TGD assigns 10 Hz biorhythm to electron as an intrinsic frequency scale
pAdic coupling constant evolution and origins of padic length scale hypothesis have remained for a long time poorly understood. The progress made in the understanding of the Smatrix of the theory (or rather, its generalization Mmatrix) (see this) has however changed the situation. The unexpected
prediction is that zero energy ontology assigns to elementary p"/public_html/articles/ macroscopic times scales. In particular, the time scale assignable to electron correspond to the fundamental biorhythm of 10 Hz.
1. Mmatrix and coupling constant evolution
The final breakthrough in the understanding of padic coupling constant evolution came through the understanding of Smatrix, or actually Mmatrix defining entanglement coefficients between positive and negative energy parts of zero energy states in zero energy ontology (see this). Mmatrix has interpretation as a "complex square root" of density matrix and thus provides a unification of thermodynamics and quantum theory. Smatrix 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 Mmatrix 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 Mmatrix, 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^{i}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^{i})=nlog(2) and with a proper choice of the coefficient of logarithm log(2) dependence disappears so that rational number results.
2. pAdic coupling constant evolution
Could the time scale hierarchy T_{n}= 2^{i}T_{0} defining hierarchy of measurement resolutions in time variable induce padic coupling constant evolution and explain why padic 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. pAdic 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. pAdic thermodynamics applies because the partonic 3surfaces X^{2} are as 2D dynamical systems random apart from lightlikeness of their orbit. For CP^{2} type vacuum extremals the situation reduces to that for a onedimensional random lightlike
curve in M^{4}. The orbits of Brownian particle would now correspond to lightlike geodesics \gamma_{3} at X^{3}. The projection of γ_{3} to a time=constant section X^{2} subset X^{3} would define the 2D 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^{i}T_{0} (the full lightlike geodesic). pAdic 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.
 pAdic primes near powers of 2 would be in preferred position. pAdic time scale would not relate to the padic 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 padic 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)≈ 1.× 10^{4} years. A deep connection between elementary particle physics and biology becomes highly suggestive.
 In the proposed picture the padic prime p≈ 2^{k} would characterize the thermodynamics of the random motion of lightlike geodesics of X^{3} so that padic prime p would indeed be an inherent property of X^{3}.
 The fundamental role of 2adicity suggests that the fundamental coupling constant evolution and padic mass calculations could be formulated also in terms of 2adic thermodynamics. With a suitable definition of the canonical identification used to map 2adic mass squared values to real numbers this is possible, and the differences between 2adic and padic thermodynamics are extremely small for large values of for p≈ 2^{k}. 2adic temperature must be chosen to be T^{2}=1/k whereas padic temperature is T_{p}= 1 for fermions. If the canonical identification is defined as
∑_{n≥0} b_{n} 2^{n}→ ∑_{m≥1}2^{km}∑_{0≤ n< m} b_{km+n}2^{n}.
It maps all 2adic integers n<2^{k} to themselves and the predictions are essentially same as for padic thermodynamics. For large values of p≈ 2^{k} 2adic real thermodynamics with T_{R}=1/k gives essentially the same results as the 2adic one in the lowest order so that the interpretation in terms of effective 2adic/padic topology is possible.
3. pAdic length scale hypothesis and biology
The basic implication of zero energy ontology is the formula T(k)≈ 2^{k/2}L(k)/c= L(2,k)/c.
This would be the analog of E=hf in quantum mechanics and together hierarchy of Planck constants
would imply direct connection between elementary particle physics and macroscopic physics. Especially
important this connection would be in macroscopic quantum systems, say for Bose Einstein condensates of Cooper pairs, whose signature the rhythms with T(k) as period would be. The presence of this kind of rhythms might even allow to deduce the existence of BoseEinstein condensates of hitherto unknown p"/public_html/articles/.
 For electron one has T(k)=.1 seconds which defines the fundamental f_{e}=10 Hz biorhythm appearing as a peak frequency in alpha band. This could be seen as a direct evidence for a BoseEinstein condensate of Cooper pairs of high T_{c} superconductivity. That transition to "creative" states of mind
involving transition to resonance in alpha band might be seen as evidence for formation of large BE condensates of electron Cooper pairs.
 TGD based model for atomic nucleus (see this) predicts that nucleons are connected by flux tubes having at their ends light quarks and antiquarks with masses not too far from electron mass. The corresponding padic frequencies f_{q}= 2^{k}f_{e} could serve as a biological signature of exotic quarks connecting nucleons to nuclear strings . k_{q}=118 suggested by nuclear string model would give f_{q}= 2^{18}f_{e}=26.2 Hz. Schumann resonances are around 7.8, 14.3, 20.8, 27.3 and 33.8 Hz and f_{q} is not too far from 27.3 Hz Schumann resonance and the cyclotron frequency f_{c}(^{11}B^{+})=27.3 Hz for B=.2 Gauss explaining the effects of ELF em fields on vertebrate brain.
 For a given T(k) the harmonics of the fundamental frequency f=1/T(k) are predicted as special time scales. Also resonance like phenomena might present. In the case of cyclotron frequencies they would favor values of magnetic field for which the resonance condition is achieved. The magnetic field which in case of electron gives cyclotron frequency equal to 10 Hz is B_{e}≈ 3.03 nT. For ion with charge Z and mass number A the magnetic field would be B_{I}= (A/Z)× (m_{p}/m_{e})×B_{e}. The B=.2 Gauss magnetic field explaining the findings about effects of ELF em fields on vertebrate brain is near to B_{I} for ions with f_{c} alpha band. Hence the value of B could be understood in terms of resonance with electronic BE condensate.
 The hierarchy of Planck constants predicts additional time scales T(k). The prediction depends on the strength of the additional assumptions made. One could have scales of form nT(k)/m with m labeling the levels of hierarchy. m=1 would give integers multiples of T(k). Integers n could correspond to ruler and compass integers expressible as products of first powers of Fermat primes and power of 2. There are only four known Fermat primes so that one has n=2^{i}∏_{i} F_{i}, F_{i} in {3,5,17,257, 2^{16}+1}. In the first approximation only 3 and 5 and 17multiples of 2adic length scales would result besides 2adic length scales. In more general case products m^{1}m^{2} and ratios m^{1}/m^{2} of ruler and compass integers and their inverses 1/m^{1}m^{2} and m^{2}m^{1} are possible.
 Mersenne primes are expected to define the most important fundamental padic time scales. The list of real and Gaussian (complex) Mersennes M_{n} possibly relevant for biology is given by n=89, 107, 113*, 127, 151*,157*, 163*, 167* ('*' tells that Gaussian Mersenne is in question). See the
table.
For background see that chapter New Physics and Qualia of "Quantum Hardware of Living Matter".
