretation as a complex conjugation in CP₂ and has therefore a geometric meaning (see this). At the space-time level holography = holomorphy principle (see this, this, and this) forces us to reconsider the TGD based view of discrete symmetries. They would relate naturally to transformations relating different holomorphic structures differing by a conjugation of some in some complex coordinates of H or of a hypercomplex coordinate of M⁴.

How do these discrete symmetries relate to those of condensed matter physics? According to Wikipedia (see this), the periodic table of topological insulators and topological superconductors, also called tenfold classification of topological insulators and superconductors, is an application of topology to condensed matter physics. It indicates the mathematical group for the topological invariant of the topological insulators and topological superconductors, given a dimension and discrete symmetry class. The ten possible discrete symmetry families are classified according to three main symmetries: particle-hole symmetry, time-reversal symmetry and chiral symmetry. The table was developed between 2008 2010 by the collaboration of Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki and Andreas W. W. Ludwig; and independently by Alexei Kitaev. The table applies to topological insulators and topological superconductors with an energy gap, when particle-particle interactions are excluded. The table is no longer valid when interactions are included. The table applies in dimensions D=1, 2, 3. The table also gives the value set of the topological invariant with the associated symmetry, which can be Z₂, Z or 2Z in these dimensions.

Three basic symmetries appear in the 10-fold periodic table for topological insulators and superconductors and various cases are classified by the symmetries T, C, and S for which the squares of these symmetries are well-defined and by the broken symmetries. T acts as time reversal, C is the analog of charge conjugation, which transforms particles and holes to each other, and chiral transformation S transforms left-and right-handed chiralities to each other. C and T are antiunitary operators involving hermitian conjugation at the level of the Hilbert space and expressible as products of a unitary operator with a complex conjugation operator K. CPT is therefore unitary and it is possible to have CPT=1. These transformations are idempotent: T²=Id, C²=Id, S²=Id. In quantum theory, one can consider the possibility that their representations are projective so that one can have T²=-Id and C²=-Id. S²=1 holds always true.

In the table of the Wikipedia article, the violation of these symmetries is expressed using the symbol X. My understanding is that for instance, the charge conjugate need not belong to the space of allowed states. The TGD counterparts of T, S, and C at the space-time level would be naturally realized in terms of various conjugations of the complex coordinates and hypercomplex coordinate of H.

1. In the TGD framework, holography = holomorphy vision involves the notion of Hamilton-Jacobi structure for M⁴ ⊂ M⁴× CP₂ (see this) as a combination of longitudinal hypercomplex structure and transversal complex structure. The hypercomplex coordinate u has coordinate lines with light-like tangent vectors and the hypercomplex conjugation transforms u to its conjugate v (both coordinates are real).

   The complex coordinate w is transversal to (u and v). The coordinate lines define local polarization direction and light-like direction and characterize massless extremals as analogs of modes of massless fields. To simplify the situation, it is convenient to consider the simplex possible hypercomplex structure with (u= t-z,v= t+z) and w identifiable as the transversal planar complex coordinate w=x+iy.

2. There are two complex CP₂ coordinates ξⁱ but the conjugation as a symmetry applies simultaneously to both of them. CP₂ complex conjugation is naturally related to C, or at least its particle physics counterpart mapping fermions with antifermions and vice versa. In condensed physics C maps creation and annihilation operators of fermions (electrons and holes) to each other and one expects that CP₂ complex conjugation is involved.

   The complex conjugation in M⁴ cannot be involved with C. Does C involve HC? Geometric picture does not favor this. CPT =1 at the space-time level does not allow the presence of HC in C since HC is involved with both P and T realized at the space-time level assuming holography= holomorphy principle.

3. Chiral transformation S changed the chirality of say DNA strand and actd as reflection P. S would therefore naturally correspond to the HC: u→ v combined with w→ -w? Also no hypercomplex structure goes to its conjugate. This definition of S and the relation S=TC which by S²=1, is assumed in the 10-fold periodic table, corresponds to CPT=1. Together with S²=1, which is always true in the periodic table, it guarantees CST=1 as a counterpart of CPT=1. Note that neither C nor P are symmetries at the level of the Dirac equation at the embedding space level while CP is.
4. For the simplest option (u,v)=(t-z,t+z), HC combined with T corresponds to u→ -u and changes the functional form of f. This does not affect the generalized complex structure but affects the space-time spacetime surface. T corresponds to HC followed but u→ -u and conjugation of the HC structure and change of the analytic form. For P the presence of HC conjugation changes the hypercomplex structure.
5. w→ -w and (u,v)→ -(u,v) as analog of PT does not affect the generalized complex structure but affect the space-time surface since the functional form of the functions (f₁,f₂) obeying generalized holomorphy is changes. P, T, and CP affect the generalized complex structure and are always accompanied by HC. These symmetries cannot leave space-time surfaces invariant unless they consist of regions with different generalized complex structures glued together.
6. The minus sign in T² and C² could be also due an additional multiplication of the complex coordinate of H with an imaginary unit. In the case of T, -1 sign cannot appear in this way but would come from the unitary operator.

A possible interpretation of these symmetries is as space-time analogs of the basic symmetries of H representable in terms of complex and hypercomplex conjugations allowing to transform space-time surfaces satisfying holography= holomorphy principle to new such surfaces.

1. If all 3 basic conjugations are performed one obtains a new preferred extremal if longitudinal and transversal M⁴ degrees are independent. Same is true if one performs either both M⁴ conjugations or both CP₂ conjugations.
2. What is new is that the holomorphy violates these symmetries locally at the level of a single space-time surface. For instance, matter and matter could correspond to space-time regions with holomorphic structures related by CP₂ conjugation C and the formation of large space-time regions with a given holomorphic structure could correspond to matter antimatter asymmetry. This globalization of C and CP could relate the mystery of antimatter symmetry. These regions could correspond to the field bodies with large size induced antimatter asymmetry at the smaller space-time sheets associated with them. Quantum coherence in astrophysical and even cosmic scales with a quantum coherence scale characterizing the space-time surface, would be essential (see this).
3. The model for elementary particles (see this) involves in an essential manner a pair of parallel Minkoskian space-time sheets connected by wormhole contacts. How could the complex structures of the Minkowskian space-time sheets relate to each other?

   Ordinary complex surfaces are invariant under complex conjugation although the complex structure changes in the conjugation. Also the space-time surface can remain invariant under the complex conjugation although it is holomorphic with respect to the complex coordinates. What about the hypercomplex conjugation? Could the parallel space-time sheets be related by a hypercomplex conjugation HC and possibly also by the conjugation w→ w and C? If they are related by charge conjugation C (see this), fermions and antifermion lines would reside at the wormhole throats of opposite space-time sheets, which has been assumed.

4. One can also consider quantum superpositions of the space-time surfaces with different complex and hypercomplex structures. The expectation is that fermion and antifermion lines reside at the opposite wormhole throats of the space-time sheets connected by wormhole contacts. Do the space-time sheets have conjugate complex structures in both CP₂ and M⁴? Interesting questions relate to the CP breaking of K⁰ and B⁰ mesons. CP involves HC conjugating the hypercomplex structure and affecting the space-time surface: at the level H and Dirac equation this kind CP violation does not occur. Could the presence of HC explain the CP violation?

See the article TGD and Condensed matter or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

How large heff states are stabilized?

The quantum critical state is unstable by definition because the h_eff> h states are more energetic than the h_eff=h states and spontaneously decay into these.

One way to avoid this would be for the h_eff> h molecule to form a bound state, for example with a molecule or a larger structure. The electric field of the larger charged structure and that in turn a state where h_eff would be stabilized. However, I do not understand the details of the mechanism. How to build a state in which h_eff> h dark protons are possible in the minimum energy state. Is this possible if only the electromagnetic interaction is involved?

This is a fundamental question. So let's start from a clean table.

1. In the case of DNA and cell membranes, h_eff stabilization is related to the presence of electric fields, but do they produce the stabilization or are they a consequence of it?

   A h_eff> h state and a state bound with another state are created so that the h_eff> h state stabilizes because the dissociation is no longer energetically favorable. It should be noted that due to their large negative charge DNA and the cell membrane are biologically completely unique. Charge separation does also occur at the level of the brain and the whole body and its sign correlates with the level of consciousness: the sign of the voltage changes during sleep. The Earth itself also has an electric field, which suggests that the biosphere is conscious.

2. In the case of DNA, the bound state would be between phosphate and deoxyribose. Would the large h_eff=h_em somehow be made possible by the longitudinal and radial electric fields of DNA or is it a consequence of a stabilization mechanism? Maintaining the electric field requires energy, so metabolic energy input is still necessary but at the level of classical fields. But do electric fields maintain dark protons at the monopole flux tubes or vice versa?

   The problem: In the case of DNA, the repulsive energy of the negative charges of the phosphates destabilizes the state. In addition, there is repulsion between the dark protons in the flux tubes. Charge separation, where the dark protons and the phosphate ions are far apart, requires energy because the neutral ground state is of minimum energy.

   The solution of the problem: Some interaction energy must compensate for the increase in interaction energy. Could strong interactions of the dark protons in the flux tubes, proposed to form dark nuclei with a scale down nuclear binding energy, be involved? The strong interaction would stabilize the repulsive energy of the negative charge of the phosphates, the same would happen for the dark protons. Long range electric field would be a consequence, not the cause.

   1. The TGD-based model of cold fusion this, this, this, and this) indeed assumes that the dark protons in the magnetic flux tubes form an analogy of the atomic nucleus and the scaled binding energy of the nucleus would produce the binding energy. Strong interactions in the TGD sense would play a key role in biology and also in electrolysis. This would be new and revolutionary.
   2. Of course, one could try to cope with just electromagnetic interactions.

      i) The negative electrostatic energy would be between the dark protons and the negative charge of the phosphates. One would expect this energy to be small, but is it for flux tubes?
      ii) What about the role of water? It can become positively charged (and for example Mg²⁺ ions do), which can produce a Coulomb bound state. Mg²⁺ ions are naturally present in monopole flux tubes, but is the contribution large enough?
      iii) The binding energy is related to the bound state between negatively charged phosphates and riboses. The problem is that ribose molecules are not permanently positively charged. This doesn't seem promising.
      

   3. In the case of the cell membrane, the electric field associated with the membrane potential should accompany large values of h_eff. A decrease in the field strength below a critical value would lead to a decrease in the value of h_em, perhaps down to h_eff=h because h_em is proportional to the field value and quantized as an integer. The scale of quantum coherence would be reduced and a nerve impulse would be generated.

      The naive Maxwellian assumption would be that a nerve impulse is generated when the voltage is too high: there would be a di-electric breakdown, just as is supposed to happen in a Tesla coil. The fact that exactly the opposite happens is a central mystery of biology. A decrease in h_em would explain the mystery. One can pose an interesting and somewhat nosy question: has it really been tested that breakdown is the correct mechanism in Tesla coils?

      Also now the strong interactions with monopole flux tubes would stabilize the state.

   4. The negative charge on the surface of the Earth's electric field and the protons and ions in the gravitational flux tubes and electric flux tubes and their strong interaction would stabilize the biosphere as a conscious system.

See the article How the genetic code is realized at the level of the magnetic body of DNA double strand? or the About honeycombs of hyperbolic 3-space and their relation to the genetic code.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.