dorsal/arxiv
View SchemaThe Pauli equation in scale relativity
| Authors | Marie-Noelle Celerier, Laurent Nottale |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0609107 |
| URL | https://arxiv.org/abs/quant-ph/0609107 |
| DOI | 10.1088/0305-4470/39/40/020 |
| Journal | J.Phys.A39:12565-12585,2006 |
Abstract
In standard quantum mechanics, it is not possible to directly extend the Schrodinger equation to spinors, so the Pauli equation must be derived from the Dirac equation by taking its non-relativistic limit. Hence, it predicts the existence of an intrinsic magnetic moment for the electron and gives its correct value. In the scale relativity framework, the Schrodinger, Klein-Gordon and Dirac equations have been derived from first principles as geodesics equations of a non-differentiable and continuous spacetime. Since such a generalized geometry implies the occurence of new discrete symmetry breakings, this has led us to write Dirac bi-spinors in the form of bi-quaternions (complex quaternions). In the present work, we show that, in scale relativity also, the correct Pauli equation can only be obtained from a non-relativistic limit of the relativistic geodesics equation (which, after integration, becomes the Dirac equation) and not from the non-relativistic formalism (that involves symmetry breakings in a fractal 3-space). The same degeneracy procedure, when it is applied to the bi-quaternionic 4-velocity used to derive the Dirac equation, naturally yields a Pauli-type quaternionic 3-velocity. It therefore corroborates the relevance of the scale relativity approach for the building from first principles of the quantum postulates and of the quantum tools. This also reinforces the relativistic and fundamentally quantum nature of spin, which we attribute in scale relativity to the non-differentiability of the quantum spacetime geometry (and not only of the quantum space). We conclude by performing numerical simulations of spinor geodesics, that allow one to gain a physical geometric picture of the nature of spin.
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"abstract": "In standard quantum mechanics, it is not possible to directly extend the\nSchrodinger equation to spinors, so the Pauli equation must be derived from the\nDirac equation by taking its non-relativistic limit. Hence, it predicts the\nexistence of an intrinsic magnetic moment for the electron and gives its\ncorrect value. In the scale relativity framework, the Schrodinger, Klein-Gordon\nand Dirac equations have been derived from first principles as geodesics\nequations of a non-differentiable and continuous spacetime. Since such a\ngeneralized geometry implies the occurence of new discrete symmetry breakings,\nthis has led us to write Dirac bi-spinors in the form of bi-quaternions\n(complex quaternions). In the present work, we show that, in scale relativity\nalso, the correct Pauli equation can only be obtained from a non-relativistic\nlimit of the relativistic geodesics equation (which, after integration, becomes\nthe Dirac equation) and not from the non-relativistic formalism (that involves\nsymmetry breakings in a fractal 3-space). The same degeneracy procedure, when\nit is applied to the bi-quaternionic 4-velocity used to derive the Dirac\nequation, naturally yields a Pauli-type quaternionic 3-velocity. It therefore\ncorroborates the relevance of the scale relativity approach for the building\nfrom first principles of the quantum postulates and of the quantum tools. This\nalso reinforces the relativistic and fundamentally quantum nature of spin,\nwhich we attribute in scale relativity to the non-differentiability of the\nquantum spacetime geometry (and not only of the quantum space). We conclude by\nperforming numerical simulations of spinor geodesics, that allow one to gain a\nphysical geometric picture of the nature of spin.",
"arxiv_id": "quant-ph/0609107",
"authors": [
"Marie-Noelle Celerier",
"Laurent Nottale"
],
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"doi": "10.1088/0305-4470/39/40/020",
"journal_ref": "J.Phys.A39:12565-12585,2006",
"title": "The Pauli equation in scale relativity",
"url": "https://arxiv.org/abs/quant-ph/0609107"
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