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.