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Toward a fundamental theorem of quantal measure theory

Published online by Cambridge University Press:  06 September 2012

RAFAEL D. SORKIN
RAFAEL D. SORKIN*
Affiliation:
Perimeter Institute, 31 Caroline Street North, Waterloo ON, N2L 2Y5, Canada and Department of Physics, Syracuse University, Syracuse, NY 13244-1130, U.S.A. Email: rsorkin@perimeterinstitute.ca
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Abstract

In this paper 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 ℤn. In both cases, the dynamics can be coded into a vector-valued measure μ on Ω, the space of all histories. Initially, μ is just defined on special subsets of Ω called cylinder events, and we 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 μ 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 that in a suitable sense converges to the event whose measure we are 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.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 22 , Special Issue 5: Computability of the Physical , October 2012 , pp. 816 - 852
Copyright
Copyright © Cambridge University Press 2012

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