The recent view is that the hierarchy of extensions of rational numbers induces a hierarchy of extensions of p-adic number fields in turn defining adele. This hierarchy gives rise to dark matter hierarchy labelled by a hierarchy of Planck constants and also evolutionary hierarchy. The hierarchy of Planck constants heff=n× h0 is and essential element of quantum TGD and adelic physics suggests the identification of n as the dimension of extension of rationals. n could be seen as a kind of IQ for the system.
What is also new is the proposal that preferred p-adic primes labelling physical systems could correspond to so called ramified primes, call them p, of extension of rationals for which the expression of the rational p-adic prime as product of primes of extension contains less factors than that the dimension n of extension so that some primes of extension appear as higher powers. This is analog for criticality as the appearance of multiple roots of a polynomial so that the derivative vanishes at the root besides the polynomial itself.
Before continuing it is good to make some confessions. Already in the earlier approach I considered two options for explaining the Shnoll effect: either in terms of p-adic fractality or in terms of quantum phase q of both. I however too hastily concluded that the p-adic option fails and choose the quantum phase option.
In the following both options are seen as parts of the story relying on a principle: the approximate scaling invariance of probability distribution P(n) for fluctuations under scalings by powers of p-adic prime p implying that P(n) is approximately identical for the divisions for which the interval Δ defining division differs by a power of p.
Second new idea is the lift of P(n) to wave function Ψ(n) in the space of counts. For quantum phase qm, m=p, Ψm would have quantum factor proportional to a wave function in finite field Fp, and the notion of counting modulo p suggests that the wave function corresponds to particle in box - standing wave - giving rise to P(n) representing diffraction pattern.
Basic facts about Shnoll effect
Usually one is not interested in detailed patterns of the fluctuations of physical variables, and assumes that possible deviations from the predicted spectrum are due to the random character of the phenomena studied. Shnoll and his collaborators have however studied during last four decades the patterns associated with random fluctuations and have discovered a strange effect described in detail in the articles of Shnoll (references can be found in
A Possible Explanation of Shnoll Effect).
- Some examples studied by Shnoll and collaborators are fluctuations of chemical and nuclear decay rates, of particle velocity in external electric field, of discharge time delay in a neon lamp RC oscillator, of relaxation time of water protons using the spin echo technique, of amplitude of concentration fluctuations in the Belousov-Zhabotinsky reaction. Shnoll effect appears also in financial time series, which gives additional support for its universality. Often the measurement reduces to a measurement of a number of events in a given time interval τ. More generally, it is plausible that in all measurement situations one divides the value range of the studied observable observable to intervals of fixed length and counts the number of events in each interval to get a histogram representing the distribution N(n), where n is the number of events in a given interval and N(n) is the number of intervals with n events. These histograms allow to estimate the probability distribution P(n), which can be compared with theoretical predictions for the spectrum of fluctuations of n. Typical theoretical expectations for the fluctuation spectrum are characterized by Gaussian and Poisson distributions.
- Contrary to the expectations, the histograms describing the distribution of N(n) has a distribution having several maxima and minima (see the figures in the article of Shnoll and collaborators (see this). Typically -say for Poisson distribution - one expects single peak. As the duration of the measurement period increases, this structure becomes gets more pronounced: standard intuition would suggest just the opposite to take place. The peaks also tend to be located periodically. According to Shnoll the smoothed out distribution is consistent with the expected distribution in the case that it can be predicted reliably.
- There are also other strange features involved with the effect. The anomalous distribution for the number n of events per fixed time interval (or more general value interval of measured observable) seems to be universal as the experiments carried out with biological, chemical, and nuclear physics systems demonstrate. The distribution seems also to be same at laboratories located far away from each other. The comparison of consecutive histograms shows that the histogram shape is likely to be similar to the shape of its nearest temporal neighbors. The shapes of histograms tend to recur with periods of 24 hours, 27 days, or 365 days. The regular time variation of consecutive histograms, the similarity of histograms for simultaneous independent processes of different nature and occurring in different geographical positions, and the above mentioned periods, suggest a common reason for the phenomenon possibility related to gravitational interactions in Sun-Earth and Earth-Moon system.
P(n|λ)= exp(-λ) λn/n! .
The mean value of n is λ>0 and also variance equals to λ. The replacement of distribution with a many-peaked one means that the probabilities P(n| λ) are modified so that several maxima and minima result. This can occur of course by the randomness of the events but for large enough samples the effect should disappear.
The universality and position independence of the patterns suggest that the modification changes slowly as a function of geographic position and time. The interpretation of the periodicities as periods assignable to gravitational interactions in Sun-Earth system is highly suggestive. It is however very difficult to imagine any concrete physical models for the effect since distributions look the same even for processes of different nature. It would seem that the very notion of probability somehow differs from the ordinary probability based on real numbers and that this deformation of the notion of probability concept somehow relates to gravitation.
Quantum group inspired model for Shnoll effect
Usually quantum groups are assigned with exotic phenomena in Planck length scale. In TGD they are assignable to a finite measurement resolution. TGD inspired quantum measurement theory describes finite measurement resolution finite measurement resolution in terms of inclusions of hyper-finite factors of type II1 (HFFs) and quantum groups related closely to the inclusions and appear also in the models of topological quantum computation quantum computation based on topological quantum field theories.
Consider first the original version of the proposed model. If I would rewrite it now correcting also the small errors, the summary would be as follows. This slightly revised model can be included as such to the new model.
- The possibility that direct p-adic variants of real distribution functions such as Poisson distribution might allow to understand the findings was discussed also in the original version. The erratic conclusion was that this cannot the case. In fact, for λ=1/pk the sum of probabilities P(n) without normalization factor is finite, and the appriximate scaling symmetry P(n)≈ P(prn) emerges for k=1. p-Adicity predicts approximate p-periodicity corresponding to the periodic variation of nR with the lowest pinary digit of n.
- It was argued that one should replace the integer n! in P(n) with quantum integer (n!)qm, q=exp(iπ/m), identified as the product of quantum integers rqm=(qr-q-r)/(q-q-1), r<n.
This however leads to problems since rqm can be negative. The problem can be circumvented by interpreting n! as p-adic number and expanding it in powers of p with pinary digits xk<p. For m=p the replacement of xk with quantum integer yields positive pinary digits.
The resulting quantum variant of p-adic integer can be mapped to its real counterpart by a generalization of canonical identification x= ∑ xnpn→ ∑ xnp-n. Whatever the detailed definition, quantum integers are non-zero and positive. The quantum replacement r→ rqm of the integers appearing in rational parameters in P(n|λi) might therefore make sense. It however does not make sense in the exponents like λn and λ=pk, k>1,2,.., is forced by convergence condition.
- I proposed also another modification of quantum integers xqm, x<p=m appearing in as pinary digits by decomposing x into a product of primes s<p and replacing s with quantum primes sqp so that also the notion of quantum prime would make sense: one might talk about quantum arithmetics. This is possible but is not necessary.
At the first re-reading the original model looked rather tricky, and this led to a revised model feeding in the adelic wisdom. One implication hierarchy of Planck constants heff/h0=n with n identified as the dimension of Galois extension.
One also ends up to the proposal that preferred p-adic primes p correspond to so called ramified primes of the extension of rationals inducing the extensions of p-adic number fields defining the adele. This kind of prime would naturally define a small-p p-adicity associated with Shnoll effect, which would thus serve as a direct signature of adelic physics.
- The first observation in conflict with the original belief is that one can actually define purely p-adic variant of the Poisson distribution P(n| λ) by replacing 1/n! with its image (n!)R under canonical identification. For instance, for Poisson distribution one must have λ= p-k, k=1,2,.. for both real and p-adic distributions to nake sense. The sum of the probabilities P(n) is finite. Poisson distribution with trivial quantum part is determined uniquely.
- One can also consider quantization P(n)=|Ψ(n)|2, suggested by the vision about quantum TGD
as complex square root of thermodynamics and hierarchy of Planck constants making possible macroscopic quantum coherence in arbitrarily long scales. The complexity of Ψ(n) could genuine quantum interpretation. Quantum factor of Ψ(n) allows interpretation as a wave function in finite field Fp representing the space of counts modulo p. The existence of quantum p-adics requires m=p. Scaling by p is not a symmetry but multiplication by 0<k<p and shift by 0≤ k<p act as symmetries analogous to rotations and translations acting on waves functions in Euclidian 3-space.
- The objections against Shnoll effect lead to an additional condition - or should one say principle - stating that the P(n) is approximately invariant under scalings n→ pkn. This could be seen as a manifestation of p-adic fractality in turn reflecting quantum criticality of TGD Universe.
- Taking n as the observable simplifies p-adicization crucially since the highly non-unique p-adicization of classical observables is avoided. One could speak of quantum measurement in the space of counts n defining universal observables. In quantum measurements the results are typically expressed as numbers of counts in given bin so that this kind of p-adicization is physically natural. The division of measurement interval would define an ensemble and n would be measured in each interval. State function reduction would localize Ψ(n) to n in each interval.
- The factorials n! appearing in P(n)=(dnf/dxn)/n! identified as coefficients of Taylor series of its generating function developed in pinary expansion for p=m. In this expansion one must invert powers of p in (n!)R and could also replace the coefficients of powers of p with quantum integers or replace even primes in their prime composition with quantum primes. For given norm (n!)R has period p approximately.
- The n:th derivative X(n)= dnf/dxn appearing as coefficient of 1/n! is replaced with X(n)R/X(n)p giving approximate periodicity and scaling invariance n→ pn.
- Quantum phase is associated with the ansatz stating P(n)= | Ψ(n)|2. In the "diffractive" situation quantum counterpart corresponds to | (kn)qm|2, 0<k<p-1. This gives rise to periodicity with period m=p.
One can imagine several explanations for the dependence of the time series distribution P(n) on the direction of the momentum of alpha particle and on the dependence of P(n) on time.
- The change of ramified prime p induced by the change of the extension of rationals would affect the periods. An interesting question is whether the effects understood in terms of the effect of the measurement apparatus on many-sheeted space-time many-sheeted space-time topology and geometry on p. Can one speak about measurement of p and of extension of rationals?
- The extension of rationals (and thus p) need not change. The "quantum factor of Ψ in P(n)=| Ψ(n)|2 has part depending on qp. The dependence on qp could change without change in p so that the extension of rationals need not change. One could speak about measurement of an observable related to the quantum factor of Ψ. A more concrete model relies on wave function proportional to (kn)qp ∝ qmkn+qm-kn - analog to a superposition of plane waves with momenta k propagating to opposite directions in the space of counts and producing in P(n) diffraction pattern proportional to (qn)qp2. Change of momentum k by scaling or shift induced by variation of the gravitational parameters or time evolution could be in question.
To sum up, I cannot avoid the thought that fluctuations regarded usually as a mere nuisance could be actually a treasure trove of new physics. While we have been busily building bigger and bigger particle accelerators, the truth would have been staring directly at our face and even winking eye to us.
See the updated article Shnoll Effect decade later.
For a summary of earlier postings see Latest progress in TGD.