Modelling of psychological behavior on the basis of ultrametric mental space: Encoding of categories by balls

In this paper we present a model of processing of mental information based on encoding by points of ultrametric space. Basic mental entities categories are encoded by ultrametric balls. Our model describes processes which take place in subconsciousness. It seems that ultrametric is a right tool for modeling of unconscious mental processes. Properties of ultrametric balls match well properties of unconscious representation of information which have been discussed in psychology.     

On the topological structure of a mathematical model of human unconscious

On the basis of two our previous works, in this paper, following Jacques Lacan psychoanalytic theory, we wish to outline some further remarks on the topological structure of a mathematical model of human unconscious.     

Pairwise correlations in a three-partite quantum system from a prequantum random field

Prequantum classical statistical field theory (PCSFT) is a model that provides the possibility to represent the averages of quantum observables (including correlations of observables on subsystems of a composite system) as averages with respect to fluctuations of classical random fields. In view of the PCSFT terminology, quantum states are classical random fields. The aim of our approach is to represent all quantum probabilistic quantities by means of classical random fields. We obtain the classical-random-field representation for pairwise correlations in three-partite quantum systems. The three-partite case (surprisingly) differs substantially from the bipartite case. As an important first step, we generalized the theory developed for pure quantum states of bipartite systems to the states given by density operators.     

An Application of the Theory of Open Quantum Systems to Model the Dynamics of Party Governance in the US Political System

The Gorini-Kossakowski-Sudarshan-Lindblad equation allows us to model the process of decision making in US elections. The crucial point we attempt to make is that the voter’s mental state can be represented as a superposition of two possible choices for either republicans or democrats. However, reality dictates a more complicated situation: typically a voter participates in two elections, i.e. the congress and the presidential elections. In both elections the voter has to decide between two choices. This very feature of the US election system requires that the mental state is represented by a 2-qubit state corresponding to the superposition of 4 different choices. The main issue is to describe the dynamics of the voters’ mental states taking into account the mental and political environment. What is novel in this paper is that we apply the theory of open quantum systems to social science. The quantum master equation describes the resolution of uncertainty (represented in the form of superposition) to a definite choice.     

Non-Kolmogorovian Approach to the Context-Dependent Systems Breaking the Classical Probability Law

There exist several phenomena breaking the classical probability laws. The systems related to such phenomena are context-dependent, so that they are adaptive to other systems. In this paper, we present a new mathematical formalism to compute the joint probability distribution for two event-systems by using concepts of the adaptive dynamics and quantum information theory, e.g., quantum channels and liftings. In physics the basic example of the context-dependent phenomena is the famous double-slit experiment. Recently similar examples have been found in biological and psychological sciences. Our approach is an extension of traditional quantum probability theory, and it is general enough to describe aforementioned contextual phenomena outside of quantum physics.     

Generalized probabilities taking values in non-Archimedean fields and in topological groups

We develop an analog of probability theory for probabilities taking values in topological groups. We generalize Kolmogorov’s method of axiomatization of probability theory, and the main distinguishing features of frequency probabilities are taken as axioms in the measure-theoretic approach. We also present a survey of non-Kolmogorovian probabilistic models, including models with negative-, complex-, and p-adic-valued probabilities. The last model is discussed in detail. The introduction of probabilities with p-adic values (as well as with more general non-Archimedean values) is one of the main motivations to consider generalized probabilities with values in more general topological groups than the additive group of real numbers. We also discuss applications of non-Kolmogorovian models in physics and cognitive sciences. A part of the paper is devoted to statistical interpretation of probabilities with values in topological groups (in particular, in non-Archimedean fields).