This is the first fundamental book devoted to non-Kolmogorov probability models. It provides a mathematical theory of negative probabilities, with numerous applications to quantum physics, information theory, complexity, biology and psychology. The book also presents an interesting model of cognitive information reality with flows of information probabilities, describing the process of thinking, social, and psychological phenomena.
The book explores mathematical models of information dynamics, focusing on consciousness through the lens of classical and quantum physics. It aims to create a formalism akin to the Newton-Descartes program, adapted for understanding mental processes. Recognizing the limitations of deterministic models, the author proposes integrating statistical approaches to better represent information flows, suggesting that these models may exhibit quantum-like characteristics. This innovative perspective seeks to bridge the gap between physical dynamics and the complexities of mental phenomena.
Exploring the realm of p-adic numbers reveals their significance beyond pure mathematics, particularly in the context of quantum physics. Initially perceived as exotic, p-adic numbers have emerged as foundational in recent quantum models. Introduced by K. Hensel in 1904, p-adic numbers are constructed through a unique valuation process, leading to various completions of rational numbers. This book delves into p-adic analysis, probability, and their applications in modern physics, challenging the traditional reliance on real and complex numbers in scientific models.
The book delves into the mathematical foundations of randomness, addressing both classical and quantum perspectives. It explores classical theories, highlighting their achievements and challenges, before transitioning to the complexities of quantum randomness. By discussing the interrelation between these two realms and tackling interpretational and foundational issues, it offers a thorough examination suitable for researchers in mathematical physics, probability, and statistics, regardless of their expertise level.
Exploring the intersection of mathematics and reality, this book delves into the significance of different number systems in shaping our understanding of the world. It challenges the traditional reliance on real analysis, highlighting the potential of p-adic numbers and non-Archimedean analysis to offer alternative perspectives. Beyond just p-adic analysis, the text addresses a broader spectrum of philosophical and physical concepts, suggesting that many limitations in our comprehension stem from the uncritical application of the Archimedean axiom.
The book delves into the development of Grassmann algebra, particularly focusing on anticommuting generators and their derivatives, highlighting Martin's innovative approach that diverged from traditional Newtonian analysis. It discusses the significant contributions of mathematicians like Berezin and Kac in forming algebraic superanalysis, leading to the emergence of supermathematics. Despite advancements in this field, the text notes the ongoing reliance on Schwinger's formalism in physics, where anticommuting variables are often treated as "real functions." The concept of superspace, introduced by Salam and Strathdee, is also explored.
Focusing on the intersection of mathematics and biology, this work explores fundamental issues in biology, decision-making, and psychology through the lens of open quantum systems theory. It provides a unique analytical perspective, employing mathematical models to deepen the understanding of complex biological and psychological phenomena.
The book presents a unifying framework for the probabilistic principles of classical statistical mechanics and quantum mechanics through the Vaxjo model. It explores how this contextual probabilistic model can bridge the gap between the two disciplines, offering a comprehensive understanding of their underlying similarities and differences. The discussion delves into the implications of this unification for both theoretical physics and practical applications, making it a significant contribution to the field.
From Psychology to Finance
Quantum-like structure is present practically everywhere. Quantum-like (QL) models, i. e. models based on the mathematical formalism of quantum mechanics and its generalizations can be successfully applied to cognitive science, psychology, genetics, economics, finances, and game theory. This book is not about quantum mechanics as a physical theory. The short review of quantum postulates is therefore mainly of historical value: quantum mechanics is just the first example of the successful application of non-Kolmogorov probabilities, the first step towards a contextual probabilistic description of natural, biological, psychological, social, economical or financial phenomena. A general contextual probabilistic model (Växjö model) is presented. It can be used for describing probabilities in both quantum and classical (statistical) mechanics as well as in the above mentioned phenomena. This model can be represented in a quantum-like way, namely, in complex and more general Hilbert spaces. In this way quantum probability is totally demystified: Born's representation of quantum probabilities by complex probability amplitudes, wave functions, is simply a special representation of this type.