How p-adic coupling constant evolution and p-adic length scale hypothesis emerge from quantum TGD proper?

What p-adic coupling constant evolution really means has remained for a long time more or less open. The progress made in the understanding of the S-matrix of theory has however changed the situation dramatically.

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=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 T_n= 2ⁿT₀ induce p-adic coupling constant evolution and explain why p-adic length scales correspond to L_p propto p^1/2R, p≈ 2^k, R CP₂ 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.

1. 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 T_n=2ⁿT₀ (the full light-like geodesic). p-Adic length scales would result as L²(k)= D T(k)= D2^kT₀ 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.

2. 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_p²/R₀= p^1/2L_p, 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.

3. In the proposed picture the p-adic prime p≈ 2^k 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.

4. 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 T₂=1/k whereas p-adic temperature is T_p= 1 for fermions. If the canonical identification is defined as

   ∑_{n≥ 0} b_n 2ⁿ→ ∑_{m ≥1} 2^-m+1∑_{0≤ n< k} b_n+(k-1)m2ⁿ ,

   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.

For background see the chapter Construction of Quantum Theory: S-Matrix of "Towards S-matrix".