Dendrogramic Representation of Data: CHSH Violation vs. Nonergodicity

Quantization of events in the event-universe and the emergence of quantum mechanics

Quantum mechanics (QM) is derived based on a universe composed solely of events, for example, outcomes of observables. Such an event universe is represented by a dendrogram (a finite tree) and in the limit of infinitely many events by the p-adic tree. The trees are endowed with an ultrametric expressing hierarchical relationships between events. All events are coupled through the tree structure. Such a holistic picture of event-processes was formalized within the Dendrographic Hologram Theory (DHT). The present paper is devoted to the emergence of QM from DHT. We used the generalization of the QM-emergence scheme developed by Smolin. Following this scheme, we did not quantize events but rather the differences between them and through analytic derivation arrived at Bohmian mechanics. We remark that, although Bohmian mechanics is not the main stream approach to quantum physics, it describes adequately all quantum experiments. Previously, we were able to embed the basic elements of general relativity (GR) into DHT, and now after Smolin-like quantization of DHT, we can take a step toward quantization of GR. Finally, we remark that DHT is nonlocal in the treelike geometry, but this nonlocality refers to relational nonlocality in the space of events and not Einstein's spatial nonlocality. By shifting from spatial nonlocality to relational we make Bohmian mechanics less exotic.

EEG-based spatio-temporal relation signatures for the diagnosis of depression and schizophrenia

The diagnosis of psychiatric disorders is currently based on a clinical and psychiatric examination (intake). Ancillary tests are used minimally or only to exclude other disorders. Here, we demonstrate a novel mathematical approach based on the field of p-adic numbers and using electroencephalograms (EEGs) to identify and differentiate patients with schizophrenia and depression from healthy controls. This novel approach examines spatio-temporal relations of single EEG electrode signals and characterizes the topological structure of these relations in the individual patient. Our results indicate that the relational topological structures, characterized by either the personal universal dendrographic hologram (DH) signature (PUDHS) or personal block DH signature (PBDHS), form a unique range for each group of patients, with impressive correspondence to the clinical condition. This newly developed approach results in an individual patient signature calculated from the spatio-temporal relations of EEG electrodes signals and might help the clinician with a new objective tool for the diagnosis of a multitude of psychiatric disorders.

Dendrographic Hologram Theory: Predictability of Relational Dynamics of the Event Universe and the Emergence of Time Arrow

Recently we started the development of Dendrographic Hologram Theory (DH-theory). It is based on the novel mathematical representation of the relational event universe (in the spirit of Smolin et al.). Elementary events are represented by branches of dendrograms, finite trees that are generated from data with clustering algorithms. In this context, we studied the dynamics of the event universe generated by the appearance of a new event. Generally, each new event can generate the complete reconstruction of the whole dendrogramic universe. However, we found (via numerical simulation) unexpected stability in this universe. Its events are coupled via the hierarchic relational structure, which is relatively stable even with respect to the random generation of new events. We also observed the regularity patterns in the location of new events on dendrograms. In the course of evolution, the dendrogram’s complexity increases and determines the arrow of time in the event universe. We used the complexity measure from particle shape dynamics, which was shown to increase in both directions away from a Janus point and thus determine the arrow of time in symmetrical manner away from a Janus point. The particle shape dynamics theory is a relational theory with close ideological resemblance to DH-theory, as both rely on Mach’s principle and Leibniz’s relationalism and principles. By using the complexity measure on dendrograms and its p-adic string representation, we demonstrate the emergence of a time arrow from the p-adic zero-dimensional field, where space and time are absent.

Towards Unification of General Relativity and Quantum Theory: Dendrogram Representation of the Event-Universe

Following Smolin, we proceed to unification of general relativity and quantum theory by operating solely with events, i.e., without appealing to physical systems and space-time. The universe is modelled as a dendrogram (finite tree) expressing the hierarchic relations between events. This is the observational (epistemic) model; the ontic model is based on p-adic numbers (infinite trees). Hence, we use novel mathematics: not only space-time but even real numbers are not in use. Here, the p-adic space (which is zero-dimensional) serves as the base for the holographic image of the universe. In this way our theory is connected with p-adic physics; in particular, p-adic string theory and complex disordered systems (p-adic representation of the Parisi matrix for spin glasses). Our Dendrogramic-Holographic (DH) theory matches perfectly with the Mach’s principle and Brans–Dicke theory. We found a surprising informational interrelation between the fundamental constants, h, c, G, and their DH analogues, h(D), c(D), G(D). DH theory is part of Wheeler’s project on the information restructuring of physics. It is also a step towards the Unified Field theory. The universal potential V is nonlocal, but this is relational DH nonlocality. V can be coupled to the Bohm quantum potential by moving to the real representation. This coupling enhances the role of the Bohm potential.