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).     

Representation Theorem of Observables on a Quantum System

An orthomodular lattice (OML) with a conditional state can be used as a model for noncompatible events (a quantum system). In this paper we will study some properties of a conditional state and an s-map which are defined on an OML. We show conditions when a quantum system has the same properties as the classical probability space.     

Von Neumann Uniqueness Theorem doesn't Hold in Hyperbolic Quantum Mechanics

It is shown that Von Neumann Uniqueness Theorem doesn't hold in Hyperbolic Quantum Mechanics.     

Some remarks on an experiment suggesting quantum-like behavior of cognitive entities and formulation of an abstract quantum mechanical formalism to describe cognitive entity and its dynamics

We have executed for the first time an experiment on mental observables concluding that there exists equivalence (that is to say, quantum-like behavior) between quantum and cognitive entities. Such result has enabled us to formulate an abstract quantum mechanical formalism that is able to describe cognitive entities and their time dynamics.     

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.