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arXiv:1807.10382

Negative probabilities, II: What they are and what they are for

Andreas R. Blass, Yuri G. Gurevich

Published Jul 26, 2018• 3 citations
arXivHTMLPDF

Abstract

A signed probability distribution may extend a given traditional probability from observable events to all events. We formalize and illustrate this approach. We also illustrate its limitation. We argue that the right question is not what negative probabilities are but what they are for.
DOI
10.48550/arxiv.1807.10382

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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.
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.