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.