**1. M-matrix and coupling constant evolution**

The final breakthrough in the understanding of p-adic coupling constant evolution came through the understanding of S-matrix, or actually M-matrix defining entanglement coefficients between positive and negative energy parts of zero energy states in zero energy ontology (see this). M-matrix has interpretation as a "complex square root" of density matrix and thus provides a unification of thermodynamics and quantum theory. S-matrix is analogous to the phase of Schrödinger amplitude multiplying positive and real square root of density matrix analogous to modulus of Schrödinger amplitude.

The notion of finite measurement resolution realized in terms of inclusions of von Neumann algebras allows to demonstrate that the irreducible components of M-matrix are unique and possesses huge symmetries in the sense that the hermitian elements of included factor N subset M defining the measurement resolution act as symmetries of M-matrix, which suggests a connection with integrable quantum field theories.

It is also possible to understand coupling constant evolution as a discretized evolution associated with time scales T_{n}, which come as octaves of a fundamental time scale: T_{n}=2^{n}T_{0}. Number theoretic universality requires that renormalized coupling constants are rational or at most algebraic numbers and this is achieved by this discretization since the logarithms of discretized mass scale appearing in the expressions of renormalized coupling constants reduce to the form log(2^{n})=nlog(2) and with a proper choice of the coefficient of logarithm log(2) dependence disappears so that rational number results.

**2. p-Adic coupling constant evolution**

One can wonder how this picture relates to the earlier hypothesis that p-adic length coupling constant evolution is coded to the hypothesized log(p) normalization of the eigenvalues of the modified Dirac operator D. There are objections against this normalization. log(p) factors are not number theoretically favored and one could consider also other dependencies on p. Since the eigenvalue spectrum of D corresponds to the values of Higgs expectation at points of partonic 2-surface defining number theoretic braids, Higgs expectation would have log(p) multiplicative dependence on p-adic length scale, which does not look attractive.

Is there really any need to assume this kind of normalization? Could the coupling constant evolution in powers of 2 implying time scale hierarchy T_{n}= 2^{n}T_{0} induce p-adic coupling constant evolution and explain why p-adic length scales correspond to L_{p} propto p^{1/2}R, p≈ 2^{k}, R CP_{2} length scale? This looks attractive but there is a problem. p-Adic length scales come as powers of 2^{1/2} rather than 2 and the strongly favored values of k are primes and thus odd so that n=k/2 would be half odd integer. This problem can be solved.

- The observation that the distance traveled by a Brownian particle during time t satisfies r
^{2}= Dt suggests a solution to the problem. p-Adic thermodynamics applies because the partonic 3-surfaces X^{2}are as 2-D dynamical systems random apart from light-likeness of their orbit. For CP_{2}type vacuum extremals the situation reduces to that for a one-dimensional random light-like curve in M^{4}. The orbits of Brownian particle would now correspond to light-like geodesics γ_{3}at X^{3}. The projection of γ_{3}to a time=constant section X^{2}subset X^{3}would define the 2-D path γ_{2}of the Brownian particle. The M^{4}distance r between the end points of γ_{2}would be given r^{2}=Dt. The favored values of t would correspond to T_{n}=2^{n}T_{0}(the full light-like geodesic). p-Adic length scales would result as L^{2}(k)= D T(k)= D2^{k}T_{0}for D=R^{2}/T_{0}. Since only CP_{2}scale is available as a fundamental scale, one would have T_{0}= R and D=R and L^{2}(k)= T(k)R. - p-Adic primes near powers of 2 would be in preferred position. p-Adic time scale would not relate to the p-adic length scale via T
_{p}= L_{p}/c as assumed implicitly earlier but via T_{p}= L_{p}^{2}/R_{0}= p^{1/2}L_{p}, which corresponds to secondary p-adic length scale. For instance, in the case of electron with p=M_{127}one would have T_{127}=.1 second which defines a fundamental biological rhythm. Neutrinos with mass around .1 eV would correspond to L(169)≈ 5 μm (size of a small cell) and T(169)≈ 10^{4}years. A deep connection between elementary particle physics and biology becomes highly suggestive. - In the proposed picture the p-adic prime p≈ 2
^{k}would characterize the thermodynamics of the random motion of light-like geodesics of X^{3}so that p-adic prime p would indeed be an inherent property of X^{3}. - The fundamental role of 2-adicity suggests that the fundamental coupling constant evolution and p-adic mass calculations could be formulated also in terms of 2-adic thermodynamics. With a suitable definition of the canonical identification used to map 2-adic mass squared values to real numbers this is possible, and the differences between 2-adic and p-adic thermodynamics are extremely small for large values of for p≈ 2
^{k}. 2-adic temperature must be chosen to be T_{2}=1/k whereas p-adic temperature is T_{p}= 1 for fermions. If the canonical identification is defined as∑

_{n≥ 0}b_{n}2^{n}→ ∑_{m ≥1}2^{-m+1}∑_{0≤ n< k}b_{n+(k-1)m}2^{n},it maps all 2-adic integers n<2

^{k}to themselves and the predictions are essentially same as for p-adic thermodynamics. For large values of p≈ 2^{k}2-adic real thermodynamics with T_{R}=1/k gives essentially the same results as the 2-adic one in the lowest order so that the interpretation in terms of effective 2-adic/p-adic topology is possible.

## 2 comments:

Your map on this is not coming up here.

http://www.tgdtheory.fi/cmaphtml.html

Thank you for informing. I try to find the reason.

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