What is it about?

The system of axioms for probability theory laid in 1933 by Andrey Nikolaevich Kolmogorov can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Therefore, we create the complex probability set C, which is the sum of the real set R with its corresponding real probability, and the imaginary set M with its corresponding imaginary probability. Hence, all stochastic experiments are performed now in the complex set C instead of the real set R. The objective is then to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory. Consequently, the corresponding probability in the whole set C is always equal to one and the outcome of the random experiments that follow any probability distribution in R is now predicted totally in C. Subsequently, it follows that, chance and luck in R is replaced by total determinism in C. Consequently, by subtracting the chaotic factor from the degree of our knowledge of the stochastic system, we evaluate the probability of any random phenomenon in C. 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.

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Why is it important?

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, the probability theory is a nondeterministic theory by nature; that means that the outcome of the events is due to chance and luck. By adding new dimensions to the event in R, we make the work deterministic and hence a random experiment will have a certain outcome in the complex set of probabilities C. It is of great importance that the stochastic system, like the problem in quantum mechanics considered here, 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 which will make the event in C = R + M deterministic. If this is found to be fruitful, then a new theory in statistical sciences and prognostic and mainly in quantum mechanics is elaborated and this is to understand absolutely deterministically those phenomena that used to be random phenomena in R. This is what I called ‘The Complex Probability Paradigm (CPP)’, which was initiated and elaborated in my nineteen previous papers.

Perspectives

Although I have taught courses on probability and statistics at the university level for many years, I consider myself a beginner in this branch of knowledge; in fact an absolute beginner, always thirsty to learn and discover more. I think that the mathematician who proves to be successful in tackling and mastering the theory of probability and statistics has made it halfway to understanding the mystery of existence revealed in a universe governed sometimes in our modern theories by randomness and uncertainties. The probabilistic aspect is evident in the theories of the quantum world, of thermodynamics, or of statistical mechanics, for example. Hence, the universe’s secret code, I think, is written in a mathematical language, just as Galileo Galilei expressed it in these words: “Philosophy is written in this very great book which is the universe that always lies open before our eyes. One cannot understand this book unless one first learns to understand the language and recognize the characters in which it is written. It is written in a mathematical language and the characters are triangles, circles and other geometrical figures. Without these means it is humanly impossible to understand a word of it. Without these there is only clueless scrabbling around in a dark labyrinth.”

Dr. Abdo Abou Jaoude
Notre Dame University Louaize

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This page is a summary of: The Paradigm of Complex Probability and Quantum Mechanics: The Infinite Potential Well Problem - The Position Wave Function, November 2022, IntechOpen,
DOI: 10.5772/intechopen.107300.
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