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Toward a fundamental theorem of quantal measure theory


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.

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Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
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