Lecture Notes

The true logic of the world is in the calculus of probabilities. James Clarc Maxwell

I have written some publications on probability theory. They differ in some aspects, focuses, and applications. All have in common a probabilistic framework for formulating classical probability theory, quantum probability, thermodynamics, diffusion, and the Wiener integral based on a set of four axioms or principles. It explains everything that conventional quantum probability and classical probability theory achieve. Interested people can download the publications via the following links:

"An Alternative Approach to Probability in Quantum Information Science"

via

https://www.intechopen.com/online-first/1176955

"Conceptual basis of probability and quantum information theory"

via

https://tore.tuhh.de/entities/publication/519250c6-1def-4b15-b493-871f6f4c6348

"A Unified treatment of classical probability, thermodynamics, and quantum information theory"

via

https://www.tuhh.de/ti3/paper/jansson/UnifiedTreatmentProbability.pdf

and

"Quantum Information Theory for Engineers: Free Climbing through Physics and Probability"

via

https://www.tuhh.de/ti3/jansson/Freeclimbing.pdf

The basis of Quantum Theory and Quantum Information Science is the underlying probability concept. Our probabilistic framework is not an interpretation of quantum mechanics, such as ``Many-Worlds'', ``Bohm's Theory'' or the ``Copenhagen interpretation''. It is much more general and can be viewed as a probability algorithm that calculates probabilities of future events in very different statistical areas. It partially supports the opinion of Fuchs and Peres:

The thread common to all the nonstandard "interpretations" is the desire to create a new theory with features corresponding to some reality independent of our potential experiments. But, trying to fulfill a classical worldview by encumbering quantum mechanics with hidden variables, multiple worlds, consistency rules, or spontaneous collapse without any improvement in its predictive power only gives the illusion of a better understanding. Contrary to those desires, quantum theory does not describe physical reality. What it does is provide an algorithm for computing probabilities for the macroscopic events ("detector clicks") that are the consequences of our experimental interventions. This strict definition of the scope of quantum theory is the only interpretation ever needed, whether by experimenters or theorists. Fuchs and Peres, 2000

We make a clear distinction between possibilities and outcomes. Moreover, we use a time concept based on the classification of future, present, and past. As a result, previously perplexing paradoxes find resolution. In particular, the superposition principle takes on a new meaning.

Our probabilistic framework stands apart from the Hilbert space formalism of quantum mechanics, held by numerous physicists and eloquently articulated in the book "Quantum Mechanics: The Theoretical Minimum" authored by Susskind and Friedman, who write on page 24:

For a classical system, the space of states is a set (the set of possible states), and the logic of classical physics is Boolean. That seems obvious, and it is not easy to imagine any other possibility. Nevertheless, the real world operates along different lines, at least whenever quantum mechanics is important. The space of states of a quantum system is not a mathematical set [6]; it is a vector space. Relations between the elements of a vector space are different from those between the elements of a set, and the logic of propositions is different as well. Susskind and Friedman, 2014

Our probability theory relies solely on elementary set theory, classical logic, and complex numbers. Consequently, this theory is accessible for instruction in educational settings.

This framework can be regarded as an axiomatic approach to probability in Hilbert's sense. In his sixth of the twenty-three open problems presented at the International Congress of Mathematicians in Paris in 1900, Hilbert called for an axiomatic probability theory.