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 Tn, which come as octaves of a fundamental time scale: Tn=2nT0. 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(2n)=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 Tn= 2nT0 induce p-adic coupling constant evolution and explain why p-adic length scales correspond to Lp propto p1/2R, p≈ 2k, R CP2 length scale? This looks attractive but there is a problem. p-Adic length scales come as powers of 21/2 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 r2= Dt suggests a solution to the problem. p-Adic thermodynamics applies because the partonic 3-surfaces X2 are as 2-D dynamical systems random apart from light-likeness of their orbit. For CP2 type vacuum extremals the situation reduces to that for a one-dimensional random light-like curve in M4. The orbits of Brownian particle would now correspond to light-like geodesics γ3 at X3. The projection of γ3 to a time=constant section X2 subset X3 would define the 2-D path γ2 of the Brownian particle. The M4 distance r between the end points of γ2 would be given r2=Dt. The favored values of t would correspond to Tn=2nT0 (the full light-like geodesic). p-Adic length scales would result as L2(k)= D T(k)= D2kT0 for D=R2/T0. Since only CP2 scale is available as a fundamental scale, one would have T0= R and D=R and L2(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 Tp= Lp/c as assumed implicitly earlier but via Tp= Lp2/R0= p1/2Lp, which corresponds to secondary p-adic length scale. For instance, in the case of electron with p=M127 one would have T127=.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)≈ 104 years. A deep connection between elementary particle physics and biology becomes highly suggestive.
- In the proposed picture the p-adic prime p≈ 2k would characterize the thermodynamics of the random motion of light-like geodesics of X3 so that p-adic prime p would indeed be an inherent property of X3.
- 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≈ 2k. 2-adic temperature must be chosen to be T2=1/k whereas p-adic temperature is Tp= 1 for fermions. If the canonical identification is defined as
∑n≥ 0 bn 2n→ ∑m ≥1 2-m+1∑0≤ n< k bn+(k-1)m2n ,
it maps all 2-adic integers n<2k to themselves and the predictions are essentially same as for p-adic thermodynamics. For large values of p≈ 2k 2-adic real thermodynamics with TR=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.