Toward a "fundamental theorem of quantal measure theory"

TL;DR

This paper explores extending quantum measures in path-integral contexts, proposing a new method for measure extension using convergent sequences, and demonstrating a case where quantum measure differs from classical measure.

Contribution

It introduces a novel approach for extending quantal measures via canonical sequences and convergence criteria, addressing challenges of unbounded variation.

Findings

A new measure extension method for quantal measures

Identification of events with zero classical but non-zero quantum measure

Application to causal sets and lattice particle models

Abstract

We address the extension problem for quantal measures of path-integral type, concentrating on two cases: sequential growth of causal sets, and a particle moving on the finite lattice Z_n. In both cases the dynamics can be coded into a vector-valued measure mu on Omega, the space of all histories. Initially mu is defined only on special subsets of Omega called cylinder-events, and one would like to extend it to a larger family of subsets (events) in analogy to the way this is done in the classical theory of stochastic processes. Since quantally mu is generally not of bounded variation, a new method is required. We propose a method that defines the measure of an event by means of a sequence of simpler events which in a suitable sense converges to the event whose measure one is seeking to define. To this end, we introduce canonical sequences approximating certain events, and we propose a measure-based criterion for the convergence of such sequences. Applying the method, we encounter a simple event whose measure is zero classically but non-zero quantally.