Negative probability in the framework of combined probability

TL;DR

This paper develops an axiomatic framework called combined probability that rigorously integrates negative probability with conventional probability, expanding the mathematical foundations relevant for physics applications.

Contribution

It constructs an axiomatic system synthesizing conventional and negative probability, extending traditional probability theory in a rigorous mathematical manner.

Findings

Axioms for combined probability are introduced and analyzed.

Relations between combined, extended, and conventional probability are established.

Mathematical properties of combined probability are studied.

Abstract

Negative probability has found diverse applications in theoretical physics. Thus, construction of sound and rigorous mathematical foundations for negative probability is important for physics. There are different axiomatizations of conventional probability. So, it is natural that negative probability also has different axiomatic frameworks. In the previous publications (Burgin, 2009; 2010), negative probability was mathematically formalized and rigorously interpreted in the context of extended probability. In this work, the axiomatic system that synthesizes conventional probability and negative probability is constructed in the form of combined probability. In a mathematically rigorous way, both theoretical concepts - combined probability and extended probability - stretch conventional probability so that it can takes negative values. After introducing axioms for combined probability, we study its properties, as well as relations to extended probability and conventional probability.