Tuesday, August 21, 2007

Back at home

I returned home from a little conference in Röros, Norway. The conference was about various anomalous phenomena and organized by the Society of Scientific Exploration. Participants were science professionals and the atmosphere very very warm. It is amazing to see that scientists can be critical without debunking, crackpotting, and casting personal insults as the discussion culture in so many physics blogs would suggest. I am starting to believe again that scientists can behave like civilized human beings rather than barking and biting like mad dogs! Lectures were absolutely excellent and I got for the first time in my life through the entire lecture of mine;-)! This was a very enjoyable event and I have a lot of new ideas to digest.

In her blog Kea mentions a continuous geometry in a sense of von Neumann. It is obtained by taking a finite field of order q=pn and taking the so called pro-finite limit of projective geometries P(1,q)→ P(2,q)→P(4,q)→...P(2n,q)→... At the limit this geometry contains subspaces of any dimension d in the interval [0,1]. For Jones inclusions the indices are quantized in as M:N= 4cos2(π/n), n=3,4,...

I should check what this continuous geometry of von Neumann really means before saying anything but I cannot avoid the temptation to say that this brings strongly in mind TGD related notions suggesting some generalizations. Well...! I should stop here but I will take the risk of making fool of myself as so many times before.

1. Another manner to see the continuous geometry

The inclusion sequence of tensor powers of Clifford algebra defining hyper-finite factors is a counterpart for the inclusion sequence defining the continuous geometry. 22n corresponds for Jones inclusions the dimension of matrix algebra obtained as a tensor power of 2×2 matrix/Clifford algebra.

Something very closely related to the spinor counterpart associated with infinite-D Clifford algebra should be in question. Complex 2-spinors define S2 as a 1-D complex projective space. Is quantum version of this space in question? For quantum spinors the dimension would vary in range (1,2) in discrete manner (square root of index 1≤M:N≤4) and for quantum S2 it would not be larger than 1 if complex numbers are involved. One could also consider a restriction to real numbers.

2. Generalizing the continuous geometry keeping q=pn fixed

I would bet that this construction generalizes considerably since Jones inclusions correspond to spinor representation of SU(2) and all compact groups and all their representations define inclusion series for all all values of dimension n of Abelian group Zn defining quantum phase. The properties of HFFs suggest that the powers of 2 could be replaced with powers of any integer and primes are especially interesting in this respect. All quantum counterparts of various projective spaces associated with spinor representations of various compact Lie groups (at least subset of ADE type groups) might be obtained by allowing n in q=pn to vary. n would also correspond to quantum phase Q= exp(i2π/n).

3. Could one replace finite fields with extensions of p-adic numbers and glue together p-adic continuous geometries?

p-Adic TGD for a given p suggests that one could generalize the continuous geometries by regarding finite field G(p,n) as n-dimensional algebraic extension of G(p,1) and replacing G(p,n) with an n-dimensional algebraic extension of p-adic numbers. This could give p-adic variants of a quantum projective geometry. For prime values of n the powers of quantum phase Q=exp(i2π/n) would define a concrete representation for the units of G(p,n).

The appearance of quantum phases Q=exp(i2π/n) also bring in mind a generalization of the notion of imbedding space involving hierarchy of cyclic groups inspired by dark matter hierarchy and hierarchy of Planck constants.

A further extension inspired by quantum TGD would be the gluing structures with different values of p together (generalization of the notion of number by gluing reals and p-adics along common rationals and perhaps also algebraics).

4. The stochastic process associated with Riemann Zeta

In my own primitive physicist's way I "know" that Riemann Zeta has a fundamental role in the construction of quantum TGD and it appears in the concrete formulas involving number theoretic braids involving also several number theoretical conjectures. Unfortunately, I have no rigorous articulation for these gut feelings.

Kea mentions also a family of stochastic processes with integer valued dynamical variable n=1,2,... The processes are parameterized by a positive integer s=1,2,... and the probabilities are given by n-s so that the partition function is Riemann Zeta for a positive integer valued argument. Thermodynamical interpretation requires that log(n) are the eigenvalues of "energy". In arithmetic quantum field theory log(n) has indeed interpretation as "energy" of a many particle state (this relates closely to infinite primes). One might hope that a proper generalization of this stochastic process might help to get additional insights also to the role of Zeta in TGD.

The generalization of the stochastic process to M-matrix inspired by the zero energy ontology is natural. s is analogous to the inverse temperature and analytic continuation would mean that also complex temperatures are considered. Partition function would have interpretation as a complex square root of density matrix with complex phases identified as elements of S-matrix in the diagonal representation. This would with with the zero energy ontology inspired unification of the density matrix and S-matrix to Matrix defining the coefficients of time like entanglement between positive and negative energy states. Zeta(s) for complex values of s would thus define naturally elements of a particular M-matrix.

There are several questions.

  1. The first questions relate to the identification of the "energy" having values log(n). Does n label the tensor powers of the 2×2 Clifford algebra appearing in the sequence of inclusions appearing in the definition of the hyper-finite factor of type II1? Or could it correspond to n in G(p,n) and thus to the quantum phase Q and more generally, to the dimension of algebraic extension of p-adic numbers?

    It would seem that only the first interpretation makes sense since there exists a large number of algebraic extensions of rationals with dimension n. Hence the first attempt to interpret would be that p(n) gives the probability that the system's state corresponds is created by the n:th tensor power of 2×2 Clifford algebra. Different n:s should define independent states and this is not necessarily consistent with the inclusion sequence in which lower dimensional tensor powers are included to higher dimensional ones unless the probabilities correspond to states obtained by identifying states which differ by a state created by a lower tensor power.

  2. For the zeros of Zeta the interpretation as a stochastic process or M-matrix obviously fails. What could this mean?

    1. In statistical physics partition function vanishes at criticality and in TGD zeros of Zeta correspond to quantum criticality at which the value of Planck constant can change. This suggests that one should consider at quantum criticality only the cutoffs with "energy" not larger than log(n). The cutoff in the sum defining Riemann Zeta would mean that only m≤ n tensor powers of 2×2 Clifford algebra create the positive and negative energy states pairing to form zero energy states.

      Physically this would mean that thermodynamical states would contain only pairs of positive and negative energy states for which positive/negative energy states have fermion number not larger than n. Note that coherent states of Cooper pairs are consistent with fermion number conservation only in zero energy ontology.

    2. If the TGD inspired conjecture that piIm(s) is algebraic number for zeros of zeta holds true, the partition function defined by zeta with cutoff would be algebraic number and cutoff M-matrix would be algebraically universal.

To sum up, these observations support my gut feeling that p-adic physics, hierarchy of Jones inclusions, and the hierarchy of Planck constants plus the generalization of imbedding space inspired by it, appear as aspects of the same very general mathematical structure obtained by gluing together a lot of structures. Also the process of gluing copies of 8-D imbedding space together along 4-D subspaces to form a larger structure should relate to this.

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