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.