Quantum Mechanics as Quantum Measure Theory

TL;DR

This paper presents a hierarchy of measure theories, showing how quantum mechanics naturally emerges as a generalized measure theory weaker than classical probability, clarifying the relationship between classical and quantum physics.

Contribution

It introduces a generalized measure theory framework that encompasses quantum mechanics, providing a natural derivation of quantum probabilities and clarifying classical-quantum relationships.

Findings

Quantum probabilities derived as squares of amplitudes

Quantum mechanics as a special case of generalized measure theory

Clarification of classical physics as a subset of quantum physics

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.