On the Connection between Nelson’s Stochastic Quantum Mechanics and Nottale’s Theory of Scale Relativity
In this paper, we review and compare the stochastic quantum mechanics of Nelson and the scale relativity theory of Nottale. We consider both nonrelativistic and relativistic frameworks and include the electromagnetic field. These theories propose a derivation of the Schrödinger and Klein–Gordon equa...
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2024-09-01
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| author | Pierre-Henri Chavanis |
| author_facet | Pierre-Henri Chavanis |
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| description | In this paper, we review and compare the stochastic quantum mechanics of Nelson and the scale relativity theory of Nottale. We consider both nonrelativistic and relativistic frameworks and include the electromagnetic field. These theories propose a derivation of the Schrödinger and Klein–Gordon equations from microscopic processes. We show their formal equivalence. Specifically, we show that the real and imaginary parts of the complex Lorentz equation in Nottale’s theory are equivalent to the Nelson equations, which are themselves equivalent to the Madelung and de Broglie hydrodynamical representations of the Schrödinger and Klein–Gordon equations, respectively. We discuss the different physical interpretations of the Nelson and Nottale theories and stress their strengths and weaknesses. We mention potential applications of these theories to dark matter. |
| format | Article |
| id | doaj-art-b8870dcd623841e486bd20af4feaf173 |
| institution | OA Journals |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-09-01 |
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| spelling | doaj-art-b8870dcd623841e486bd20af4feaf1732025-08-20T01:56:00ZengMDPI AGAxioms2075-16802024-09-0113960610.3390/axioms13090606On the Connection between Nelson’s Stochastic Quantum Mechanics and Nottale’s Theory of Scale RelativityPierre-Henri Chavanis0Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, 31062 Toulouse, FranceIn this paper, we review and compare the stochastic quantum mechanics of Nelson and the scale relativity theory of Nottale. We consider both nonrelativistic and relativistic frameworks and include the electromagnetic field. These theories propose a derivation of the Schrödinger and Klein–Gordon equations from microscopic processes. We show their formal equivalence. Specifically, we show that the real and imaginary parts of the complex Lorentz equation in Nottale’s theory are equivalent to the Nelson equations, which are themselves equivalent to the Madelung and de Broglie hydrodynamical representations of the Schrödinger and Klein–Gordon equations, respectively. We discuss the different physical interpretations of the Nelson and Nottale theories and stress their strengths and weaknesses. We mention potential applications of these theories to dark matter.https://www.mdpi.com/2075-1680/13/9/606stochastic quantum mechanicstheory of scale relativitySchrödinger equationKlein–Gordon equation |
| spellingShingle | Pierre-Henri Chavanis On the Connection between Nelson’s Stochastic Quantum Mechanics and Nottale’s Theory of Scale Relativity Axioms stochastic quantum mechanics theory of scale relativity Schrödinger equation Klein–Gordon equation |
| title | On the Connection between Nelson’s Stochastic Quantum Mechanics and Nottale’s Theory of Scale Relativity |
| title_full | On the Connection between Nelson’s Stochastic Quantum Mechanics and Nottale’s Theory of Scale Relativity |
| title_fullStr | On the Connection between Nelson’s Stochastic Quantum Mechanics and Nottale’s Theory of Scale Relativity |
| title_full_unstemmed | On the Connection between Nelson’s Stochastic Quantum Mechanics and Nottale’s Theory of Scale Relativity |
| title_short | On the Connection between Nelson’s Stochastic Quantum Mechanics and Nottale’s Theory of Scale Relativity |
| title_sort | on the connection between nelson s stochastic quantum mechanics and nottale s theory of scale relativity |
| topic | stochastic quantum mechanics theory of scale relativity Schrödinger equation Klein–Gordon equation |
| url | https://www.mdpi.com/2075-1680/13/9/606 |
| work_keys_str_mv | AT pierrehenrichavanis ontheconnectionbetweennelsonsstochasticquantummechanicsandnottalestheoryofscalerelativity |