Has LUX-ZEPLIN found a scaled up variant of electron with mass of 256 GeV behaving like dark matter?

LUX-ZEPLIN (LZ) experiment at the Sanford Underground Research Facility in Lead, South Dakota reports 3 sigma evidence for a dark matter particle candidate at mass above 200 GeV, WIMP as it is called. See the popular article, the Nature commentary and the research article.

TGD predicts a hierarchy of hadron physics labelled by ordinary Mersenne primes M_k= 2^k/-1 and their Gaussian counterparts. k=107 corresponds to ordinary hadron physics and k=89 to hadron physics with mass scales of hadrons which are expected to be by factor 512 higher. They should be reachable at LHC and there exist pieces of evidence for resonances identifiable as M₈₉ mesons (see for instance this, this and this).

These particles are predicted to be dark in the sense that the value of the effective Planck constant h_eff is larger than the value of ordinary Planck constant so that they do not appear in the same vertices as ordinary particles. M₈₉ hadrons would be created as their dark variants having the same Compton length as their M₁₀₇ counterparts. The strange properties of quark gluon plasma would reflect the quantum criticality for a transitions from M₁₀₇ hadron physics to M₈₉ hadron physics (see this and this).

This hierarchy applies also to leptons. The M₈₉ counterpart of electron would have mass 2^(127-89)/2 m_e= 256 GeV. If has h_eff/h equal to the ratio (M₁₂₇ /M₈₉)^1/2= 2¹⁹, it would behave like a dark matter particle and have the same Compton length as ordinary electron: this would be a prerequisite for quantum criticality. Note that also the existence of M₁₀₇ dark electron with mass of 500 MeV and M_G,113 dark electron with mass of 64 MeV is suggested.

See the article Comparing the S-matrix descriptions of fundamental interactions provided by standard model and TGD 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.

Is Nature a mathematician proving theorems by holography

We had an interesting discussion with Tuomas Sorakivi about metamathematics yesterday. For a mainstream physicist, mathematics is "maths": a box from which you can draw calculational methods. The physical reality behind the models is real to the physicist, and mathematics is just a tool for creating models for it.

For a Platonist, on the other hand, mathematics is real. What the mathematician himself is, remains an open question in both views. A theory of consciousness is needed. TGD proposes such a theory (see for instance this and this).

In TGD, physics represents mathematics rather than vice versa. The view of the standard physicist is turned upside down. Mathematics is Platonia identified as the "world of classical worlds" (WCW) consisting of space-time surfaces in H=M⁴× CP₂. In zero energy ontology, space-time surfaces obeying holography would represent numbers and more general mathematical objects, even theorems. Theorems can be identified as space-time surfaces which obey an almost deterministic holographic time evolution. If determinism were perfect, there would be only a single theorem - not very interesting mathematics - but this is not the case, so several theorems corresponding to different time evolutions follow from the given premises. This happens naturally. In the TGD Universe, the task of the mathematician is to become aware of the proofs that Nature produces.

This solves the problem caused by the combinatorial explosion encountered in attempts to mechanize mathematical proofs. Theorems are deduced from the axioms by rules that are analogous to the dynamic equations of classical physics. One can speak of truth-preserving dynamics. Boolean logic crystallizes these traffic rules. The problem is that these traffic rules are very weak. From given premises, a huge number of theorems can be deduced in an infinite number of ways. In practice, a mathematical machine is therefore impossible.

The combinatorial explosion can be illustrated by the set-theoretic representation of Boolean algebra. The premises correspond to the set A. The logical implication A→B corresponds to the fact that B is a subset of A. The number of subsets increases exponentially as the number of elements of A increases. There are 2^N elements among the subsets if A has N elements. The result is a combinatorial explosion.

How are human mathematicians able to derive any theorems at all? Could one think that physics comes to rescue and replaces the Boolean traffic rules with holographic dynamics for proving. Could the laws of Nature define not only the axioms but also proofs of the theorems as holographic time evolutions. Given premises A would lead to a finite number of consequences B, provided that the holographic dynamics is slightly nondeterministic. The quantum-platonism of TGD would realize this dream.

See the article G&oum;del, Lawvere, and TGD 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.

Cosmic birefringence in TGD framework

In Big Think Ethan Siegel has an article titled "Cosmic birefringence: the signal that could upend cosmology" (see this). I have considered the generalization of birefringence in the TGD framework in case of neutrinos (see this). The original proposal for bireferengence for neutrinos came from Anca Tureanu (see this).

What makes this so interesting is that from Atama Cosmology Telescope (ACT) collaboration, a new set of analyses has arrived (see this). From ACT alone, the significance rules out zero rotation angle (β=0°) at 2.9σ, and by jointly combining WMAP, Planck, and ACT data together, a new paper accepted in Physical Review D in August of 2026 claims to see β=0.277° +/- 0.057°, which is non-zero at 4.8σ significance.

Ordinary birefringence in condensed matter occurs because orthogonal linear polarizations travel with different velocities. This causes phase difference between polarizations visible as a rotation of the polarization plane.

Cosmic birefringence occurs because circular polarizations travel with different velocities.

1. The reason why for circular polarizations is that initial polarizations are linear with a fixed direction caused by the density perturbations (E-mode) with quadrupole character.

   E-mode polarization is generated by density perturbations such as sound waves, which are scalars. Quadrupole (4 poles) is realized as two pairs of sources of radiation in orthogonal directions and photons coming from these sources experience Thomson scattering from charged particles such as electrons. The scattering is dominantly in right angles. If the other source is more intense (hotter) the polarization (as statistical parameter) is linear and along the axis connecting the more intense source pair.

2. Also B-mode polarization is in principle possible and would be created by gravitational waves interacting with matter and inducing to the energy momentum tensor non-diagonal components. This would cause B-modes for which the distribution of scattering angles are characterized by spherical harmonics Y₂^+/-2 rather than Y₂^+/-1,0 as for E-modes. Y₂^+/-2 has characteristic sin(2φ) or cos(2φ) dependence implying cross-shaped polarization pattern with 4 angle maxima.
3. Cosmic birefringence would emerge in the following way. The original linear polarization can be expressed as the sum of circular polarizations. These travel with different velocities and this causes a phase shift. When the polarizations combine in a telescope, the resulting linear polarization is rotated by some angle by interference. Quantum effect in cosmic scales is in question and would require a revolution in cosmology and entire quantum physics.

The TGD explanation involves two new elements.

1. Quantum coherence is required in arbitrarily long, perhaps even cosmological length scales. The hierarchy of Planck constants, which includes gravitational Planck constant, makes this possible. So called massless extremals (MEs) as geometric counterparts of massless radiation realize precisely targeted propagation of photons along MEs as dark photons. Darkness means a very large value of gravitational Planck constant making possible very long quantum coherence length. The MEs can be also associated with the monopole flux tubes arriving from distant sources.
2. Also a polarization dependent light-velocityd ue to interactions with environment is required.

The possibility of warping is one of the basic differences between GRT and TGD. It makes possible reduced light velocity and the polarization dependence of the light velocity.

1. Warped space-time surfaces are flat like Minkowski space but the light-velocity is reduced because 3-space can be said to rotate along CP₂ geodesic. This notion emerged already during the first years of TGD but are only during last year (see this and this)). I have started to realize how deep its implications are. Warping effect occurs also for the fermion lines associated with the 3-D light-like parton orbits containing them.
2. Fermion line is a light-like geodesic of H = M⁴×CP₂ and also of space-time surface but its M⁴ projection is time-like geodesic of M⁴ and characterized by mass. Therefore warping gives rise to a space-time description of particle massivation.

3. Also Allais anomaly and the variation of gravitational constant could be understood (see this). In condensed matter physics refraction and reflection of light would involve change of the reduced light-velocity at the boundary between differently warped space-time regions.
4. Space-time surfaces are small deformations of these warped gravitational vacua (for warped gravitational vacua the g_tt component of the induced metric is constant deviating from unity and defines constant gravitational potential as analog of constant electric potential). Warping can be generalized to the level of Hamilton-Jacobi structure (see this).

Cosmic birefringence would emerge in the following way.

1. The MEs assigned to the photons arriving from a distant source are different from opposite circular polarizations because ME allows only single polarization which can be local. The linear polarized ME can be decomposed to a superposition of MEs with circular polarizations and the photons associated with these MEs propagate with different velocity and cause the rotation of the polarization plane observed in the measurement in which linear polarization is the outcome.
2. Cosmic birefrience also implies parity violation in cosmic scales. The hierarchy of effective Planck constants allows the weak parity violation to occur in arbitray long scales.

Consider now objections against this picture. In the standard picture the quantum superposition for linear polarizations corresponds to the superpositions for gauge potentials and this correspondence fails since the YM type field equations of the standard model are not linear. Can one accept this approximation?

TGD is strongly non-linear and linear superposition for the induced gauge potentials fails in the general case. Massless extremals (MEs) are however an exception.

1. MEs are characterized by two parameters: light-like wave vector k satisfying k· k=0 and polarization vector ε orthogonal to it satisfying k· ε=0. For the simplest MEs k and ε are constant. The inner product u=k·m of k with the vector m defined by linear Minkowski coordinates appears in the plane phase factor exp(ik·m).
2. This picture generalizes. One can assume a local light-like vector k(m) expressible as a gradient k(m)= ∇ u and thus defining tangents for coordinate curves of u. k· m→ u= ∫ ^m k(m)dm. Also ε can be made local polarization vector ε (m) as tangent vector to coordinate curves orthogonal to the local plane M² defined by light-like vector u.

   In the same way, the coordinate line parallel to ε (m) corresponds to v= ∫ ^m ε(m)dm. This would make possible local variants of polarization and light-like vector and make possible curvilinear photons.

   This picture was one motivation for introducing the notion of Hamilton-Jacobi (H-J)structure (see this) as a generalization of complex structure to M⁴. H-J structure involves a integrable distribution of hypercomplex planes defined by the local light-like vector and its dual and orthogonal complex planes defined by a complex local polarization vector and its conjugate.

3. For MEs, the superposition for waves propagating along ME in the same direction holds true and is only restricted by the condition that the amplitudes are small so that the CP₂ characterizing the induced fields are in the allowed range (say angle variable cos(θ) for geodesic sphere of CP₂. This restriction can be removed by using complex coordinates for CP₂. These coordinates are always possible and especially natural for CP₂ if holography = holomorphy hypothesis is satisfied (see this and this). Also waves with the same fixed local linear polarization can be superposed.
4. What about circular (elliptic) polarizations, which are superpositions of two linear polarizations which are not parallel? Are these possible?
   1. They could correspond to superpositions of solutions corresponding to two different polarization vectors ε₁(w) and ε₂(w) giving rise to complex polarization vectors ∫ ^mε₁(w)dw and ∫ ^mε₂(w)dw. Can these polarization vectors have phase difference?
   2. The second way to obtain circular polarization is as a rotating linear polarization. Could one assume that the local polarization ε depends on both w and u: ε ε(w,u)?
   I am unable to invent any obvious reason preventing either option.

One should also understand what happens when the linear superposition of two orthogonal linear polarizations assignable to a single ME transforms to superposition of differently warped MEs with different reduced light velocities.

1. The linear polarization can be expressed for ME with standard light velocity as a superposition of two circular polarizations with phase difference. The interaction with the environment must distinguish between these polarizations so that they must correspond to different MEs with different warping and reduced light-velocity.
2. This interaction must induce the entanglement of the circular polarization states with the environment. Entanglement must induce the warping of MEs and reduce the maximal signal velocity by giving to the photon a very small mass (see this). p-Adic thermodynamics indeed predicts this possibility (see this and this. The opposite circular polarizations would propagate with slightly different velocities so that the MEs must be different.

There is also the question about whether the photons from the remote source propagate in an entangled state all the way or whether state function reductions are possible during the travel.

1. Gravitational Planck constant characterizes a pair M,m of masses. The associated Compton length is proportional to the Schwartschild radius of the larger mass M associated with the pair of masses. For the Milky Way it is .47 ly and much shorter than the path travelled by photon in the case of cosmic birefringence. Therefore it seems that entanglement cannot be preserved during the entire trip. Quantum measurements during the travel can destroy it and interaction with the environment can transform it to superposition of circular polarizations re-entangling with the environment.
2. Zero energy ontology (ZEO) is forced by the slight non-determinism of the classical time evolution in TGD. ZEO solves the basic paradox of quantum measurement theory. In ZEO one can distinguish between two kinds of state function reductions (SFRs). The "small" ones (SSFRs) generalize the Zeno effect so that repeated quantum measurements do not leave the state invariant anymore. SSFRs are crucial for TGD inspired theory of consciousness and cognition and, somewhat surprisingly, also for the TGD description of fundamental interaction. The "big" SFRs (BSFRs) correspond to ordinary quantum measurements and they change the arrow of time. Either SSFRs or pairs of BSFRs changing the arrow of time temporarily could occur during the travel.

See the article Comparing the S-matrix descriptions of fundamental interactions provided by standard model and TGD 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.

Challenging the interpretation of Zero Energy Ontology

In holography = holomorphy (H-H) principle the basic objects are not 3-D surfaces but their slightly non-deterministic Bohr orbits. This forces zero energy ontology (ZEO), which solves the basic problem of quantum measurement theory. Classical non-determinism is associated with the edges of the space-time surface and plays a fundamental role in both the TGD inspired theory of conscious experience and basic quantum TGD. It is assumed that edges correspond to "small" state function reductions (SSFRs) in which quantum decoherence is assumed to occur. This also occurs in "big" SFRs (BSFRs) as the TGD counterparts of ordinary state function reductions in which also the arrow of geometric time changes. There are. however several open questions related to ZEO.

1. Are SSFRs really SFRs associated with the edges of the space-time surface or does their sequence define a TGD finite counterpart of Feynman path integral so that no decoherence would occur?
2. There are also questions related to the details of the periods consisting of SSFRs. The increase of the size of causal diamond (CD) by scalings occurring in SSFRs would allow us to understand how subjective a sequence of SSFRs corresponds to geometric time as distance between the tips of CD.

   The alternative proposal is that CD shifts gradually to the direction of the active boundary of CD at which the states do not remain fixed in SSFRs. Which option is correct? Does the size of the CD increase indefinitely or does it reduce in BSFRs so that conscious entities would experience childhood after every BSFR? Is the breaking of T symmetry (and also CP symmetry) necessary for the gradual shift of CD to a fixed time direction?

3. The feed of metabolic energy is necessary for consciousness. Sleep and wakeup can be identified as conscious states with opposite arrows of geometric time. Does the periodically varying feed of the metabolic energy from the Sun as light ,making also possible direct visual perceptions, give rise to a stochastic resonance in which the state of consciousness corresponds to flips between the potential wells of a bistable system.

See the article Challenging the interpretation of Zero Energy Ontology or the chapter Zero Energy Ontology.

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.

Could expolanet K2-18b serve as a seat of life in TGD Universe?

I have just developed a model for the evolution of geology and biology of Earth and this leads to a criterion for whether life can evolve on the planet (or moon) (see this). The criterion relies heavily on new physics predicted by TGD. The margin is very narrow. The criterion is rather tight and not satisfied for planets with too large Schwartschild radius to radius, for instance giant gas planets of the Earth.

1. The p-adic transition k=139→ k=137 from ordinary atomic physics to scaled down atomic physics (for which Mills has found evidence), having 4 times higher atomic binding energy scale than ordinary atomic physics, would reduce the atomic radius by factor 1/2 and increase the density by factor 8. This transition would induce scaling of the radius of Earth by factor 1/2 and subsequent Cambrian Explosion in which it increases by factor 2. This transition would bring life from underground oceans to the surface of the Earth.
2. The liberated atomic and gravitational binding energies make it possible in the case of Earth to throw out a surface layer condensing to form the Moon and metabolic energy burst generating dark atoms, analogous to Rydberg atoms with very large size, making possible large scale quantum coherence inside Earth. This generates p-Adic length scales, which correspond to biologically important scales. Large scale quantum coherence makes possible the evolution of life in the interior of the Earth. The condition that the CE in which expansion to the original radius occurs, can take place requires that gravitational binding energy per proton is larger than the atomic binding energy. This gives for the lower bound of atomic weight A as A>.3 which is satisfied.

The recently found exoplanet K2-18b is regarded as a candidate for a seat of life. In particular, there are indications that it has water at its surface. Its distance is 124 light years. Its radius and mass are R=2.61R_E and M= 8.6M_E. This gives r_s/R= 8.6/2.61∼ 3.3. The condition for A in the case of the Earth is scaled up from A>.3 to estimate A>.99 satisfied already for hydrogen atoms. K2-18b would be just at the border line but the condition making possible the expansion and burst of life to the surface of K2-18b, would be satisfied.

See the article Could the notions of quantum geology and quantum biology make sense? 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.

Has solar system lost two planets or is the standard model for the formation of the solar system totally wrong?

Sabine Hossenfelder (see this) told about a possible solution of a problem plaguing the standard model of the solar system. The model cannot explain the formation of planets as a gravitational condensation of a proto disk. The narrative discussed by Sabine is that actually 2 additional planets would have formed but would have been lost later due to collisions of the planets.

TGD suggests an alternative solution to the problem. Instead of two planets missing from the model, the entire view about the formation of planets by gravitational condensation from the primordial matter disk would be wrong. The TGD proposal is that the planets were formed in explosions throwing out a surface layer of the Sun. There is an obvious analogy with supernova explosions (see for instance this, this and this).

This kind of explosion could have also led to the formation of the moons of the Earth and Mars.

1. It is now known that the Moon consists of same stuff as the Earth, which strongly disfavors the earlier model, which assumes that Moon was formed as a planet, christened as Theia, having a mass about mass of Mars collided with the Earth and threw out mass that condensed to form the Moon (see this).
2. By taking the formation of Moon and the mysterious Cambrian explosion in which photosynthesizing multicellular life suddenly popped up out of nowhere, as starting points, one ends up to a rather detailed quantum view of the geological and biological evolution of the Earth (see this).
3. The formation of the Moon would be associated with the contraction of the radius of the Earth by factor 1/2 about 4.5 Gy ago. Much later an expansion of radius by factor 2 would have given rise to Cambrian Explosion (CE) .5 Gy ago. Life would have evolved in underground oceans and bursted to the surface in CE.

There are several questions to be answered.

1. The energetics of the expansion is not possible in the standard physics. How metabolism and evolution of photo-synthesizing multicellular life were possible in underground oceans? How were the underground oceans possible in the first place? TGD based new physics suggests answers to these questions.
2. p-Adic length scale hypothesis applied in condensed matter scales makes the model quantitative. A crucial step would have been the reduction of atomic radius by factor 1/2. There is evidence for this kind of transition in lab scale (experiments of Mills (see this) and Holmlid (see this). The liberated atomic binding energy makes possible the formation of the Moon and also provides the metabolic energy making possible the formation of quantum coherent large h_eff phases even in the scale of the Earth.
3. The large scale quantum entanglement reduces the number of translational degrees of freedom and makes possible low pressure and temperature essential for life. Pollack effect is also an essential aspect of the model. This could make possible the evolution of life in underground oceans or water reservoirs under paradize-like conditions in the womb of Mother Gaia.

This kind of model could apply also to the formation of planets by explosions throwing out a surface layer of the Sun and even to supernova explosions. The physics in the interior of the Sun could be completely different from the standard view.

1. I have already earlier proposed a model for solar wind and solar energy production, which would not be based on fusion in the solar core but to a transformation of nuclei of M₈₉ hadron physics with mass scale by a factor 512 higher than ordinary hadron mass scale to ordinary hadrons. The unexpected strange features of quark gluon plasma provide evidence for M₈₉ hadron physics (see this).
2. The model suggests that the interior of the Sun, about which we have only speculations, is actually more analogous to a huge living cell rather than a superhot hell inside which hot fusion takes place. See (see this).

See the article About the recent TGD based view concerning cosmology and astrophysics 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 counterparts of blackhole entropy and temperature

Gyanprakash Raj asked about the TGD view of blackhole entropy. In particular, does TGD predict blackhole entropy excess predicted by various models of blackhole trying to understand blackhole information paradox?

TGD forces us to give up the standard view of blackholes as systems having mass concentrated at a single point. The QFT limit of TGD replaces the topologically extremely complex space-time surface with a single region of M⁴ made slightly curved.

1. Outside the horizon, the blackhole-like object (BH) would be like an ordinary blackhole. Inside BH the description as a volume filling flux tube spaghetti, giving rise to maximal density, would be more appropriate.
2. For the TGD counterparts of the ordinary blackholes the flux tube would have thickness of nucleus size scale but an entire hierarchy corresponding to predicted hierarchy of hadron physics labelled by ordinary and Gaussian Mersenne primes is highly suggestive (see this). The difference between stars and blackholes would not be so dramatic as in GRT.

In TGD, the formulas for BH entropy predict dramatic results if ordinary Planck constant is replaced by gravitational Planck constant.

1. The standard blackhole entropy is extremely small. It is proportional to 1/ℏ and if ℏ is replaced with its gravitational counterpart ℏ_gr it becomes even smaller.
2. The standard blackhole temperature is proportional to ℏ and increases equally dramatically. The explanation would be gravitational quantum coherence which increases the geometric pixel proportional to Planck length squared and increases the energy scale.

Just for fun one can look for the standard formulas for S_B and T_B in the TGD framework.

Consider first the blackhole entropy.

1. Blackhole entropy S_B is essentially the area A of black hole divided by the area unit l_P² propto ℏG:

   S_B propto A/l_P² .

2. Suppose one replaces ℏ with gravitational Planck constant

   ℏ_gr= GMm/β₀ = r_Sm/2β₀ .

   Here the mass m would be naturally the mass of the nucleon characterizing monopole flux tubes of spaghetti as gravitationally dark giant nuclei.

3. This scales up the l_P² l_P²→(r_sm/2β₀*ℏ)l_P² = (Λ_gr/λ)l_P², where Λ_gr=r_s/2β₀ is mass independent gravitational Compton length (Equivalence Principle). λ is the ordinary Compton length.
4. β₀ is a velocity parameter whose interpretation I have finally understood. TGD is distinguished from GRT in that it allows flat warped space-time surfaces for which light velocity c in M⁴ is reduced to β₀. See (see this) .
5. The black hole entropy is scaled dramatically downwards so that one cannot speak of excess entropy. One could say that the scaling of Compton length to gravitational one increases the size of the pixel defining the bit. The interpretation of small entropy could be in terms of gravitational quantum coherence. A BH-like object would be an extremely ordered quantum coherent system, totally unlike the standard BH. Also the centers of the stars could be these kinds of objects.

What about blackhole temperature?

1. The blackhole temperature is given by T_B= ℏ/(8πr_s) and is extremely low.
2. In the replacement ℏ→ℏ_gr , T_B would be replaced by

   T_B= m/8πβ₀, where m is the nucleon mass.

3. For β₀=1, one one would have T_B ∼ 37.4 MeV. However, the Nottale's Bohr orbit model for planets predicts for the Sun the value β₀ ∼ 2^-11. This would give

   T_B= 79m_p ,

   which is not far from the mass of Z boson 85.7 m_p, which would correspond to M₈₉ hadron physics TGD indeed predicts that solar wind and solar energy production is due to the decay of M₈₉ hadrons (basically nucleons) at the surface of the Sun. The mass of the M₈₉ pion is 512 times that of the ordinary pion and equals 71.7 GeV.

4. Could one think of detecting gamma radiation from the Sun with this energy? Note that the decay of M₈₉ would produce gamma rays with energy 39.5 GeV. There is indeed a mysterious gamma ray emission from the Sun around 40 GeV energy (see this).

See the article About the recent TGD based view concerning cosmology and astrophysics 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 energetics of the proposed compression and expansion of Earth

One should also understand the energetics of compression and expansion. Under what conditions these transitions are possible? Gravitational and atomic binding energies are involved. Gravitational binding energies per atom are proportional to the ratio r_s/R_P for the planet. The binding energy liberated in the transition k=139→ 137 is independent of planetary parameters. In the case of Earth, this transition should provide the energy making possible formation of the Moon and the proposed formation of Rydberg atoms making possible large scale quantum coherence.

One can ask whether the transition is possible for all planets and whether the Earth and Venus, with almost the same value of r_s/r_P, are in a special position. For large values of r_s/r_P the electric binding energy needed for the formation of a moon might be too large. r_s/r_P is indeed large for Jupiter and Saturn which suggests that the transition k=139→ 137 essential for the emergence of life and for the formation of the Moon cannot reduce the radius by factor 1/2. The mechanism for the formation of moons must be different. For small values of r_s/r_P electromagnetic binding energy would be more than needed. A rough estimate for r_s/R is as r_s/R∼ (R/R_E)² (r_s(E)/r_E) and decreases with the size of the object.

Consider next estimates for the changes of the gravitational and electric binding energies per atom. For simplicity, restrict the consideration to the case of the Earth first.

1. The gravitational binding energy per atom with mass number A increases in the transition R_E→ R_E/2.

   Δ E_gr=(r_s/2R_E) A m_p ∼ A × 3.14 ~eV .

2. The change Δ E_B of the binding energy in the transition k=139→ 137 can be estimated by using the expression for the ground state binding energy of atom with nuclear charge Z for k=139 assigned for ordinary atoms

   E_B(139)= Z² ×α_em²/8m_e ∼ Z² ×13.6 eV .

   One has E_B(137)= 4E_B(139) so that the increase of the binding energy is

   Δ E_B= 3E_B(139)∼ 3Z² ×13.6 ~eV .

   From Z≤ A/2 one has

   Δ E_B≤ A²× (3/4)× 13.6 ∼ ~eV

3. The condition Δ E_gr= Δ E_B gives a lower bound A≥ .3 .

   This condition is true for all atoms in the case of Earth and there is also surplus energy to throw out a layer forming the Moon.

For a general planet, the lower bound for A scales as the ratio x_P=(r_s/r_P)/(r_s/R_E). The values of x_P for Venus Mercury and Mars (.85,0.14 .20). It has been proposed that Venus has had a moon but has lost it. Also the nearness of the Sun is unfavourable for having a moon. The formation of the moon could have led to a compression of the Venus and if the life evolved in the interior it would not have survived at the surface after the expansion since the atmosphere (95 percent CO₂ and 3.5 percent nitrogen) has temperature of 467 ^_ºC and pressure, which is 93 times that on the Earth.

For the gas giants Jupiter resp. Saturnus with ratio x_J∼ 29.0 resp. x_s∼ 10 the lower bound for A is 7.7 resp. 3.3. Moon resp. Titan has x_J∼ .07 resp. x_T∼ .056.

See the article Could the notions of quantum geology and quantum biology make sense? 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.