Dendrogramic Data Representation and p-Adic Physics Insights

From Dendrogramic Representation of Data to p-adic Universe and CHSH Violation from NonergodicityI Oded Shor 1,2, Felix Benninger 1,2,3 and Andrei Khrennikov 4* 1 Felsenstein Medical Research Center, Beilinson Hospital, Petach Tikva, Israel 2 Sackler Faculty of Medicine, Tel Aviv University, Tel Aviv 6997801, Israel 3 Department of Neurology, Rabin Medical Center, Petach Tikva 4941492, Israel 4 Faculty of Technology, Department of Mathematics, Linnaeus University, 351 95 V xj , Sweden

P-adic theoretical physics Applications of p-adic numbers in physics were intiated in Steklov Mathematical Institute in 1980s by Vladimirov and Volovich with studies in non-Archimedean superanalysis and the conference in Tashkent. The latter was the big event in my life, since after it I started to work in p-adic analysis. Paper: Volovich, I. V., p-adic string. Classical and Quantum Gravity, 1987, 4, 83-87, stimulated intensive research in theory of p-adic strings, Volovich, Vladimirov, Aref eva, Dragovich, Witten, Parisi, Framton, Missarov, Chehov, Freud, Olson, . 1. And the book: Vladimirov, V.S.; Volovich, I.V.; Zelenov, E.I. p-Adic Analysis and Mathematical Physics; World Scientific: Singapore, 1994, attracted attention of physicists and mathematicians world-wide. 2.

Success and problems of p-adic theoretical physics P-adic modelling in physics was very successful, see, e.g., the reviews B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, and I.V. Volovich, On p-adic mathematical physics, p-Adic Numbers, Ultrametric Anal., Appl. 1, 1 17 (2009). 1. B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich, and E. I. Zelenov, p- Adic mathematical physics: The first 30 years, p-Adic Numbers, Ultrametric Anal., Appl. 9, 87 121 (2017). 2. The main problem (from my viewpoint) was the difficulty of coupling p-adic theoretical models with experimental data. Personally I spent a lot of time by thinking on it; since 1990s I discussed this problem many times with Vladimirov, Volovich, Dragovich, and Zelenov. One conversation with Zelenov was the dead-end of my thinking: Evgenii told me at that occasion: All measurement devices are designed to get outputs calibrated in real metric . 3.

We present fundamental results from

Shor, O., Glik, A., Yaniv-Rosenfeld, A., Valevski, A., Weizman, A., Khrennikov, A., & Benninger, F. (2021). EEG p-adic quantum potential accurately identifies depression, schizophrenia and cognitive decline. Plos one, 16(8), e0255529. Our starting point was representation of data colelcted in medical studies by dendrograms with the aid of clustering algorithms, then we encoded brancehs of trees by p-adic numbers (sequences of 0/1), natural numbers with p-adic distance between them, then contructed the Bohmian potential on p-adics and it was used for the medical diagnoses.

Instead of time series of real numbers, we represent experimental data by dentrograms Denrograms, finite trees, are generated by hierarchic clustering algorithms. Thus we interested mainly in relations between data sets, not simply order w.r.t. position on the real line. Our starting point was pure data analyse, analyse of hierarchic structuring in data. This analysis led us to deep theoretical and philosophical conclusions on foundations of classical and quantum physics and their interrelation. This interrelation is not so sharp as in the standard theory. The degree of quantumness is checked with the aid of different tests, e.g., violation of Bell type inequalities. The situation is similar to formalization of random sequence as passing a batch of statistical tests, say NIST-tests.

Dendrogramic Representation of the Universe ontic-epitemic Universe According to Bohr, the outcomes of measurements are not the objective properties of systems.They quantitively represent interrelation between a system S and an observer O (measurement device of O). We shall use the ontic-epistemic structuring of scientific theories. Ontic description is observer independent description of reality - as it is. Epistemic description is based on knowledge which O can extract within experiments. Ontic theory is not verifiable experimentally. It is unapproachable by the observer; O constructs its approximate epistemic representation by collecting data.

We extend the Copenhagen (Bohrs) ideology: In our theory, observer O has the free choice not only to perform or not perform some observation, but even the free choice to decompose the collected data into blocks and to treat these blocks as the (epistemic) systems representation.

Event universe (also Wheeler, Smolin, Rovelli and if we go deeper Leibnitz) The fundamental entities of the epistemic theory are events, not systems. Systems are composed of a certain number of events (clusters of events), for example, several clicks of detectors (or other detection events). In our DH-theory, systems are extracted from data with the aid of clustering algorithms Different algorithms can generate different decompositions of epistemic universe into system. But numerical simulation shows that properties of systems generated by different clustering algorithms do not differ.

P-adic representation of events on dendrograms The standard p-adic ultrametric, where p>1 is a natural number, is introduced on the set of sequences Zp in the following way. Let x = (a0, a1, a2, ..., an, ), y = (b0, b1, b2, ..., bn, ) Zp, i.e., aj , bj =0,1, , p-1. We set rp(x, y) = l/pk if aj= bj, j = 0, 1, ..., k 1, and ak bk. This is not only metric on the set of sequences, but even utrametric, i.e., it satisfies the strong triangle inequality, for any three sequences x, y, z: rp(x, y) <= max [rp(x, z), rp(z, y)]

Treelike representation of 2-adic space

Unsual properies of p-adic geometry This strange geometry, e.g., all triangles are isosceles. This is a consequence of the strong triangle inequality. Define open and closed balls, B(R, a) = { x: rp(a, x) <= R} and B-(R, a) = { x: rp(a, x) <R}. Both balls are at the same time closed and open sets of the metric space, each point in a ball can be selected as its center. Geometrically a ball is a batch of infinite branches having the finite common root-branch. Such spaces are disordered, totally disconnected, and having zero topological dimension. However, such geometries arise very naturally from data series with application of clustering algorithms.

Algebra on Zp: operations of multiplication, addition, subtraction. Sequences can be represented as numbers given by series w.r.t. power of p, X= a0+ a1 p + a2 p^2 + + an p^n+ , aj=0,1, , p-1. They converge in p-adic metric. Zp, the limit of growing dendrograms, is the ontic representation of Universe, finite dendrograms are epistemic representations of growing complexity. The points of Zp are elementary ontic events, subsets of Zp are ontic events. End-points of branches of a finite dendrogram are epistemic elementary events, sets of these points are epistemic events. A huge batch of ontic events is identified with an epistemic point-event. By extendin dendrograms size we spit epistemic points of n-level to numerous events of m-level, m >>n.

David Bohm: implicate and explicate order Ontic-epistemic structuring of science matches Bohm s methodology based on the notions of implicate and explicate order. Explicate order is expressed in the abstractions that humans normally perceive. Implicate order is more fundamental order of reality: In the enfolded [or implicate] order, space and time are no longer the dominant factors determining the relationships of dependence or independence of different elements. Rather, an entirely different sort of basic connection of elements is possible, from which our ordinary notions of space and time, along with those of separately existent material particles, are abstracted as forms derived from the deeper order. These ordinary notions in fact appear in what is called the "explicate" or "unfolded" order, which is a special and distinguished form contained within the general totality of all the implicate orders (Bohm 1980

All Events in the universe of the observer O collected as data And then clustered Implicate Or the observer O can decide to decompose the universe to systems system There must be a correspondence Between the two representations It is the same Universe after all. Is there??? Explicate

Implicate Order Present in the Explicate Order YES THERE IS SUCH CORRESPONDANCE upon correlating the unit simple dendrogram (or its p-adic numbers representative of its edges) in terms of cross-correlation coefficient, it was found to be best matched to all of its corresponding components in the Universal dendrogram or to its corresponding components in the more complex and larger dendrograms. BLUE correlations of simple systems to its representations in more complex systems ORANGE correlations of simple systems to representations of other systems in the more complex systems

From Epistemic (Explicate) to Ontic (Implicate): p-Adic Universe By considering the data blocks of increasing size, in the limit, O can reconstruct the ontic description. The dendrogram-based epistemic theory leads to the p-adic geometry of the ontic universe. In the limit, dendrograms of increasing size generate the infinite p-adic tree endowed with the p-adic ultrametric.

From Ontic (Implicate) to Epistemic (Explicate) Our previous epistemic-to-ontic pathway centralized the role of an observer O. This can make the impression that the personal observer perspective plays a crucial role in DH theory. Now, by starting with the p-adic ontic model, we diminish the subjective component. The points of the p-adic tree Z_p represent all possible events that can happen in the Universe. Thus, as well as the epistemic universe, the ontic universe is represented as a set of events. However, these are not observational events; they cannot be associated with, say, the clicks of detectors. We consider the p-adic points absolute events. An absolute event is the endpoint of the infinitely long path of the p-adic tree Each finite cutoff of the infinite path a is an element of the explicate order, and it can be interpreted as the characteristic of a physical system

Dendrogram Viewpoint on Classical Quantum Interrelation In our epistemic model, the sharp classical quantum separation disappears. The degree of classicality is based on a system s complexity the size and topological complexity of its dendrogram representation. Quantum systems are characterized by a low complexity of their dendrograms. Thus, electrons and atoms can behave as quantum in some measurements because they have a very simple hierarchic structure of interrelation between their components. In our approach, even classical physical systems can exhibit quantum(-like) behavior within a hierarchic representation of experimental data.

CHSH Violations for 2-Slit Diffraction Experiment Data From a single ccd frame of the diffraction experiment we find the column in which detectors ticked for example in frame 2 columns 1 2 3 4 and 5 ticked we then calculate the single data point corresponding to frame 2 as log10(1*2*3*4*5) From a fixed number of consecutive data points we construct a single dendrogram and then 2 series of dendrograms.

Algorithm for computation of CHSH values on dendrogram series We constructed two time series of dendrograms one for Alice A=(A1A2A3 An) and one for Bob B=(B1B2B3 Bn). For Alice we select two pairs of numbers a=[a1 a2] a =[a1 a2 ] the two pairs aren t identical. Analogs of two vectors orientations of polarization beam splitters or Stern- Gerlach magnets. For Bob we select two pairs of numbers b=[b1 b2] b =[b1 b2 ] the two pairs aren t identical.

Observables as functions of dendrograms If for dendrogram Ai we chose pair a we have the following: If Ai had both of the numbers in a we give Sai=1 else Sai=-1. We proceed in the same for selection of a and b, b . Then we calculate the correlations: Cab=( Sai*Sbi)/length(a and b are selected together ) Cab =( Sai*Sb i)/length(a and b are selected together ) Ca b=( Sa i*Sbi)/length(a and b are selected together ) Ca b =( Sa i*Sb i)/length(a and b are selected together ) C=Cab - Cab +Ca b+Ca b

Our data denostrated that for some selections of parameters a, a , b, b |c| >2. Our ideology. All systems have special relational hidden variables. These hidden variables are the hierarchical relations between constitutes of the system. . These relational properties are simply ignored/or not being accounted in the formalism of classical physics. It seems that in quantum mechanics in some sort of way they do. So, it seems that real quantum data already represents some sort of hidden relational hierarchy.

CHSH inequality violation: The Role of Nonergodicity DH theory provides the possibility of introducing hidden variables (of the special type) beyond any experimental data and considering a class of realistic observables, those represented as functions of hidden variables. Hence, straightforwardly, it seems that Bell type inequalities cannot be violated under the assumption of locality. However, we showed the violation of the CHSH inequality for special selection of settings that determine the observables. The reason for this is precisely the violation of ergodicity; the measure-theoretic and frequency averages do not coincide

Two ways for non-ergodicity: first way We start with one sequence A = (A1A2....An), where each d-vector Aj = (u1,u2,...,ud) represents a dendrogram. We fix a=[a1 a2] and define Sa = Sa(u)(1) We continue to calculate Sa. We stress that (1) is the condition of realism, the value of hidden variable u determines the outcome of the observable Sa. Then < ?? >= ( ?=? ??(??))/? (2) Now we compute the probability distribution of hidden variables. Let us fix one d-vector u which is present in our dendrograms and calculate proportion number of occurrences of u ? ? =?????? ?? ??????????? ?? ? ? then ?? = ??? ? ? ? (4) And non ergodicity implies << Sa >> < Sa > ? (3)

Two ways for non-ergodicity: second way Then, let us proceed without the random choice and calculate for A = (A1...An),B = (B1...bN) Sa and Sb. Thus our frequency correlation: ? ??(??)??(??))/? (1) < ??,?? >= ( ?=1 we should complete this analysis by the measure-theoretic framework. We consider all possible pairs of hidden variables, d-vectors, u,v which are present in the pairs Aj,Bj and find the probability number of occurrences of u,v ? ?,? =?????? ?? ??????????? ?? ?,? ? And ??,?? = ??? ? ??(?)? ?,? and for non-ergodic correlations we expect that ??,?? < ??,?? >

And in our data we really see both types of ergodicity violation, for averages and for correlations. Moreover, we see coupling with degree of violation of CHSH. There exists quadruples (a,a , b, b ) such that, for avreages no violation of ergodicity, i.e., <Sa>=<<Sa>> ., but CHSH is violated. But we have violation of ergodicity for correlations.

Tsirelson bound A priory there is no reason to expect that C would be bound by 2 sqrt{2}, in principle, the bound can be up to 4. However, we have never found violation exceeding Tsierelson bound, which is derived in the canonical quantum formalism. See also Khrennikov, A. Buonomano against Bell: Nonergodicity or nonlocality? Int. J. Quant. Inf. 2017,15, No. 08, 1740010.

Why dendrogramic representation can lead to violation of ergodicity? May be any local realistic theory with hidden variables and violation of Bell type inequalities would lead to violation of ergodicity? So, may be there is nothing special in dendrogam- representation

The starting point was my old book: A. Khrennikov, Information Dynamics in Cognitive, Psychological, Social, and Anomalous Phenomena, Kluwer, 2004. Oded Shor, Amir Glik, Amit Yaniv-Rosenfeld, Avi Valevski, Abraham Weizman, Andrei Khrennikov , Felix Benninger EEG p-adic quantum potential accurately identifies depression, schizophrenia and cognitive decline https://journals.plos.org/plosone/article/authors?id=10.1371/journal.pone.02 55529