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Updates

Here are some updates (mini blog) on the paper Counting and Cohomology (as well as also at the paper on particles on primes). More is also on the Quantum calculus blog.

• A Cern press release about Penta quarks. If the number theory picture should point to the right direction, then penta quarks would look like this and not like that. It would be a molecule, and not a particle:
• A short description of the core insights is on this blog entry. There is also something about Kustaanheimo.
• August 30. A question: One question is this that between all SU(n) symmetries why we only see U(1), SU(2) and SU(3) in standard model? Could your approach answer to this question?. The group U(1) is the unit sphere in C, the group SU(2) is the unit sphere in H. The action of the group SU(3) is described in section 2.7 of the paper but it is a nonlinear action. One could let other Lie groups act by on the unit sphere of H but then it appears not natural. Here the group action of SU(3) or U(3) is related to permutations of the coordinates which are "strong" symmetries. Maybe its just coincidence that basic symmetry groups appear naturally together in the algebras C and H. Most likely the structural analogies are of the same kind as in this paper which made a lot of sense to me, even before learning abstract algebra. As many other kids at that time, I also cracked the rubik cube without any knowledge of abstract group theory but we all knew by experience (playing with the cube nonstop for weeks, and even dreaming about it) that one can not turn one corner by 120 degrees without affecting other cubes. It is a "quark" situation. Analogies between particles and primes is far from new. Popular is the primon gas or "arithmetic gas" which was mentioned in section 7.4 of this note. It is a nice model because the Riemann zeta function is the partition function in a Boson framework or its reciprocal in a Fermion frame work. By the way, this entire string of analogies has started with the following reformulation of quadratic reciprocity. If (p|q) is the Jacobi symbol then for two odd primes (p|q) = -(q|p) if and only if both p,q are Fermions (primes of the form 4k+3, which I associate with neutrini because they are neutral and are light as the square root of the arithmetic norm is prime, not the norm like for 4k+1 primes). The primes of the form 4k+1 are Bosons as they are the composite of two Gaussian primes (a+i b)(a- i b) which I dub electron-positron pair. This is at least for me to never forget or mix up the statement of quadratic reciprocity and to remember the Fermat 2 square theorem or see the Lagrange 4 square theorem as the statement that there are no neutrini type particles among hadrons. It is evident in many places of number theory that the prime p=2 plays a special role, not only because it is even. It is ramified. It is splits into (1+i)(1-i) but the two factors are conjugate -i(1+i) = 1-i. So, its charge is zero and it is Boson like. What particle comes closest with this feature? It is the Higgs boson. Now this sounds wacky but it is not so much if you see 2 also a fundamental scaling symmetry of space especially in the context of universality: in the Barycentric refinement (doubling the grid) we see a limit which always spectrally unique.
  The paper cautions at various places about the danger of analogies. For me, that paper is a motivation to look closer at the structure of primes in H and especially at their symmetries. The actual physics is certainly much more complicated. Its a bit unfortunate that the document was bumped into physics, since it does not do any physics. But yes, the relations between mathematical and physical structures are exciting. Still, for getting physics, one has to be able to predict something new quantitatively. So far, discrete approaches have not delivered, except for numerical approximations of continuous models (PDE's an exmple being Regge calculus or gauge theories like lattice gauge theories). I believe this will only change if a model can predict quantitatively some new phenomenon which then can be confirmed and measured.
• August 29: the following picture shows the connected component of the prime graph for n = 2*3*5*7*11 = 2310. The vertices are all the integers between 2 and n which are square free. Two integers are connected if one is a factor of the other:
• August 28: I initially tried to make a worksheet and animation for Extension class but it turned out to be better to go with a geometric theme related to Pythagoras. In the following animation, the unit spheres are painted yellow or green. Yellow, if it is a sphere. In that case, the sphere and the newly added point produce a new "handle" to the space and the Euler characteristic changes by 1. The case when a new prime is added is a situation where the handle is zero dimensional (the unit sphere of the point is -1 dimenisional). Here is the movie: