Extended Probabilities: Mathematical Foundations

TL;DR

This paper introduces extended probabilities that include negative values, providing a mathematical foundation that generalizes classical probability and addresses longstanding issues in physics related to negative probabilities.

Contribution

It offers an axiomatic framework for extended probabilities, establishing a rigorous mathematical basis and linking them to classical probability as a positive subset.

Findings

Extended probabilities can take negative values.

Classical probability is a positive section of extended probability.

Provides axiomatic foundations for negative probabilities.

Abstract

There are important problems in physics related to the concept of probability. One of these problems is related to negative probabilities used in physics from 1930s. In spite of many demonstrations of usefulness of negative probabilities, physicists looked at them with suspicion trying to avoid this new concept in their theories. The main question that has bothered physicists is mathematical grounding and interpretation of negative probabilities. In this paper, we introduce extended probability as a probability function, which can take both positive and negative values. Defining extended probabilities in an axiomatic way, we show that classical probability is a positive section of extended probability.