Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-19T23:59:00.407Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bombelli, L., Lee, J., Meyer, D. and Sorkin, R. D. (1987) Spacetime as a Causal Set. Physical Review Letters 59 521–524.CrossRefGoogle ScholarPubMed
Brightwell, G., Dowker, F., García, R. S., Henson, J. and Sorkin, R. D. (2003) ‘Observables’ in Causal Set Cosmology. Physical Review D 67 084031. (Preprint available at http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/.)CrossRefGoogle Scholar
Brightwell, G., Dowker, H. F., García, R. S., Henson, J. and Sorkin, R. D. (2002) General Covariance and the ‘Problem of Time’ in a Discrete Cosmology. In: Bowden, K. G. (ed.) Correlations, Proceedings of the ANPA 23 conference, Alternative Natural Philosophy Association, London1–17. (Preprint available at http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/.)Google Scholar
Dowker, F., Johnston, S. and Surya, S. (2010) On extending the Quantum Measure. (Available at arXiv:1007.2725 [gr-qc].)CrossRefGoogle Scholar
Dowker, F. (2006) Causal sets and the deep structure of Spacetime. In: Ashtekar, A. (ed.) 100 Years of Relativity – Space–time Structure: Einstein and Beyond, World Scientific 445–464.Google Scholar
Gudder, S. and Sorkin, R. D. (2012) Two-site quantum random walk (in preparation).CrossRefGoogle Scholar
Henson, J. (2009) The causal set approach to quantum gravity. In: Oriti, D. (ed.) Approaches to Quantum Gravity – Towards a new understanding of space and time, Cambridge University Press 393–413. (An extended version of this review is available at arXiv:gr-qc/0601121v2.)CrossRefGoogle Scholar
Isham, C. J. (2002) Some Reflections on the Status of Conventional Quantum Theory when Applied to Quantum Gravity. (Available at arXiv:quant-ph/0206090v1.)Google Scholar
Kechris, A. S. (1995) Classical Descriptive Set Theory, Springer-Verlag.CrossRefGoogle Scholar
Kolmogorov, A. N. and Fomin, S. V. (1961) Measure, Lebesgue Integrals, and Hilbert Space (Translated by Brunswick, Natascha Artin and Jeffrey, Alan), Academic Press.Google Scholar
Martin, X., O'Connor, D. and Sorkin, R. D. (2005) The Random Walk in Generalized Quantum Theory. Physical Review D 71 024029. (Preprint available at http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/.)CrossRefGoogle Scholar
Moschovakis, Y. N. (2009) Descriptive Set Theory (second edition), American Mathematical Society.CrossRefGoogle Scholar
Pearle, P. (1973) Finite-Dimensional Path-Summation Formulation for Quantum Mechanics. Physical Review D 8 2503–2510.CrossRefGoogle Scholar
Rideout, D. P. and Sorkin, R. D. (2000) A Classical Sequential Growth Dynamics for Causal Sets. Physical Review D 61 024002. (Preprint available at http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/.)CrossRefGoogle Scholar
Sorkin, R. D. (1994) Quantum Mechanics as Quantum Measure Theory. Modern Physics Letters A 9 (33)3119–3127. (Preprint available at http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/.)CrossRefGoogle Scholar
Sorkin, R. D. (1997) Quantum Measure Theory and its Interpretation. In: Feng, D. H. and Hu, B.-L.Quantum Classical Correspondence: Proceedings of the 4th Drexel Symposium on Quantum Nonintegrability, International Press 229–251. (Preprint available at http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/.)Google Scholar
Sorkin, R. D. (2005) Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School). In: Gomberoff, A. and Marolf, D. (eds.) Lectures on Quantum Gravity: Series of the Centro De Estudios Científicos), proceedings of the Valdivia Summer School, Plenum 305–328. (Preprint available at http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/.)CrossRefGoogle Scholar
Sorkin, R. D. (2007) Quantum dynamics without the wave function. Journal of Physics A: Mathematical and Theoretical 40 3207–3221. (Preprint available at http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/.)CrossRefGoogle Scholar
Sorkin, R. D. (2012) Logic is to the quantum as geometry is to gravity. In: Murugan, J., Ellis, G. F. R. and Weltman, A. (eds) Foundations of Space and Time, Cambridge University Press. (Preprint available at http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/.)Google Scholar
Surya, S. (2011) Directions in Causal Set Quantum Gravity. (Available at arXiv:1103.6272v1 [gr-qc].)Google Scholar
Varadarajan, M. and Rideout, D. (2006) A general solution for classical sequential growth dynamics of Causal Sets. Physical Review D 73 104021.CrossRefGoogle Scholar