dorsal/arxiv
View SchemaQuantum-classical transition in Scale Relativity
| Authors | Marie-Noëlle Célérier, Laurent Nottale |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0609161 |
| URL | https://arxiv.org/abs/quant-ph/0609161 |
| DOI | 10.1088/0305-4470/37/3/026 |
| Journal | J.Phys.A37:931-955,2004 |
Abstract
The theory of scale relativity provides a new insight into the origin of fundamental laws in physics. Its application to microphysics allows us to recover quantum mechanics as mechanics on a non-differentiable (fractal) spacetime. The Schrodinger and Klein-Gordon equations are demonstrated as geodesic equations in this framework. A development of the intrinsic properties of this theory, using the mathematical tool of Hamilton's bi-quaternions, leads us to a derivation of the Dirac equation within the scale-relativity paradigm. The complex form of the wavefunction in the Schrodinger and Klein-Gordon equations follows from the non-differentiability of the geometry, since it involves a breaking of the invariance under the reflection symmetry on the (proper) time differential element (ds <-> - ds). This mechanism is generalized for obtaining the bi-quaternionic nature of the Dirac spinor by adding a further symmetry breaking due to non-differentiability, namely the differential coordinate reflection symmetry (dx^mu <-> - dx^mu) and by requiring invariance under parity and time inversion. The Pauli equation is recovered as a non-relativistic-motion approximation of the Dirac equation.
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"abstract": "The theory of scale relativity provides a new insight into the origin of\nfundamental laws in physics. Its application to microphysics allows us to\nrecover quantum mechanics as mechanics on a non-differentiable (fractal)\nspacetime. The Schrodinger and Klein-Gordon equations are demonstrated as\ngeodesic equations in this framework. A development of the intrinsic properties\nof this theory, using the mathematical tool of Hamilton\u0027s bi-quaternions, leads\nus to a derivation of the Dirac equation within the scale-relativity paradigm.\nThe complex form of the wavefunction in the Schrodinger and Klein-Gordon\nequations follows from the non-differentiability of the geometry, since it\ninvolves a breaking of the invariance under the reflection symmetry on the\n(proper) time differential element (ds \u003c-\u003e - ds). This mechanism is generalized\nfor obtaining the bi-quaternionic nature of the Dirac spinor by adding a\nfurther symmetry breaking due to non-differentiability, namely the differential\ncoordinate reflection symmetry (dx^mu \u003c-\u003e - dx^mu) and by requiring invariance\nunder parity and time inversion. The Pauli equation is recovered as a\nnon-relativistic-motion approximation of the Dirac equation.",
"arxiv_id": "quant-ph/0609161",
"authors": [
"Marie-No\u00eblle C\u00e9l\u00e9rier",
"Laurent Nottale"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/37/3/026",
"journal_ref": "J.Phys.A37:931-955,2004",
"title": "Quantum-classical transition in Scale Relativity",
"url": "https://arxiv.org/abs/quant-ph/0609161"
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