### TGD assigns 10 Hz biorhythm to electron as an intrinsic frequency scale

p-Adic coupling constant evolution and origins of p-adic length scale hypothesis have remained for a long time poorly understood. The progress made in the understanding of the S-matrix of the theory (or rather, its generalization M-matrix) (see this) has however changed the situation. The unexpected prediction is that zero energy ontology assigns to elementary particles macroscopic times scales. In particular, the time scale assignable to electron correspond to the fundamental biorhythm of 10 Hz.

**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^{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. p-Adic coupling constant evolution**

Could the time scale hierarchy T_{n}= 2^{i}T_{0} defining hierarchy of measurement resolutions in time variable 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 \gamma_{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^{i}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)≈ 1.× 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^{-km}∑_{0≤ n< m}b_{km+n}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.

**3. p-Adic 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 Bose-Einstein condensates of hitherto unknown particles.

- For electron one has T(k)=.1 seconds which defines the fundamental f
_{e}=10 Hz bio-rhythm appearing as a peak frequency in alpha band. This could be seen as a direct evidence for a Bose-Einstein condensate of Cooper pairs of high T_{c}super-conductivity. 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 anti-quarks with masses not too far from electron mass. The corresponding p-adic 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 B-E 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 17-multiples of 2-adic length scales would result besides 2-adic 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 p-adic 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".

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