# Negative Probabilities, Fine's Theorem and Quantum Histories

Johnathon Haliwell ( Imperial College London )

- 14:00 20th June 2014 ( week 8, Trinity Term 2014 )Hoare Room, RHB

Many situations in quantum theory and other areas of physics lead to quasi-probabilities
which seem to be physically useful but can be negative. The interpretation of such
objects is not at all clear. In this paper, we show that quasi-probabilities naturally
fall into two qualitatively different types, according to whether their non-negative
marginals can or cannot be matched to a non-negative probability. The former type,
which we call viable, are qualitatively similar to true probabilities, but the latter
type, which we call non-viable, may not have a sensible interpretation.
Determining the existence of a probability matching given marginals is a non-trivial
question in general. In simple examples, Fine's theorem indicates that inequalities
of the Bell and CHSH type provide criteria for its existence. A simple proof of Fine's theorem
is given. Our results have consequences for the linear positivity condition of Goldstein and
Page in the context of the histories approach to quantum theory. Although it is a very weak
condition for the assignment of probabilities it fails in some important cases where our results
indicate that probabilities clearly exist. Some implications for the histories approach to quantum
theory are discussed.