Non-commutative geometry: warmly recommended

I love to visit the good old This Week's Finds . Down-to-earth explanations of mathematical concepts and ideas, good humor, and no putting down mentality. What more one could hope for.

This Week's Finds is not anymore the only one of its kind. A new highly interesting blog has appeared in the blogsphere at the boundary layer between mathematics and physics: Noncommutative Geometry. I warmly recommend it to myself and theoretical physicists who might still dare to dream about the day when M theory is not anymore the only possible theory of even more than everything and that the proponents of competing theories are not anymore regarded as mindless crackpots and human waste.

Number theory as generalized physics

Amusingly, and rather satisfyingly, the basic theme during last days has been number theory as generalized quantum physics whereas my own basic theme (at much lower level of sophistication) has been physics as a generalized number theory.

One of very interesting articles briefly discussed in the blog is Quantum Statistical Mechanics and Class Field Theory.

1. A physical model allowing to construct the generators of the maximal Abelian extension of rationals and identify the action of corresponding Galois group (call it Gal) on these generators, is proposed. Someone might build an association with Hilbert's twelth problem.
2. The idea is to assign a C^* algebra to so called Hecke algebra appearing in Langlands program. The quantal time evolution and thermodynamics of this simple number theoretical quantum system is essentially unique since it corresponds to a state in a hyperfinite factor of type III.
3. What is amazing is that the numerical values of states of this system at the infinite temperature limit code for the algebraic numbers generating the maximal Abelian extension of rationals. This kind of condition is fundamental in TGD program of physics as generalized number theory having universality with respect to number field. Furthermore, Gal acts as symmetries of the system.
4. Partition function is nothing but Riemann Zeta as a function of temperature. Rieman Zeta and more general zetas have become an essential element of quantum TGD although I am very far from rigorous formulations yet.
5. There are two phases: low temperature phase corresponding to 1/T> 1 and high temperature phase corresponding to critical strip 1/T≤ 1. At T=1 partition function diverges: a signature of phase transition. At low temperature phase the action of Gal is non-trivial and thermal expectations of observables transforming non-trivially under Gal are non-vanishing. At high temperature phase the action of Gal is trivial meaning that thermal expectations of observables transforming non-trivially under Gal vanish. There is complete analogy with non-trivial action of gauge group on Higgs vacuum expectation value below critical temperature.

6. In TGD framework somewhat different breaking of gauge symmetry occurs for the Galois group of closure or rationals identified as infinite symmetric group whose group algebra is..., you guessed correctly: hyper-finite factor of type II₁! The finite Galois groups to which symmetry breaking occurs correspond are analogous to unbroken subgroups of gauge groups and directly related to Jones inclusions.

The article inspires interesting questions which only an innocent novice blessed with deep ignorance about basics can articulate.

1. At the critical line at which partition function is real. Could one consider quantum field theory at a finite temperature 1/T= Re(s)=1/2 and finite time interval t=Im[s]?

2. The partition function vanishes for the values of t corresponding to non-trivial zeros of zeta. What could this mean physically? Could it be that along critical line symmetry is broken in step wise manner. At first zero the thermal expectations for some non-singlets under some subgroup H of Gal become non-vanishing. At the next zero H is extended. In accordance with some earlier TGD inspired speculations this process would gradually lead to the entire Abelian Galois group and emergence of all phases generating maximal abelian extension of rationals. This suggests also a possible connection with Jones inclusions coded by phases exp(iπ/n) and cyclic group Z_n.

3. The reckless number theoretic speculations inspired by TGD inspire further questions. Could it be that for zeros the numbers p^iy appearing in the product decomposition of zeta are algebraic numbers in maximal Abelian extension? Could also zeros be algebraic numbers as the most stringent speculations suggest?

Physics as generalized number theory

I find this article highly interesting for many reasons.

1. In TGD framework finite subgroups of Galois group of Galois group of algebraic closure of rationals play a fundamental role as group of symmetries of number theoretic braids (see earlier postings).

2. Number theoretic universality requires that everything reduces to algebraic numbers belonging to the algebraic extension of rationals or p-adics corresponding to the level of predicted number theoretic hierarchy. For instance, the maxima of Kähler function coding for the geometry of the world of classical worlds would be algebraic number (also Neper number e and its roots could be allowed in p-adic context if extensions of p-adics are required to be finite-dimensional). S-matrix elements would be algebraic numbers and so on. The model discussed realizes this dream for thermodynamical states at infinite temperature limit.

3. The notion of number theoretic braid defined as a subset of the intersection of real parton and its p-adic counterpart would consist of algebraic points so that the construction of S-matrix would reduce to discrete number theory.

4. It seems that number theoretical QFT at finite temperature based on hyperfinite factors of type III could be associated with the strands of number theoretic braids representing incoming partons propagating in generalized Feynman diagrams generalizing the braid diagrams. This would bring in p-adic thermodynamics at the fundamental level. Coupling constant evolution would be at the level of free states and no summation over loops would be needed.

Some ideas relating to physics, number theory, and biology

Before closing I want to repeat myself by recalling some associated ideas suggesting deep connections between number theory, quantum computation, and biology.

1. Vertices are direct generalizations of those for Feynman diagrams with point-like vertex being replaced with 2-dimensional partonic surface. They are totally different from string diagrams.

2. At the vertices number theoretic braids replicate and also DNA replication might involve this process as a deeper level process. Isomorphisms between tensor products of HFFs of type II₁ associated with incoming resp. outgoing lines would define vertices. Everything would be unitary and definitely non-trivial.

3. I ended up with braids from TGD inspired model for topological quantum computation using braids. Universe could be topological quantum computer at all levels of fractal hierarhcy. Copying of information represented by number theoretic braids would occur at vertices, its communication would takes place at propagator lines, and quantum computation would be performed at incoming lines.

To conclude, I think that Noncommutative Geometry might provide an excellent opportunity for a theoretical physicists to enjoy simple explanations of ideas which are difficult to extract from formal mathematical papers.