The Paradigm of Complex Probability and Isaac Newton’s Classical Mechanics: On the Foundation of Statistical Physics

The concept of mathematical probability was established in 1933 by Andrey Nikolaevich Kolmogorov by defining a system of five axioms. This system can be enhanced to encompass the imaginary numbers set after the addition of three novel axioms. As a result, any random experiment can be executed in the complex probabilities set C which is the sum of the real probabilities set R and the imaginary probabilities set M. We aim here to incorporate supplementary imaginary dimensions to the random experiment occurring in the “real” laboratory in R and therefore to compute all the probabilities in the sets R, M, and C. Accordingly, the probability in the whole set C = R + M is constantly equivalent to one independently of the distribution of the input random variable in R, and subsequently the output of the stochastic experiment in R can be determined absolutely in C. This is the consequence of the fact that the probability in C is computed after the subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic experiment. We will apply this innovative paradigm to Isaac Newton’s classical mechanics and to prove as well in an original way an important property at the foundation of statistical physics

The Paradigm of Complex Probability and the Theory of Metarelativity: A Simplified Model of MCPP

All our work in classical probability theory is to compute probabilities. The original idea in this research work is to add new dimensions to our random experiment, which will make the work deterministic. In fact, probability theory is a nondeterministic theory by nature; which means that the outcome of the events is due to chance and luck. By adding new dimensions to the event in the real set of probabilities R, we make the work deterministic, and hence a random experiment will have a certain outcome in the complex set of probabilities and total universe G = C. It is of great importance that the stochastic system, like in real-world problems, becomes totally predictable since we will be totally knowledgeable to foretell the outcome of chaotic and random events that occur in nature, for example, in statistical mechanics or in all stochastic processes. Therefore, the work that should be done is to add to the real set of probabilities R the contributions of M, which is the imaginary set of probabilities that will make the event in G = C=R+Mdeterministic. If this is found to be fruitful, then a new theory in statistical sciences and in science, in general, is elaborated and this is to understand absolutely deterministically those phenomena that used to be random phenomena in R. This paradigm was initiated and developed in my previous 21 publications. Moreover, this model will be related to my theory of Metarelativity, which takes into account faster-than-light matter and energy. This is what I called “The Metarelativistic Complex Probability Paradigm (MCPP),” which will be elaborated on in the present two chapters 1 and 2

The Paradigm of Complex Probability and Thomas Bayes’ Theorem

The mathematical probability concept was set forth by Andrey Nikolaevich Kolmogorov in 1933 by laying down a five-axioms system. This scheme can be improved to embody the set of imaginary numbers after adding three new axioms. Accordingly, any stochastic phenomenon can be performed in the set C of complex probabilities which is the summation of the set R of real probabilities and the set M of imaginary probabilities. Our objective now is to encompass complementary imaginary dimensions to the stochastic phenomenon taking place in the “real” laboratory in R and as a consequence to gauge in the sets R, M, and C all the corresponding probabilities. Hence, the probability in the entire set C = R + M is incessantly equal to one independently of all the probabilities of the input stochastic variable distribution in R, and subsequently the output of the random phenomenon in R can be evaluated totally in C. This is due to the fact that the probability in C is calculated after the elimination and subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic phenomenon. We will apply this novel paradigm to the classical Bayes’ theorem in probability theory

The Paradigm of Complex Probability and Quantum Mechanics: The Infinite Potential Well Problem – The Momentum Wavefunction and the Wavefunction Entropies

The mathematical probability concept was set forth by Andrey Nikolaevich Kolmogorov in 1933 by laying down a five-axioms system. This scheme can be improved to embody the set of imaginary numbers after adding three new axioms. Accordingly, any stochastic phenomenon can be performed in the set C of complex probabilities which is the summation of the set R of real probabilities and the set M of imaginary probabilities. Our objective now is to encompass complementary imaginary dimensions to the stochastic phenomenon taking place in the “real” laboratory in R and as a consequence to calculate in the sets R, M, and C all the corresponding probabilities. Hence, the probability is permanently equal to one in the entire set C = R+M independently of all the probabilities of the input stochastic variable distribution in R, and subsequently, the output of the random phenomenon in R can be determined perfectly in C. This is due to the fact that the probability in C is calculated after the elimination and subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic phenomenon. My innovative Complex Probability Paradigm (CPP) will be applied to the established theory of quantum mechanics in order to express it completely deterministically in the universe C=R+M

The Paradigm of Complex Probability and the Theory of Metarelativity: The General Model and Some Consequences of MCPP

Calculating probabilities is a crucial task of classical probability theory. Adding supplementary dimensions to nondeterministic experiments will yield a deterministic expression of the theory of probability. This is the novel and original idea at the foundation of my complex probability paradigm. As a matter of fact, probability theory is a stochastic system of axioms in its essence; that means that the phenomena outputs are due to randomness and chance. By adding novel imaginary dimensions to the nondeterministic phenomenon happening in the set R will lead to a deterministic phenomenon and thus a stochastic experiment will have a certain output in the complex probability set and total universe G = C. If the chaotic experiment becomes completely predictable, then we will be fully capable to predict the output of random events that arise in the real world in all stochastic processes. Accordingly, the task that has been achieved here was to extend the random real probabilities set R to the deterministic complex probabilities set and total universe G = C=R+M and this by incorporating the contributions of the set M, which is the complementary imaginary set of probabilities to the set R. Consequently, since this extension reveals to be successful, then an innovative paradigm of stochastic sciences and prognostic was put forward in which all nondeterministic phenomena in R was expressed deterministically in C. This paradigm was initiated and elaborated in my previous 21 publications. Furthermore, this model will be linked to my theory of Metarelativity, which takes into consideration faster-than-light matter and energy. This is what I named “The Metarelativistic Complex Probability Paradigm (MCPP),” which will be developed in the present two chapters 1 and 2