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.