Article
| Title: | Non-differentiability of Feynman paths (English) |
| Author: | Muldowney, Pat |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 1 |
| Year: | 2025 |
| Pages: | 123-139 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | A well-known mathematical property of the particle paths of Brownian motion is that such paths are, with probability one, everywhere continuous and nowhere differentiable. R. Feynman (1965) and elsewhere asserted without proof that an analogous property holds for the sample paths (or possible paths) of a non-relativistic quantum mechanical particle to which a conservative force is applied. Using the non-absolute integration theory of Kurzweil and Henstock, this article provides an introductory proof of Feynman's assertion. (English) |
| Keyword: | Feynman path integral |
| Keyword: | quantum mechanics |
| Keyword: | Brownian motion |
| Keyword: | Kurzweil-Henstock integration |
| MSC: | 26A27 |
| MSC: | 28A25 |
| MSC: | 60J65 |
| MSC: | 81Q30 |
| DOI: | 10.21136/CMJ.2024.0493-22 |
| . | |
| Date available: | 2025-03-11T15:58:28Z |
| Last updated: | 2025-03-19 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152900 |
| . | |
| Reference: | [1] Dvoretzky, A., Erdős, P., Kakutani, S.: Nonincrease everywhere of the Brownian motion process.Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probabability. Vol. 2 University of California Press, Berkeley (1961), 103-116. Zbl 0111.15002, MR 0132608 |
| Reference: | [2] Feynman, R. P.: Contents of Feynman's office. Group II, Section I (Correspondence), Box 26, Letter from P. Muldowney (1982).Available at \brokenlink{https://oac.cdlib.org/findaid/{ark:/13030/kt5n39p6k0/}}. |
| Reference: | [3] Feynman, R. P., Hibbs, A. R.: Quantum Mechanics and Path Integrals.International Series in Pure and Applied Physics. McGraw-Hill, New York (1965). Zbl 0176.54902, MR 2797644 |
| Reference: | [4] Henstock, R.: The General Theory of Integration.Oxford Mathematical Monographs. Clarendon Press, Oxford (1991). Zbl 0745.26006, MR 1134656 |
| Reference: | [5] Muldowney, P.: A Modern Theory of Random Variation: With Applications in Stochastic Calculus, Financial Mathematics, and Feynman Integration.John Wiley & Sons, New York (2012). Zbl 1268.60002, MR 3087034, 10.1002/9781118345955 |
| Reference: | [6] Muldowney, P.: Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.John Wiley & Sons, Hoboken (2021). Zbl 1477.60003, MR 4484972, 10.1002/9781119595540 |
| Reference: | [7] Muldowney, P., Skvortsov, V. A.: Improper Riemann integral and Henstock integral in $\Bbb R^n$.Math. Notes 78 (2005), 228-233. Zbl 1079.26007, MR 2245044, 10.1007/s11006-005-0119-7 |
| Reference: | [8] Paley, R. E. A. C., Wiener, N., Zygmund, A.: Notes on random functions.Math. Z. 37 (1933), 647-668. Zbl 0007.35402, MR 1545426, 10.1007/BF01474606 |
| Reference: | [9] Rota, G.-C.: Indiscrete Thoughts.Modern Birkhäuser Classics. Birkhäuser, Boston (1997). Zbl 1131.00005, MR 2374113, 10.1007/978-0-8176-4781-0 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
