Basic Probability Concepts

Byron P. Roe

Byron P. Roe

Jiří Souček

In the paper the basic concepts of extended probability theory are introduced. The basic idea: the concept of an event as a subset of Ωis replaced with the concept of an event as a partition. The partition is any set of disjoint non-empty subsets of Ω(i.e. partition=subset+its decomposition). Interpretation: elements inside certain part are indistinguishable, while elements from different parts are distinguishable. There are incompatible events, e.g {{e1},{e2}} and {e1,e2}. This is logical incompatibility analogical to the impossibility to have and simultaneously not to have the which-way information in the given experiment. The context is the maximal set of mutually compatible events. Each experiment has associated its context. In each context the extended probability is reduced to classical probability. Then the quadratic representation of events, partitions and probability measures is developed. At the end the central concept of quadratic probability spaces (which extend Kolmogorov probability spaces) is defined and studied. In the next paper it will be shown that quantum mechanics can be represented as the theory of Markov processes in the extended probability theory (Einstein's vision of QM).

Brian P. Anderson

Nijalingappa Umakantha

A new approach to probability theory is presented with reference to statistics and statistical physics. At the outset, it is recognized that the “average man” of a population and the “average particle” of a gas are only objects of thought, and not real entities which exist in nature. The concept of average (man) is generalized as a new concept of represental (man) whose epistemological status is intermediate between those of the particular (the man) and the universal (a man). This new concept has become necessary as a result of emergence of statistics as a new branch of human knowledge at the beginning of the nineteenth century. Probability is defined with reference to the represental. The concept of probability is the same in probability theory and in physics. But whereas in statistics the probabilities are estimated using random sequences, in statistical physics they are determined either by the laws of physics alone or by making use of the laws of probability also. Thus in physics we deal with probability at a more basic level than in statistics. This approach is free from most of the controversies we face at present in interpreting probability theory and quantum mechanics.

Christof Wetterich