paper

Quantum Mechanics as Quantum Measure Theory

arXiv:gr-qc/9401003 · doi:10.1142/S021773239400294X

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 {\it 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.

13 pages (The only revision is the correction of a typo in equation 3.)

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