Adaptive Dynamics and General Approach to Non-Kolmogorov Probability Theory

In this chapter, we will discuss the non-Kolmogorov probability theory. Our aim is to explain where and why the usual probability theory is broken. The basic example from quantum physics is presented in very detail including the corresponding illustrations. This is the two slit experiment demonstrating interference for quantum systems. Further, we discuss the mathematical foundation of lifting theory (the basic element of quantum information theory used in this book) and the concept of adaptive dynamics with its mathematical description. Then we apply this mathematical basis to the study of the non-Kolmogorov probability theory. One of the main messages is that, although biological probabilistic behaviors (including cognition) are nonclassical, i.e., it cannot be described by the standard Kolmogorov model, it is not always possible to represent them in the canonical quantum framework. More general quantum-like models have to applied. The last part of this chapter is devoted to quantum informational foundations of theory of adaptive dynamics (AD). Here lifting of quantum states plays the fundamental role. This model generalized the canonical quantum model based on theory of open quantum systems.

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