Quantum Mechanics as Quantum Measure Theory

Abstract

The additivity of classical probabilities is only the first in a hierarchy of possible sum-rules, each of which implies its successor. The first and most restrictive sum-rule of the hierarchy yields measure-theory in the Kolmogorov sense, which physically is appropriate for the description of stochastic processes such as Brownian motion. The next weaker sum-rule defines a generalized measure theory which includes quantum mechanics as a special case. The fact that quantum probabilities can be expressed "as the squares of quantum amplitudes" is thus derived in a natural manner, and a series of natural generalizations of the quantum formalism is delineated. Conversely, the mathematical sense in which classical physics is a special case of quantum physics is clarified. The present paper presents these relationships in the context of a "realistic" interpretation of quantum mechanics.An attitude toward Quantum Mechanics which is suitable for quantum gravity in general, and for its application to cosmology in particular, is not so easy to find.Understanding the early universe requires us to reason about a time in the distant past in which observers in the ordinary sense of the word can hardly have been present. For such a situation, a philosophically "realistic" attitude toward quantum mechanics would seem to be more effective than one based on operators which must find their physical meaning in terms of "measurements". If the reality in question 1

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