Is the brain in some sense 11-D or is it something different?

Shamoon Ahmed gave a link to a popular article (see this) claiming that the brain is in some sense 11-dimensional. Probably the only thing that M-theory predicts is that the target space of strings is 11-D so that this finding might provide some confirmation of faith for frustrated M-theorists.

In the sequel I will discuss this finding from TGD viewpoint and propose a modified interpretation based on the geometry of icosahedron, one of the 5 platonic solids, which play a key role in TGD, and TGD inspired quantum biology and theory of consciousness.

The dimension 11 in this context looked to me a rather formal notion but one could give it a mathematical meaning.

1. In 3-D one can take tetrahedra, 4-simplexes as building bricks of a discretized manifold. In dimension 11 one has 12-simplexes. These are glued together, which means that n-faces with n varying from 1 to 11 are glued together along n-1-D faces.
2. In the case of the brain, one would have groups of neurons, with 12 neurons connected in such a way that one has a connectedness of a 12-simplex. There would be 11- edges meeting at each 12 vertices. Each neuron would be connected to all the other 11 neutrons and would have maximal connectedness, which is very natural if one wants a maximally coherent functional unit.

   The notion of orientation is essential: axons are oriented by the direction of nerve signals which is always the same. The orientation of axons could induce orientations of n-faces. 2-face would correspond to a loop in which signals can rotate in a single direction.

   Since axons must be present, each neuron must be connected with every other neuron. The geometric connectedness possible in the case of neurons since the axon from a given neuron can branch and have a synaptic contact with the dendrites of several neurons: for n=11-simplex with all other (11) neurons (see this). Note that also a synaptic contact with the neuron itself (autapse) is possible.

   Could one consider also a generalization of this geometric view of a simplex. Could functional coherence of the neuron group serve as a criterion for whether neurons form an n-face?

3. The interpretation in terms of 11 real dimensions might assume too much and I am reluctant to believe that it has anything to do with M-theory. However, one could realize n-simplexes in this way in 3-space and the orientation of the axon, determined by the preferred directions of signals, would define orientations of higher level simplexes. The idea that these structures could have something to do with geometric cognition allowing us to imagine higher dimensional geometric structures is attractive.

Can TGD add anything interesting to this picture? The appearance of number 12 creates an overwhelming temptation to associate this finding with one particular Platonic solid, icosahedron, having triangular faces. I am not claiming that the proposed interpretation of the findings is wrong but asking whether Platonic solids could add something interesting to the proposal.

1. The 12 vertices of the argued 11-simplex could be also identified as vertices of icosahedron, one particular Platonic solid appearing repeatedly in molecular biology. For an icosahedron, the Hamilton cycle, going through all vertices just once, has 12 vertices. It would connect each vertex to all other vertices by a unique path having a varying number of edges: 1,2,... The selection of this Hamilton cycle could raise one particular edge path among all possible closed edge paths possible in the maximally connected 12-neutron network in a special position.
2. This icosahedron need not correspond to ordinary Platonic solid in the Euclidian 3-space. The definition of nearness can be defined also in terms of functional nearness. Indeed, hyperbolic 3-space has been suggested to play a role in neuroscience for neutrons: neurons resembling each other functionally would be near to each other in the hyperbolic metric and in TGD framework this metric is assigned with hyperbolic 3-space H³ as Lorentz invariant light-cone proper time = constant surface to which the magnetic body (MB) of the brain is assigned as 3-D surface. The signals from neurons, which are near each other in functional sense, would be sent to nearby points of the MB so that functional nearness would be geometric nearness at the level of MB.
3. Also tetrahedron with 4 vertices and faces and octahedron with 6 vertices and and 8 faces are Platonic solids which have triangular faces representing 2-simplex and could correspond to dimensions d=3 and d=5. Cube with 6 square faces and d=8 vertices is the dual of octahedron and dodecahedron with d=20 vertices and 12 pentagonal faces is the dual of icosahedron. It might be also possible to assign to them dimension as the number of vertices by using maximal axonal connectedness of vertex neurons as a criterion.

   Platonic solids and Hamiltonian cycles as path going once through each vertex of the Platonic solid and identified as nuclear strings play a key role in the "Platonization" of nuclear and atomic physics leading to quite precise quantitative vision about basic numbers of nuclear and atomic physics and even hadron physics. The key observation is that the states of j=l+/-1/2-blocks of atoms and nuclei correspond to Platonic solids for l<6 (a highly non-trivial fact), which therefore provide geometric representation for the j-block (see this).

Icosahedron is a very special Platonic solid and deserves a separate discussion.

1. Icosahedron is unique among Platonic solids in the sense that it allows a large number of Hamiltonian cycles. Icosahedron, tetrahedron and their Hamiltonian cycles play a fundamental role in the TGD inspired model of genetic code (see for instance this) involving the notion of icosa-tetrahedral tessellation of hyperbolic 3-space involving all Platonic solids with triangular faces.

   Each combination of 3 icosahedral Hamiltonian cycles with symmetries Z_n, n=6,4,2 defines a particular realization of the genetic code predicting correctly the number of DNA codons coding for a given amino acid.

2. The model of the code emerged originally as a model of musical harmony. The faces of icosahedron are triangles and would define 3-chords realized as cyclotron frequencies assignable to the vertices of the triangle. Each Hamiltonian cycle would define 20 chords defining a particular harmony whereas the 12 vertices along Hamiltonian cycles would define a 12-note scale, with neighboring vertices representing frequencies related by scaling by 3/2 (quint) modulo octave equivalence.

   One could speak of music of light and since music creates and expresses emotions, the proposal is that different bio-harmonies correspond to different emotional states realized already at DNA and RNA level. Could these 12 neuron units and possible tessellations (hyperbolic crystals) associated with them relate to the realization of emotions at the level of the brain?

   Physically, the Hamiltonian cycle as a representation of 12-note scale is an analog of a closed string made of flux tubes representing the edges (pipes of organ!)

3. What is fascinating is that hyperbolic 3-space H³ (mass shell in particle physics), playing a key role in TGD, has a unique tessellation/lattice involving all Platonic solids, whose faces are triangles (icosahedron, octahedron, tetrahedron) and also provides a model of DNA making quantitatively correct predictions. I have proposed that this tessellation defines a universal realization of the genetic code realized in all scales at the level of the MB of the system. Could the 12-neuron unit interpreted as 11-simplex relate to one particular realization of this tessellation.
4. Also cubic, icosahedral, and dodecahedral regular tessellations are possible in H³ (Euclidean 3-space E³ allows only cubic regular tessellation) and they would define the analog of a homology of dimension n= 7, 11, or 19 space at neuronal level.

See the chapter TGD Inspired Model for Nerve Pulse.

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.

A proposal for an explicit realization of the Hamilton-Jacobi structures

A proposal for an explicit realization of the Hamilton-Jacobi structures

I have not been able to propose an explicit realization of the Hamilton-Jacobi structures hitherto. The Hamilton-Jacobi structure should define a slicing of the space-time surface, not necessarily orthogonal, partonic 2-surfaces and 2-D string worlds sheets. The Hamilton-Jacobi structure could be seen as a 4-D generalization of 2-D complex structure. Could Hamilton-Jacobi structures be convolutions of 2-D complex structures with their Minkowskian analogs such that the conformal moduli of the partonic 2-surfaces and strings world sheets (possibly making sense) depend on the point space-time surface?

1. The conformal structures of partonic 2-surfaces are classified by their conformal moduli expressible in terms of Teichmueller parameters (see this). Could one consider some kind of analytical continuation of the moduli spaces of the partonic 2-surfaces with different topologies to moduli spaces of time-like string world sheets?

   This would give a direct connection with p-adic mass calculations in which the (p-adic counterparts of) these moduli spaces are central. In p-adic mass calculations (see this and this), one assumes only partonic 2-surfaces. However, the recent proposal for the construction of preferred extremals of action as minimal surfaces, realizing holography in terms of a 4-D generalization of the holomorphy of string world sheets and partonic 2-surfaces, assumes a 4-D generalization of 2-D complex structure and this generalization could be just Hamilton-Jacobi structure.

2. The boundaries of a string world sheet can also have space-like portions and they would be analogous to the punctures of 2-D Euclidian strings (now partonic 2-surfaces are closed).
3. Can Minkowskian string world sheets have handles? What would the handle of a string world sheet look like? String world sheets have ends at the boundaries of the causal diamond (CD) playing a central role in zero energy ontology (ZEO).

   If the 1-D throat of the handle is a smooth curve, it must have portions with both space-like and time-like normal: this is possible in the induced metric but the portion with a time-like normal should carry vanishing conserved currents. For area action this is not possible. Intuitively, the throat of the handle would look physically to a splitting of a planar string to two pieces such that the conserved currents go to the handle. If this is the case, the 1-D throat would have two corners at which two time-like halves of the throat meet.

   The handle would be obtained by moving the throat along a space-like curve such that its Minkowskian signature and singularity are preserved. CP₂ contribution to the induced metric might make this possible.

See the article Hamilton-Jacobi Structure as a 4-D generalization of 2-D complex structure or the appendix of the chapter Identification of the Preferred extremals of K\"ahler Action.

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.

Superconducting computers and the TGD based model of nerve pulse

I have discussed the possibility of conscious computers from the TGD viewpoint in the article Could neuronal system and even GPT give rise to a computer with a variable arrow of time?. I do not believe that computers as deterministic Turing machines could be conscious. The new view of space-time, time, quantum gravitation and quantum ontology provided by TGD allows however to consider the possibility that computers would give rise to life forms enjoying at least rudimentary consciousness. The basic notions are magnetic body (MB) carrying dark matter as phases of ordinary matter with large Planck constant and controlling the "biological body" (BB)consisting of ordinary matter. Communications between MB and BB are essential for both the transfer of sensory information to MB and the control of BB by MB. Josephson radiation consisting of dark photons could make possible the communication of sensory information and cyclotron radiation from MB induced by the receival of the Josephson radiation by cyclotron resonances would realize the control.

It is not clear whether MOSFET based technology could allow the communications from transistors to the magnetic body (MB) of the system. Biological analogy strongly suggests that Josephson junctions are required and communications take place by Josephson radiation modulated by the Josephson frequency modulations induced by changes of the voltage of the junction. Dark magnetic flux tubes with large enough value of h_eff are needed to define the Josephson junction and it is far from clear whether they can be realized spontaneously for transistors.

Superconducting computing, which could be involved with both classical and quantum computation, is however a technology, which might provide at least a starting point in attempts to understand how conscious computers might be created in the TGD Universe.

Rapid single flux quantum (RSFQ) is the basic active element in the circuitry and corresponds to single Josephson junction. The presence/absence of quantized magnetic flux defines the bit. SFQ voltage pulses of duration about picoseconds are produced by switching of bits in this way. This would allow THz clock frequency f_cl.

If f_cl corresponds to Josephson frequency f_J=ZeV/h, where Z is the charge of the superconducting charge carrier, one obtains an estimate for the voltage as ZeV∼ .05 eV. For the cell membrane one has eV∼ .05 eV, which is near the thermal threshold at room temperature. The superconducting computations require a temperature of order 10 K so that the value of frequency does not seem to emerge from thermal considerations. The thermal criterion is expected to be true also for the TGD based generalization of superconducting computers if realized using the same principles as in living matter.

Flux Quantization in Josephson Junctions

1. The presence/absence of flux quantum through the junction represents a bit. Switching of the bit in RSFQ means that the flux changes by the unit Φ₀ of magnetic flux. In the simplest situation, the value of flux through the Josephson junction connecting the super conductors, which could have planar or cylindrical geometry, is equal to 0 or Φ₀.
2. When the flux through junction is changed by one unit, Faraday law Δ Φ= +/-Φ₀= Ze∫ Vdt implies a generation of voltage pulse propagating along the superconducting wire formed by the coupled cylindrical superconductors. For a constant voltage V=V₀, this condition fixes the duration T= Φ₀/ZeV of the process and this defines Josephson frequency, in turn defining the clock frequency.

TGD View of Electromagnetic Fields

1. Quantum criticality is essential for the appearance of large values of h_eff labelling the scales of long length scale quantum fluctuations. Quantum criticality combined with ZEO would make possible the emergence of life-like features.
2. The gravitational Planck constants ℏ_gr= GMm/β₀ assignable to the gravitational flux tubes of the Earth and Sun are excellent candidates in this respect. The value of ℏ_gr/ℏ is GM_Em/ℏβ₀ =(r_S(E)/2L_m), r_S denotes the Schwartzchild radius of Earth about 1 cm and L_m denotes Compton length of particle with mass m β₀∼ 1.
3. In TGD, two kinds of magnetic fields are possible. Monopole flux tubes are something new and rather remarkably, can exist in absence of currents: this makes them ideal for computation. Monopole flux tubes have closed 2-surfaces as cross sections. Flux quantization follows from the homology of CP_2. Monopole flux tubes explain the presence of long range magnetic fields appearing in even cosmological scales and also the stability of the Earth's magnetic field.
4. Also in TGD, the topological half of Maxwell's equations, that is Faraday law and the vanishing of the divergence of magnetic field, hold true. Therefore the basic argument for the outcome of the switching of the flux are not affected when ordinary flux tubes are replaced with monopole flux tubes.

RSFQ Generalization in the TGD Framework

How could the notion of RSFQ generalize in the TGD framework? The hint comes from the TGD based model of cell membrane and nerve pulse assigning to the ionic channels of the cell membrane dark Josephson junctions with a large value of h_eff making possible high T_c superconductivity.

1. Concerning the numbers assigned to RSFQ, the cell membrane looks ideal for the seat of analogues of RSFQs. I have that the cell membrane acts as a sequence of dark Josephson junctions associated with membrane proteins acting as channels and pumps. The membrane resting potential ∼ .05 eV corresponds to the frequency of 5 THz and is in the same range as the Josephson frequencies assigned with RSFQs. The large value of h_eff makes possible high temperature superconductivity and scales up the value of Josephson frequency to f_J= ZeV/h_eff so that Josephson frequencies even in EEG scales would be made possible by quantum gravitation in TGD sense.
2. No currents are needed to maintain monopole magnetic fields so that they are ideal for technological purposes. Cell membrane would be a superconductor and membrane proteins would define Josephson junctions. Membrane potential could realize the Josephson frequency f_J=ZeV/h_eff.
3. The TGD view of the basic active unit would differ from RSFR. In TGD, the absence of flux quantum in RSFQ corresponds to two U-shaped monopole flux tubes at opposite sides of the junction associated with the counterpart of the cell membrane and transversal to it. The U-shaped monopole flux tubes can reconnect to form a pair of flux tubes with opposite magnetic fluxes. This topological process is fundamental in the TGD inspired view of biocatalysis and water memory.
4. What is the effect of the generation of a pair of opposite fluxes on the membrane potential? The answer depends on whether the two opposite fluxes go through the junction or only one of them does so. In the latter case the junction acts like RSFQ in reconnection. This is a natural looking working hypothesis. The difference comes from the presence of the flux tube with opposite flux.
5. Faraday's law should apply to both flux tubes. The appearance of flux tubes would correspond to a generation of opposite fluxes Δ Φ= Φ₀= ∫ Vdt. In the simplest situation the voltage values associated with the flux quanta have opposite values \pm V₀. This is very much like in the case of nerve pulse in which the resting potential changes its sign during the first half of the nerve pulse. When the reconnection disappears, the situation would become "normal". The analog of nerve pulse would be generated and propagate along the counterpart of the axon and induce a similar process in all membrane proteins defining Josephson junction.
6. In zero energy ontology (ZEO), the identification of the generation of nerve pulse as a pair of "big" state function reductions (BSFRs) changing the arrow of time temporarily is attractive and would correspond to quantum tunnelling in standard quantum theory. An interesting question is whether pump proteins act as channel proteins in reversed time direction and whether the flux tube pairs are associated with pairs of channel and pump proteins.

Objections

1. How the very low Josephson frequencies ZeV/h_eff associated with the large values of h_eff, say h_eff=h_gr, can be consistent with the very large values of clock frequency f_cl =f_J= ZeV/h needed by a fast operation. It would seem that both h_eff and h are needed. Is this possible or are these computers doomed to be very slow?
2. Is a scale hierarchy of space-time sheets involved having various values of h_eff and could it correspond to the onion-like hierarchical structure of magnetic body (MB) involving increasing time scales as Josephson frequencies. Could the fast Josephson frequencies near the the level of ordinary matter define a hierarchy of computer clocks? Could the pulses of short duration induced by RSFQs induce a hierarchy of frequency modulations of scaled up Josephson oscillations for various values of h_eff? This could also make the computer conscious by bringing in the hierarchy of time scales.
3. High temperature superconductors would be needed. The TGD proposal is that the cell membrane is a high temperature superconductor. Biosystem would be an open system and a metabolic energy feed would take care that the distribution for the values of h_eff is preserved. Also the fact that the dark matter as h_eff>h phases of ordinary matter at the space-time sheets of the flux tubes has very weak interactions with the other sheets, in particular the sheet of the ordinary matter, would be decisive. Zero energy ontology (ZEO) would in turn be highly relevant for maintaining the quantum criticality in making possible homeostasis in which time reversal changes attractor to repulsor and vice versa.

See the article Could neuronal system and even GPT give rise to a computer with a variable arrow of time? 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.

Could the notions of Platonization and tensegrity make sense in atomic physics?

It is easy to invent an objection against the idea of holographic correspondence between nuclear and atomic physics, which requires that the notions of Platonization and tensegrity make sense also in atomic physics. Tensegrity requires that electrons in atoms are connected by monopole flux tubes. Can one invent any justification for this kind of crazy idea?

1. Hydrogen atom was the brilliant success of atomic physics. It was generalized by treating in the lowest order approximation electrons as independent entities experiencing only the Coulomb force of the nucleus. However, at the classical level this does not seem to make sense since the mutual Coulomb interaction at the shell with the same value of the principal quantum number n and same value of l should have roughly the same orbital radius r_n= n²/Z²a₀. The order of magnitude for their total repulsive Coulomb energy has a lower bound about Z(Z-1)×/r_n= Z(Z-1)Z²/(2a₀n²). Here only the nearest neighbor interactions are counted.

   The classical interaction energy behaves like Z⁴ for large values of Z whereas the attractive interaction with the nucleus behaves classically like Z²! Does it really make sense to assume that the interactions of electrons can be treated as a small perturbation?

2. If one takes quantum classical correspondence seriously, one must ask whether there exists an interaction, which would work against the repulsive Coulomb interaction and prevent the explosion of the energy shell (or the angular momentum shell j=l+/- 1/2). Here the string tension of the monopole flux tubes could come in rescue and provide the force preventing the explosion. It would contribute to the total energy a constant amount and the effect would be visible for atoms with Z>1. In a good approximation the system would behave like a rigid body. String tension would also give rise to vibrational modes whose existence would serve as a killer test for the proposal. One has a good reason to expect that these energies are rather small as compared to atomic energies.

In the Platonic model it is possible to calculate the repulsive Coulomb interaction energy also exactly since the tesselation contains the points of the Hamiltonian cycle with V vertices and its dual with F-2 vertices at free edges connecting neighboring vertices of the Hamiltonian cycle which are not nearest neighbors along the cycle.

1. For a full electron electron shell as a Platonic solid defining a tessellation of the sphere, the distances of electrons at full shell would be constant, which would make the estimation of the contribution of electron interaction energy very simple. The dominating contribution to the Coulomb interaction would come from nearest neighbor interactions between electrons of the Hamiltonian cycle and between Hamiltonian electrons and electrons of dual edges. The sum of the repulsive interaction of the shell containing only Hamiltonian edges would be constant.

2. A lower bound for this contribution to the repulsive interaction energy would come from nearest neighbor interactions and would be of order E_Coul= k₁Z²/2a₀n² + k₂Z²/2a₀n², where one has (k₁,k₂) = (2(l+1)x₁,2lx₂). Where x_i=1 for the Hamiltonian cycle and x_i=1/2 for its dual if the distance between electron of cycle and electron of its dual is half of the distance between Hamiltonian electrons. For l=1 one has (k₁,k₂)=(4,2) that is the Hamilton cycle and its dual for the tetrahedron. For l=2 one has (6,4) that is Hamiltonian cycle for octahedron and its dual for cube. For l=3 ones (8,6) that is the Hamiltonian cycle for the cube and its dual for octahedron.

3. There is also the interaction energy between different shells and if the wave functions for the orientation of the shells are allowed, the calculations are more complex. The selection of a common quantization axis of angular momentum eliminates this degree of freedom.

In the lowest order approximation, the dynamics would effectively reduce to single particle level since the Platonic tessellation would be a rigid body-like system having only the radial degree of freedom plus degrees of freedom related to orientation. The wave functions for electrons at the vertices of the Platonic solid would be obtained by the operations of the symmetric group of the Platonic solid, which is a discrete subgroup of the rotation group and the rotation would give a superpositions of the harmonics belonging to the multiplet j=l+/-1/2. Antisymmetrization would leave only the products of wave functions with different values of j_z.

The monopole flux tubes could serve as correlates for the pairing of valence electrons. p-Adically scaled down electropions could also appear as molecular bonds. Note that electropion mass is rather precisely 2 electron masses. Evidence exists for muo-pions and tau-pions and also their p-adically scaled down variants could appear as bonds. Chemical bonds could correspond to these scaled down pions. Tensegrity is indeed a very natural concept in molecular physics.

See the article About Platonization of Nuclear String Model and of Model of Atoms 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.

What Platonic vision allows to say about nuclear dynamics?

Only the static aspects of Platonic vision are discussed hitherto. Platonicity allows the orbital angular momenta l ≤ 5 and as a special case also l=9 (dodecahedron and the dual edges in icosahedron) but one cannot completely exclude also some higher values of l. The interesting question is that Platonicity could predict selection rules for nuclear reactions.

How to describe the interacting many-nucleon and many-atom states encountered in scattering? It is rather easy to guess the basic principle from the vision of the interaction of biomolecules in biocatalysis and from the interaction of closed strings in string models.

1. Biomolecules must first find each other and after this they must become close to each other to react and the energy needed to overcome the potential energy wall must come from some source. U-shaped flux tube tentacles with a large value of h_eff would reconnect and form a pair of flux tubes connecting the molecules after which the value of h_eff would be reduced and force the molecules near each other. In this process energy would be liberated and kick the molecules over the potential energy wall and the reaction could proceed.

2. The extended flux tube edges connecting two subsequent nucleons of the Hamilton cycle assigned with a Platonic shell with a given value of n and j would define the tentacle-like entities. A similar extension is possible also for the dual edges. The lengthening of a tentacle preserving its magnetic energy would involve a temporary increase of h_eff and a reduction of string tension. The reconnection of the edge tentacles would allow for the nuclei to find each other. This would induce a fusion of the Hamiltonian cycles to a Hamiltonian cycle of the composite graph. After that h_eff would be reduced and the liberated energy would allow the systems to overcome the potential wall and the nuclear reaction would proceed.

3. Can one assume that the entire Platonic solid and the Hamiltonian cycle are present for partially filled j-blocks? This would require that the free vacancies are realized as pre-existing geometric entities. The information j fixes the j-block as Platonic solid and fermion statistics and energy minimization forces the nuclei to fill a fixed j-block. Fermion statistics could thus force the existence of Platonic solid as its geometric counterpart. Hamilton sub-cycles, possibly even several, for partially filled shells must be assumed. Subcycles must define connected regions, which poses strong constraints on the order in which the free vacancies.

What can one conclude from these assumptions?

1. The initial states of the nuclear reaction can be regarded as tensor products j_i,1 ⊗ j_i,2 of j-blocks. Final states are tensor products j_f,1 ⊗ j_f,2 ... ⊗ j_f,n, of n ≥ 2 j-blocks. These tensor products must contain common states for the reaction to proceed and the assumption that the values of j_f,k are consistent with the Platonic solids, poses conditions on the values of j_f,k.

2. l=5 and l=9 and possibly some higher values of l, define elementary shells and at the fundamental level the reactions would occur between pairs of elementary shells and proceed by the proposed re-connection mechanism. The composite Hamiltonian cycle formed in the reaction is not in general elementary but should transform to a union of elementary cycles belonging to outgoing nuclei. In "topologically elastic scattering", the shells would fuse temporarily and emerge as unchanged.

3. The conservation of the sum V_a,1 + V_a,2 for the numbers V_a,i of "active" vertices containing a nucleon and the number E_a,1 + E_a,2 of "active" free edges containing a nucleon corresponds to the conservation of nucleon numbers. Active vertices and free edges would be shared by different final state Platonic solids. The total numbers of active vertices and active free edges is conserved in the reconnection but after that topological reactions modifying the face types of tessellations could occur as analogs of phase transitions changing the face type of solid lattice. These conditions pose constraints on the Platonic solids possible in the final states.

4. Can one assume something about the dynamics of faces? The assumption that the total number of faces is conserved implies that the number of free edges equal to F-2 is conserved. The face type (triangle, pentagon, or square) is the same for a given Platonic solid. An even stronger assumption would be that the total number of faces of a given type is conserved and looks unrealistic.

   The reactions in which a single Platonic solid appears in the final state would be strongly restricted by the conservation of the vertex number. Tetrahedron, octahedron, and icosahedron have triangular faces. If V₁ + V₂ is conserved, the reactions icosahedron (12 vertices, 18 free edges) ↔ 2 octahedrons (6 vertices, 6 free edges), icosahedron ↔ 3 tetrahedrons (4 vertices, 2 free edges). The number of free edges is not conserved. It seems that the most general option is the most realistic one.

   If the numbers of different face types are conserved, in reactions involving incoming Platonic solids with different faces the outcome should consist of similar solids or contain Archimedean solids, which can have several face types. There is however no deep reason for the conservation of numbers of different face types.

See the article About Platonization of Nuclear String Model and of Model of Atoms 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.

Platonization of physics leads to deep connections between atomic, nuclear and hadron physics

The motivation for the article About Platonization of Nuclear String Model and of Model of Atoms, which I have been workin last weeks, came from the discovery of a doubly magic form of oxygen (see this) with 8 protons and 28 neutrons combined with the fact that it is difficult to understand the presence of so large number of neutrons in the existing view about strong interactions. A more general challenge of understanding atomic nuclei and atoms in the TGD framework. General coordinate invariance implies that particles as 3-surfaces obey almost deterministic holography so that they give rise to analogs of Bohr orbits. This gives a rather precise content for the idea of what atoms and nuclei are classically but is not enough: one must also understand their shell structure geometrically.

The outcome was a significant advance in the understanding of the nuclear string model (see this). Nuclear string, consisting of flux tubes connecting the neighboring nucleons of the tessellation, provides a new element to the description of bound states. This element is not present in the standard nuclear physics where only short range strong interactions, assumed to be described in terms of meson exchanges at the low energy limit, are assumed to be present at the fundamental level. The flux tube has a string tension and it literally binds together the objects that it connects. This in turn led to a progress in the understanding of the model of hadrons.

The earlier proposal that string(s) form a kind of 3-D flux tube spaghetti was replaced with the proposal that energy shells have as space-time counterpart a hierarchy of 2-D spaghettis analogous to magnetic bubbles proposed in the model for the evolution of astrophysical objects (see this). These 2-D surfaces can be identified as lattices/tessellations of 2-D surfaces having nucleons at their nodes. For the sphere as the geometric counterpart of the energy shell, energy minimization suggests Platonic solids for which the maximal nucleon numbers are fixed. Platonic solid as an analog of solid state lattice naturally represents the energy minimum and their construction is analogous to crystal growth.

Nuclear string is assumed to define a Hamilton cycle connecting the V vertices of the Platonic solid and therefore having V edges. Energy minimization and the phenomena of magic proton and neutron numbers and neutron surplus suggest that neutrons are associated with the V edges of the Hamilton cycle and protons with the F-2 (F is the number faces of the Platonic solid and number of vertices for its dual) free edges of the tessellation as a graph as analogs to lattice impurities. One could speak of the dual tessellation. The roles of protons could be changed in the case of icosahedron.

These considerations inspire several questions.

1. Is there a holographic relationship between nuclear quantum states with say Z=N and electronic states of atoms as the identical structures of the states spaces for atomic electrons and nucleons moving in harmonic oscillator potential suggests?
2. The description of electron states in atomic physics and proton and neutron states in spherically symmetric harmonic oscillator model nuclei predict essentially the same spectroscopies. Does the structure of the Periodic Table reflect directly the proposed geometric shell structure of nuclei modelled as unions of 2-D spherical shells as lattices (tessellations) consisting of either neutrons and protons? Genuinely 2-D tessellations correspond to Platonic solids and behave like dynamical units. Could one say that Platonia is realized both at the level of nuclear and atomic physics?

Platonization of atomic and nuclear physics

1. By starting from atomic physics, one ends up with a number theoretic decomposition of the angular momentum states of protons, neutrons and electrons to subsets using finite field mathematics. The highly non-trivial fact is that for orbital angular momenta not larger than 5, the states with j=l+1/2 and j=l-1/2 correspond to the points of Platonic solids providing classical space-time correlates of j-blocks as geometric energy shells. j=1/2 and j=-1/2 correspond to dual Platonic solids for l≥ 1. The edges of Hamilton cycle and its dual consisting of the free edges not belonging to the cycle define what I call dual representations. This applies also in nuclear physics.
2. The first objection is that the geometric representation of half-odd integer angular momentum states looks strange. This natural in the twistor representation of angular momentum states as partial waves in the twistor sphere. The dual representation in momentum twistor space corresponds to discrete subsets of points of the twistor sphere identifiable as Platonic solids. These representations have counterparts at the level of M^4 since the twistor sphere corresponds to the space of light-like rays emanating from a given point.
3. The Platonization of the quantum numbers might generalize. The twistor space would be replaced with the space for the selections of quantization axes identified as a coset space of Lie group with its Cartan group and the connection with McKay correspondence, discussed from the TGD point of view here, here, and here, is highly suggestive. The reason is that McKay graphs emerge from the reduction of the representations of the rotation group by restricting them to finite discrete subgroups and Platonic solids indeed define this kind of reduction. Finite measurement resolution can be described in terms both number theoretically and in terms of inclusions of hyperfinite factors and also these involve McKay graphs.

A new view of nuclear physics

The Platonization of nuclear physics suggests that nuclei are engineered from nucleons by connecting them by monopole flux tubes.

1. The notion of tensegrity applies and predicts that in ground states the distances between neutrons (protons) at the energy shell are
2. constant so that one has a Platonic solid as an analog of ordinary 3-D lattice. Neutron surplus for the nuclei suggests that neutrons are associated with the free edges of Platonic cycles outside the Hamilton cycle. For all Hamiltonian cycles except tetrahedron and icosahedron,the number of dual edges is larger than the number edges of the cycle.
3. Nuclear strings as monopole flux tubes connecting the 3-surfaces representing nucleons have a string tension. This gives rise to an attractive interaction between the nucleons and stabilizes the nucleon configurations to Platonic tessellations or their duals involving nucleons at the free edges of the tessellation as a graph. The longitudinal and transversal vibrational degrees of freedom for individual nuclei are the TGD counterparts of vibrational degrees of the harmonic oscillator model of nuclei.
4. Also the dark classical Z^0 and W force could be involved. Weak isospin would replace strong isospin and the counterparts of pions would be scaled down pions corresponding to electron length scale and having mass scale of 1 MeV. This would require a large value of h_eff so that intermediate boson Compton length would correspond to nuclear length scale. With an inspiration coming from PCAC and CVC and the fact that CP_2 geometry implies a very intimate relationship between color and electroweak interactions, I have already earlier considered the possibility that strong interactions involving strong isospin could have a dual description as dark electroweak interactions. I have also asked whether dark weak bosons are possible even in longer scales and whether this could explain chirality selection in living matter. The neutron halo and maybe even other nucleon shells could be dark. A picture of the nucleus emerges in which the flux tube bonds correspond to pions but with a p-adic length scale of the electron. This explains the MeV scale of nuclear excitations and predicts a new scale of order 10 keV assignable to the mass differences of these pions. This leads to a detailed understanding of the tritium anomaly and the correlation of the nuclear decay with the solar X ray flux.
5. Energy minimization implies that in the ground state the neutrons would be pseudo-neutrons, that is protons connected by negatively charged meson-like flux tubes to each other and also to the protons of the inner shells. Pseudo-neutrons would have a dipole moment and the Coulomb potential of the protons of the nucleus would orient them radially and stabilize the position of the halo. Dipole dipole interactions would bind the pseudo-neutrons to the lower energy shells. This mechanism might also work for halos as neutron shells. It turns out that the dipole moments must be of order nuclear size, which requires h_eff/h~100 so that weak Compton length is of order nucleon Compton length so that weak interactions are as strong as em interactions in nucleon scale.
6. One should also understand the dynamics behind neutron halos (see this) located to the periphery of the nucleus. The first guess is that it corresponds to possibly partially filled Platonic tessellations or their duals but having no protons. The reason would be that the proton charge implies instability. Note however that there also protonic halos have been found. In the case of the oxygen isotope that motivated these considerations neutron halo would correspond to two shells 8 and 20 (cube and icosahedron). Nuclear strings would provide the needed stabilizing force.
7. The notion of electroweak confinement is a very attractive generalization of color confinement and leads to the notion of dark weak atoms in which the surplus Z^0 charge of the nucleus associated with protons is neutralized by opposite charges of electrons. The size of these atom cannot be smaller than the Compton length of dark weak bosons, and should be at most 10 nm, which corresponds to p-adic length scale L(151) defining a fundamental length scale of biology (cell membrane thickness, the thickness of DNA coil, the size of nucleosomes of DNA,...). This would conform with the chiral selection of living matter.

Nucleus-atom holography

Platonization and the similarity of the spectra of atoms and nucleons inspires a radical but testable strengthening of nucleus-atom holography allowed by the hierarchy of Planck constants, predicting that atomic electron shells are accompanied by neutrino shells and that there is weak confinement in long scales such that electrons and protons and atomic neutrinos and neutrons are paired to weak singlets. The tritium beta decay anomaly (see this) having no generally accepted explanation in the standard model, finds a possible quantitative explanation and thus gives a direct support for the proposal. This would mean that the myth of elusive neutrinos must be given up. The technological implications are obvious.

The obvious objection is that the existence of Hamiltonian tessellations of atomic electrons looks implausible in the standard physics framework. In particular, the existence of monopole flux tubes connecting electrons does not lok plausible. Their string tension would however allow quantum classical correspondence for many electrons not possible in standard view due to the fact the repulsive interaction energy between electrons with the same orbital radius behaves like Z^4 for large Z and is much larger than the attractive binding energy with nucleus.

Recipe for hadrons

The successful construction of nuclei encourages the question whether hadrons could be obtained by a similar lego construction. Ordinary mesons would correspond to monopole flux tubes. Assume that

1. quark masses are given by by p-adic mass calculations (see this),
2. the p-adic length scale hypothesis is satisfied but quarks and mesons can correspond several p-adic length scales,
3. family replication phenomenon corresponds to the topological mixing of 2-D partonic 2-surfaces for quarks (see this),
4. only baryonic (but not mesonic) quarks suffer topological mixing.

With these assumptions one ends up to a model predicting with a few percent accuracy the masses of light mesons and hadrons.

See the article About Platonization of Nuclear String Model and of Model of Atoms 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.

A new view of hadrons inspired by the nuclear physics model

The proposed revision of the view of the construction of nuclei makes it possible to get rid of the non-perturbative nuclear nightmare and is extremely simple. The recent QCD based view of hadron physics leads to a similar nightmare. Could the proposed lego brick picture of nuclear physics extend to the level of hadrons?

1. Could the original Gell-Mann model baryons consisting of heavy constituent quarks find a justification. Could the difference between constituent quarks and current quarks be due to different p-adic mass scales?
2. Could the masses of hadrons be just sums of quark masses and masses of bonds which are not far from pion masses. p-Adic mass calculations (see this) lead to a formula for the mass squared of leptons and quarks. For a given p-adic mass scale mass squared is in the lowest order approximation integer: m²= Am_p², where m_p is the p-adic mass scale. Apart from the effects due to CKM mixing, the values of these integers for leptons (e,ν_e) and the lightest quarks (u,d,s) are

   (A(e),A(ν),A(u),A(d),A(c),A(s))= (5,4,5,8,14,17) .

   These values represent lower bounds and perturbative p-adic second order contribution to A can be at most one unit. What is remarkable is that the masses of electron and u quark for the same p-adic length scale are identical in the lowest order and electron and neutrino masses are nearly identical. The large electron-neutrino mass difference would be due to different p-adic mass scales.

   If the p-adic length scales of u,d, and s quarks are same, their mass ratios in the lowest order are m(u)/m(d)=(5/8)^1/2 and m(s)/m(d)= (17/8)^1/2.

3. Can one predict the masses of pions and lightest baryons p, n,Λ from this input by assuming that the masses of quarks and pion-like bonds are additive and selecting the p-adic mass scale to be that associated with Gaussian Mersenne prime corresponding to k=113, to k=109 or to k=107 assigned with light baryons?

Consider first the masses of pions.

1. If quark masses are additive, one obtains identical masses m(π^-/+)/m_p= m(π⁰)=(8^1/2 +5^1/2)× m(k)= ((8/5)^1/2 +1)m_e , m_e= .5 MeV in lowest for neutral and charge pions. m(k is the p-adic mass scale p≈ 2^k, most naturally k=113. Note that in the case of neutral pions averaging for pairs uu^* and dd^* is involved.
2. Since one has A(u)= A(e)=5, for k=113, which has been identified a the nuclear p-adic length scales, the mass of u quark is obtained by scaling the mass of electron by the factor 2^(127-113)/2= 2⁷. This gives m(u(113))≈ 64 MeV. The mass of the d quark would be m(d)= (8/5)^1/2m(u)≈ 80.9 MeV. For k= 113, pion mass would be m(π)= m(u)+m(d)= 144.9 MeV, which is quite near to 140 MeV for neutral pions. Note that there would be no Cabibbo mixing for pions.

Consider next the masses of the proton, neutron and Λ.

1. The flux tube contribution to the mass would be the sum of the masses identifiable as pions and would be given by m(tubes)= 3× 140=420 MeV, that is m_p/2, exactly one half of proton mass. The simplex model assumes that the quark contribution is the same as the meson contribution. This suggests mass equipartion or a kind of dynamical supersymmetry relating pion and quark contributions to the mass of the nucleon. The masses of proton and neutron m(p)=2m(u)+ m(d)= (2+ (17/5)^1/2)m_e and m(n)= m(u)+2m(d)=(1+25^1/2) m(u) so that neutron proton mass splitting would be quite too large.
2. The first candidate for the solution of the problem is provided by the same mechanism as used to minimized energy in the construction of nuclei: n-π_L⁰ with a larger mass were replaced by p-π_L^- pairs, where π_L^- has the mass of electropion (quarks correspond to k=127 characterizing also electron). One can replace d quarks with u-π^- pairs so that the masses uud and udd are identical. The contribution of quarks to the total mass of the nucleon would be 3m(u)/2= 193 MeV for k=113. For k=111 the contribution is 384 MeV and by Δ m=36 MeV smaller than the nucleon mass ≈ 940 MeV.

   Intriguingly, if the mass equals to the average mass m(u,k=111)+m(d,k=111))/2= m(π(k=113)) of u and d quarks, k=111 gives the same contribution as pions and one obtains proton mass correctly. The masses of nucleons would come out correctly apart from differences relating to pion charge which is 4 MeV. The masses of n resp. p is 939.57 MeV resp. 938.27 and the mass difference is 1.3 MeV.

3. Could this be achieved by the TGD counterpart of CKM mixing (see this, this, this, this, and this), which is certainly present. In TGD, CKM mixing is caused by the different topological mixings of the partonic 2-surfaces at which quarks reside. In a good approximation, the mixing is present only for the lowest quark genera (g=0 (sphere), which corresponds to u (d) and g=1 (torus), which corresponds to c (s). CKM mixing would be essentially the difference of the topological mixings. In the case of Cabibbo mixing, the mixing angle θ_c would be different θ_c= θ_u-θ_d of the topological mixing angles θ_u and θ_d.

   The condition is that the mixing of u quark and scaled down c quarks is such that the light mixed state has mass m(π(k=111))=2m(π). One would have

   c_u² +s_c²(14/5)^1/2m(u(k=111)) = m(π(k=113))== m(π) .

   Here one has (c_u,s_u)=(cos(θ_u), sin(θ_u) and m(u(k=111))=128 MeV and m(π)=140 MeV. This gives s_u² = m(π)-m(u(111))/m(u(111)/((14/5)^1/2-1) giving s_u=-/+ .1392. For the Cabibbo angle s_c= .2250 this gives s_d= s_u+/- s_c. For positive s_u this gives s_d= -0.0858. In (see this){padmass3} I have discussed a model for the topological mixing of quarks assuming that mass squared values are averages of different mass squared values of the topologically mixed particles with a given p-adic length scale. In the recent case, the mixing cannot occur for the mass squared values: this would lead to a negative value for s_c².

4. This proposal resembles the Gell-Mann model in which constituent quarks would give the entire mass of the nucleon. The situation is the same now if the constituent quarks are identified as quark-flux tube pairs. The QCD inspired view replaces constituent quarks with current quarks and divides them to valence quarks and sea quarks. Due to the technical problems of the non-perturbative QCD one cannot build a concrete model. Current quark masses would be in the range 5-10 MeV.

   In the TGD framework, valence quarks could correspond to the quarks with mass scale k=111 and sea quarks would have small p-adic mass scale. Nuclear physics suggests electron mass scale as a mass scale of sea quarks: in this case the current quark masses would be m(u)=m_e and m(d)= (8/5)^1/2m_e. The total sea quark mass would be measured in few MeVs: of order .1 per cent.

5. In case that the topological mixing does not completely take care of the equipartition of the pion and quark contributions to the mass, the missing Δ m≤ 36 MeV could be assigned to the light sea quarks and corresponds to 3.8 per cent of the total mass of the nucleon. The estimates for this contribution vary but are few percent of nucleon mass. It is also known that sea quarks carry only a very small longitudinal momentum fraction and valence quarks carry 1/3 of longitudinal momentum. This would conform with the interpretation of the valence quarks as q+π-structures and sea quarks as light quarks of mass of order electron mass appearing as bonds in nuclei. They could correspond to flux loops with length of order electron's p-adic length scale L(127), which is of the order of electron Compton length.

6. Can one understand the mass m(Λ)= 1116 MeV of Λ baryon containing also strange quark s? The mass difference m(Λ) -m(n)≈ 178 MeV cannot correspond to the mass difference m(s)-m(d), which in absence of topological mixing would be maximal and in this case given by ((17/5)^1/2 -(8/5)^1/2) m(u,111)≈ 81 MeV. This is too small to explain the Λ-n mass difference.

   Could energy minimization be achieved by replacing the s-π⁰ pair with p-K^- pair solve the problem? Kaon mass is 493 MeV so that m(Λ)-m(p) would be equal to m(K)-m(π)≈ 353 MeV for k=113. This is too large by a factor 2. For k=115 one would obtain mass difference 176.1 MeV to be compared with real mass difference m(Λ) -m(n)≈ 178 MeV!

If this picture is correct, the p-adic length scale hypothesis would make it possible to build mesons from quarks with masses predicted by and various mesons whose masses would be sums of quark masses in the case of charged mesons and averages of sums in the case of neutral mesons. It is essential that quarks and mesons can exist in several p-adic mass scales related by a power of 2. Different topological mixings for U and D type quarks would explain CKM mixing. In mesons topological mixing would not occur.

See the article About Platonization of Nuclear String Model and of Model of Atoms 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.

The duality between descriptions of strong interactions as dark weak interactions and color interactions

The assumption of pseudo-neutrons as analogs of p-π^- pairs could provide insights about the values of nuclear binding energy and also the 10 keV energy scale associated with low energy excitations of nuclei required by the proposal for the explanation of the tritium beta decay anomaly.

It is good to start from the problems of existing models.

1. The old-fashioned model of nuclei assumes pion exchanges as the origin of nuclear force. The basic problem of this view is that since gluons have vanishing electroweak quantum numbers and it is difficult to understand why charged pions as mediators of strong interactions could appear.
2. Second problem is that the mass of the pion is about 140 MeV and much larger than the 1 MeV scale of the nuclear excitations. In the harmonic oscillator model this scale is just assumed. Intriguingly, the neutron proton mass difference is also near 1 MeV. Also the electropions, explaining in the TGD framework (see this) the anomalous production of electron-positron pairs in heavy ion collisions discovered already at seventies, have mass very nearly equal to 2m_e= 1 MeV. There is also evidence for muopions and taupions but all this is forgotten since there is no room for these particles in the standard model since their existence would have been observed in weak boson decays. This problem is circumvented if electropions are dark in the TGD sense.

In TGD color is realized as color partial waves in CP₂ degrees of freedom rather than as spin-like quantum numbers and both quarks electrons also allow colored excitations. In particular, leptons allow octet excitations. The anomalous electron-positron pairs would result in the decays of electropions identified as the bound states of colored electron and positron. Also the triplets of color octet leptons allow color singlets analogous to baryons and in the model of nucleus these triplets would occur at vertices from which free edge begins. If this were the case, then colored lepton states could be an essential part of nuclear physics. Note however that one can also consider a model of electropions based on scaled variants of quarks with mass scale defined by the p-adic length scale L(127) of electron.

An attractive proposal is that the pion-like color bonds in nuclei correspond to electropions or their quark-antiquark counterparts and that the mass of the neutral electropion is 1 MeV. In the following I will talk about electronpion also when it consists of quark and antiquark.

1. One can estimate the mass of the charged electropion from the mass of neutral electropion by scaling the mass difference of pion which is Δ m(π)=m(π⁰)- m(π^+/-)= (139.6 -135.0)=4.6 MeV. By scaling this with m(π_L)/m(π)≈; 1/140 one obtains Δ m(π_L)≈; 33 keV, which could correspond to the 10 keV energy scale.
2. Nucleon- flux tube pairings of type n-π_L⁰ and p-π_L^- (pseudo-neutron) and p-π⁰_L and n-π⁺_L (pseudo-proton) pairings are possible. The energy differences between these nucleon-flux tube pairs would naturally correspond to the n-p mass difference of about 1 MeV.

   The first guess is that in nuclear ground states with a minimum energy pseudo neutron p-π_L^- and genuine proton p-π_L⁰ are favored. The excited states with excitation energies in 10 keV scale contain genuine protons p-π_L⁰ and pseudo protons n-π_L^-. Also the excited states n-π_L⁺ can be excited to states n-π_L⁰ with excitation energy in the 10 keV range.

3. Dark W exchanges between nucleons and leptopions would transform the genuine and pseudo variants of nucleons to each other. These transformations involving energy change of order 1 MeV could be behind the excitations of nuclei usually assigned with strong interactions. If W bosons are dark and thus effectively massless in the scale of nuclei these transitions would be fast. This would also concretize the PCAC and CVC inspired idea that somehow strong interactions are dual to weak interactions. What is remarkable is that the value of L_W is not now fixed by the condition L_W= a_W stating that the weak interaction range is at least the Bohr radius of the weak atom! The value of ℏ_eff/ℏ\sim m_W/m_e ≈; 10⁵ would make the leptopions dark with respect to weak interactions.

Tritium beta decay anomaly again

There are several questions to be answered. Does the already proposed mechanism possibly explaining the tritium anomaly have alternatives? What really happens in the tritium beta decay? Can one understand the 10 keV scale in the anomalous tritium beta decay? How can the X-ray flux from the Sun amplify the beta decay anomaly? One can also ask whether the proposed idea about duality between dark weak interactions and strong interactions could allow a concrete quantitative formulation.

1. One should transform π_L^+/- bond to π_L⁰ bond or vice versa by emission of W boson but this changes the charge of H³ unless the W decays to e-ν pair. The exchange of W boson between nucleon and leptopion that is quark/lepton of the corresponding bond involves energy change of about 1 MeV in the process and is considerably larger than 10 keV scale for X rays. A possible mechanism inducing transition between these states would be a variant of beta decay involving a spontaneous beta decay of pseudo-neutron decaying as n-π_L⁰ → p- π_L⁰+ +W^- followed by W^-→ e^- +ν^*.

   In the spontaneous decays of p-π_L⁰ → p-π_L^- + +W⁺ and n-π_L⁺ → n- -π_L⁰ +W⁺ genuine and pseudo proton the scale of energy change is is 10 keV and these transitions could be involved with the tritium beta anomaly.

2. I have already considered a possible mechanism for tritium beta decay involving neutrino atoms and transition n+ν→ p+e. Could one consider alternative mechanisms or at least analogous transitions.

   The spontaneous decay n-π_L⁰ → n- π_L⁺ +W^- is kinematically possible whereas p-π_L^-→ p- π_L⁺ +W^- is not allowed by energy conservation.

   One can imagine also a variant of beta decay involving a spontaneous beta decay of pseudo-neutron decaying as p-π_L⁰ → p-π_L^- +W⁺ and n-π_L⁺ → n-π_L⁰ +W⁺ followed by W⁺→ e⁺ +ν. Now one would however have W⁺ rather than W^- in the final state.

   The energy of W^- and of e^-+ν^* is constrained by the mass difference Δ m(π_L)≈; 33 keV and by energy conservation. The mass difference m(H³)-m(He³) =18 keV is but pseudo neutrons . This beta decay could explain the tritium anomaly instead of n+ν → p+ e^- for the generalized atom.

Why tritium beta anomaly correlates with the X-ray flux from the Sun?

The proposed model of the beta anomaly in terms of a decay p-π_L⁰→ p-π_L^- +W⁺ does not yet explain the correlation of the beta decay anomaly with X ray emission from the Sun. Could X ray absorption with X ray energy equal to the excitation energy induce reverse dark weak transitions p-π_L^- → p-π_L⁰ ? A possible mechanism would be following:

1. The absorption of X-ray by π_L^- (π_L⁺) occurs first and increases the energy of p-π_L^- but does not induce its decay if the energy of X-ray is not much larger than Δ m(π_L). X ray can be also absorbed by n-π_L⁺.
2. After this the exchange of dark W^- between π_L^- and induces the transition p-π_L^-→ n-π_L⁰. In the same way, n-π_L⁺ can be transformed to p-π_L⁰.

See the article About Platonization of Nuclear String Model and of Model of Atoms 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.