dorsal/arxiv
View SchemaThe hidden geometric character of relativistic quantum mechanics
| Authors | Jose B. Almeida |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0606123 |
| URL | https://arxiv.org/abs/quant-ph/0606123 |
| DOI | 10.1063/1.2406055 |
| Journal | J. Math. Phys. 49(1), pp. 012301, (2007) |
Abstract
The presentation makes use of geometric algebra, also known as Clifford algebra, in 5-dimensional spacetime. The choice of this space is given the character of first principle, justified solely by the consequences that can be derived from such choice and their consistency with experimental results. Given a metric space of any dimension, one can define monogenic functions, the natural extension of analytic functions to higher dimensions; such functions have null vector derivative and have previously been shown by other authors to play a decisive role in lower dimensional spaces. All monogenic functions have null Laplacian by consequence; in an hyperbolic space this fact leads inevitably to a wave equation with plane-like solutions. This is also true for 5-dimensional spacetime and we will explore those solutions, establishing a parallel with the solutions of the Dirac equation. For this purpose we will invoke the isomorphism between the complex algebra of 4x4 matrices, also known as Dirac's matrices. There is one problem with this isomorphism, because the solutions to Dirac's equation are usually known as spinors (column matrices) that don't belong to the 4x4 matrix algebra and as such are excluded from the isomorphism. We will show that a solution in terms of Dirac spinors is equivalent to a plane wave solution. Just as one finds in the standard formulation, monogenic functions can be naturally split into positive/negative energy together with left/right ones. This split is provided by geometric projectors and we will show that there is a second set of projectors providing an alternate 4-fold split. The possible implications of this alternate split are not yet fully understood and are presently the subject of profound research.
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"abstract": "The presentation makes use of geometric algebra, also known as Clifford\nalgebra, in 5-dimensional spacetime. The choice of this space is given the\ncharacter of first principle, justified solely by the consequences that can be\nderived from such choice and their consistency with experimental results. Given\na metric space of any dimension, one can define monogenic functions, the\nnatural extension of analytic functions to higher dimensions; such functions\nhave null vector derivative and have previously been shown by other authors to\nplay a decisive role in lower dimensional spaces. All monogenic functions have\nnull Laplacian by consequence; in an hyperbolic space this fact leads\ninevitably to a wave equation with plane-like solutions. This is also true for\n5-dimensional spacetime and we will explore those solutions, establishing a\nparallel with the solutions of the Dirac equation. For this purpose we will\ninvoke the isomorphism between the complex algebra of 4x4 matrices, also known\nas Dirac\u0027s matrices. There is one problem with this isomorphism, because the\nsolutions to Dirac\u0027s equation are usually known as spinors (column matrices)\nthat don\u0027t belong to the 4x4 matrix algebra and as such are excluded from the\nisomorphism. We will show that a solution in terms of Dirac spinors is\nequivalent to a plane wave solution. Just as one finds in the standard\nformulation, monogenic functions can be naturally split into positive/negative\nenergy together with left/right ones. This split is provided by geometric\nprojectors and we will show that there is a second set of projectors providing\nan alternate 4-fold split. The possible implications of this alternate split\nare not yet fully understood and are presently the subject of profound\nresearch.",
"arxiv_id": "quant-ph/0606123",
"authors": [
"Jose B. Almeida"
],
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"doi": "10.1063/1.2406055",
"journal_ref": "J. Math. Phys. 49(1), pp. 012301, (2007)",
"title": "The hidden geometric character of relativistic quantum mechanics",
"url": "https://arxiv.org/abs/quant-ph/0606123"
},
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