The two little observations to be made require some background about quantum p-adics.
- The work with quantum p-adics leads to the notion of arithmetic Feynman diagrams with +q and ×q representing the vertices of diagrams and having interpretation in terms of direct sum and tensor product. These vertices correspond to TGD counterparts of stringy 3-vertex and Feynman 3-vertex. If generalized Feynman diagrams satisfy the rules of quantum arithmetics, all loops can be eliminated by move representing the basic rules of arithmetics and the diagrams are invariant under permutations of outgoing resp. incoming legs and incoming legs involve only vertices and outgoing legs only co-vertices in the canonical representation of the generalized Feynman diagram. Possible modifications are possible and would be due to braiding meaning that the exchange of particles is not a mere permutation represented trivially. These symmetries are consistent with the prediction of zero energy ontology that virtual particles are pairs of on mass shell massless particles. The kinetic mass shell constraints indeed imply enormous reduction in the number of allowed diagrams. This means also a far reaching generalization of the duality symmetry of the old fashioned hadronic string model. I proposed this idea for years ago but gave it up as too "romantic".
- A beautiful connection with infinite primes emerges and p-adic primes characterizes collections of collections .... of quantum rationals which describe quantum dimensions of pairs of Hilbert spaces assignable to time-like and space-like braids ending at partonic 2-surfaces.
- The interpretation for the decomposition of quantum p-adic integers to quantum p-adic prime factors is in terms of a tensor product decomposition to quantum Hilbert spaces with quantum prime dimensions lq and can be related to the singular coverig spaces of imbedding allowing to describe the many-valuedness of the normal derivatives of imbedding space coordinates at space-like ends of space-time sheets at boundaries of CD and at lightlike wormhole throats. The further direction sum decompositions corresponding to different quantum p-adic primes assignable to l>p and represented by various quantum primes lq projecting to l in turn have interpretation in terms of p-adicity. The decomposition of n to primes corresponds to braid with strands labeled by primes representing Hilbert space dimensions.
- This gives a connection with the hierarchy of Planck constants and dark matter and quantum arithmetics. The strands of braid labeled by l decompose to strands correponding to the different sheets of covering associated with the singular covering of imbedding space: here one has however quantum direct sum decomposition meaning that particles are delocalized in the fiber of the covering.
The conservation of number theoretic multiplicative momenta at ×q vertex allows to deduce the selection rules telling what happens in vertices inolving particles with different values of Planck constant. There are two options depending on whether r= hbar/hbar0 satisfies r=n or r=n+1, where n characterizes the Hilbert space- dimension assignable to the covering of the imbedding space. For both options one can imagine a concrete phase transition leading from in-animate matter to living matter interpreted in terms of phases with non-standard value of Planck constant.
Consider now the two little observations.
- The first little observation is that these selection rules mean a deviation of the earlier proposal that only particles with same values of Planck constant can appear in a given vertex. This assumption explains why dark matter identified as phases with non-standard value of Planck constant decouples from ordinary matter at vertices. This explanation is however not lost albeit being weakened. If ×q vertex contains two particles with r=n+1 for r=n option (r=1 or 2 for r=n+1 option), also the third particle has ordinary value of Planck constant so that ordinary matter effectively decouples from dark matter. For +q vertex the decoupling of the ordinary from dark matter occurs for r=n+1 option but not for r=n option. Hence r=n+1 could explain the virtual decoupling of dark and ordinary matter from each other.
- Second little observation relates to the inclusions of hyper-finite factors which should relate closely to quantum p-adic primes because finite measurement resolution should be describable by HFFs. For prime p=2 one obtains quantum dimension 2q= 2cos(2π/n) in the most general case: n=p corresponds to p-adicity and more general values fo n-adicity. The interesting observation concerns the quantum dimension [M:N] obtained as quantum factor space M/N for Jones inclusion of hyper-finite factor of type I1 with N interpreted as an algebra creating states not distinguishable from each other in the measurement resolution used. This quantum dimension is 2q2 and has interpretation as dimension of 2× 2 quantum matrix algebra. This observation suggests the existence of infinite hierarchy of inclusions with [M:N]= pq2 labelled by primes p. The integer n would correspond to n-adicity meaning p-adicity for factors of n.
For details see the new chapter Quantum Adeles of "Physics as Generalized Number Theory".