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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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L. Bombelli , J. Lee , D. Meyer and R. D. Sorkin (1987) Spacetime as a Causal Set. Physical Review Letters 59 521524.

G. Brightwell , F. Dowker , R. S. García , J. Henson and R. D. Sorkin (2003) ‘Observables’ in Causal Set Cosmology. Physical Review D 67 084031. (Preprint available at

A. S. Kechris (1995) Classical Descriptive Set Theory, Springer-Verlag.

X. Martin , D. O'Connor and R. D. Sorkin (2005) The Random Walk in Generalized Quantum Theory. Physical Review D 71 024029. (Preprint available at

Y. N. Moschovakis (2009) Descriptive Set Theory (second edition), American Mathematical Society.

P. Pearle (1973) Finite-Dimensional Path-Summation Formulation for Quantum Mechanics. Physical Review D 8 25032510.

D. P. Rideout and R. D. Sorkin (2000) A Classical Sequential Growth Dynamics for Causal Sets. Physical Review D 61 024002. (Preprint available at

R. D. Sorkin (1994) Quantum Mechanics as Quantum Measure Theory. Modern Physics Letters A 9 (33) 31193127. (Preprint available at

R. D. Sorkin (2005) Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School). In: A. Gomberoff and D. Marolf (eds.) Lectures on Quantum Gravity: Series of the Centro De Estudios Científicos), proceedings of the Valdivia Summer School, Plenum305328. (Preprint available at

R. D. Sorkin (2007) Quantum dynamics without the wave function. Journal of Physics A: Mathematical and Theoretical 40 32073221. (Preprint available at

R. D. Sorkin (2012) Logic is to the quantum as geometry is to gravity. In: J. Murugan , G. F. R. Ellis and A. Weltman (eds) Foundations of Space and Time, Cambridge University Press. (Preprint available at

M. Varadarajan and D. Rideout (2006) A general solution for classical sequential growth dynamics of Causal Sets. Physical Review D 73 104021.

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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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