Some Identities for the Quantum Measure and its Generalizations

TL;DR

This paper explores a hierarchy of measure theories extending classical probability to include quantum mechanics, introducing algebraic relations and discussing potential generalizations of quantum theory.

Contribution

It introduces a hierarchy of sum-rules that generalize probability measures, unifying classical and quantum probability theories, and discusses algebraic relations and future research directions.

Findings

Hierarchy of sum-rules connects classical and quantum probability

Algebraic relations among sum-rules are established

Potential for higher-order generalizations of quantum mechanics

Abstract

After a brief review of classical probability theory (measure theory), we present an observation (due to Sorkin) concerning an aspect of probability in quantum mechanics. Following Sorkin, we introduce a generalized measure theory based on a hierarchy of ``sum-rules.'' The first sum-rule yields classical probability theory, and the second yields a generalized probability theory that includes quantum mechanics as a special case. We present some algebraic relations involving these sum-rules. This may be useful for the study of the higher-order sum-rules and possible generalizations of quantum mechanics. We conclude with some open questions and suggestions for further work.