An essay on the completion of quantum theory. II: unitary time evolution
In this second part of the ``essay on the completion of quantum theory'' we define the {\em unitary setting of completed quantum mechanics}, by adding as intrinsic data to those from Part I (https://arxiv.org/abs/1711.08643) the choice of a north pole N and south pole S in the geometric sp...
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ftccsdartic:oai:HAL:hal-01822440v1 2023-05-15T17:39:54+02:00 An essay on the completion of quantum theory. II: unitary time evolution Bertram, Wolfgang Institut Élie Cartan de Lorraine (IECL) Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS) 2018-06-25 https://hal.archives-ouvertes.fr/hal-01822440 https://hal.archives-ouvertes.fr/hal-01822440/document https://hal.archives-ouvertes.fr/hal-01822440/file/WB-TB-2.pdf en eng HAL CCSD info:eu-repo/semantics/altIdentifier/arxiv/1807.04650 hal-01822440 https://hal.archives-ouvertes.fr/hal-01822440 https://hal.archives-ouvertes.fr/hal-01822440/document https://hal.archives-ouvertes.fr/hal-01822440/file/WB-TB-2.pdf ARXIV: 1807.04650 info:eu-repo/semantics/OpenAccess https://hal.archives-ouvertes.fr/hal-01822440 2018 octahedral symmetry Jordan-Lie algebras (geometry of) quantum mechanics (self) duality Cayley transform unitary group Lie torsor projective line AMS : 46L89 51M35 ,58B25 81P05 81R99 81Q70. [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] info:eu-repo/semantics/preprint Preprints, Working Papers, . 2018 ftccsdartic 2021-11-07T02:53:26Z In this second part of the ``essay on the completion of quantum theory'' we define the {\em unitary setting of completed quantum mechanics}, by adding as intrinsic data to those from Part I (https://arxiv.org/abs/1711.08643) the choice of a north pole N and south pole S in the geometric space. Then we explain that, in the unitary setting, a complete observablecorresponds to a right (or left) invariant vector field (Hamiltonian field) on the geometric space, and {\em unitary time evolution} is the flow of such a vector field. This interpretation is in fact nothing but the Lie group-Lie group algebra correspondence, for a geometric space that can be interpreted as the Cayley transform of the usual, Hermitian operator space. In order to clarify the geometric nature of this setting, we realize the Cayley transform as a member of a natural octahedral group that can be associated to any triple of pairwise transversal elements. Report North Pole South pole Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) Cayley ENVELOPE(-60.833,-60.833,-64.417,-64.417) North Pole South Pole |
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Open Polar |
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Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) |
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ftccsdartic |
language |
English |
topic |
octahedral symmetry Jordan-Lie algebras (geometry of) quantum mechanics (self) duality Cayley transform unitary group Lie torsor projective line AMS : 46L89 51M35 ,58B25 81P05 81R99 81Q70. [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] |
spellingShingle |
octahedral symmetry Jordan-Lie algebras (geometry of) quantum mechanics (self) duality Cayley transform unitary group Lie torsor projective line AMS : 46L89 51M35 ,58B25 81P05 81R99 81Q70. [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] Bertram, Wolfgang An essay on the completion of quantum theory. II: unitary time evolution |
topic_facet |
octahedral symmetry Jordan-Lie algebras (geometry of) quantum mechanics (self) duality Cayley transform unitary group Lie torsor projective line AMS : 46L89 51M35 ,58B25 81P05 81R99 81Q70. [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] |
description |
In this second part of the ``essay on the completion of quantum theory'' we define the {\em unitary setting of completed quantum mechanics}, by adding as intrinsic data to those from Part I (https://arxiv.org/abs/1711.08643) the choice of a north pole N and south pole S in the geometric space. Then we explain that, in the unitary setting, a complete observablecorresponds to a right (or left) invariant vector field (Hamiltonian field) on the geometric space, and {\em unitary time evolution} is the flow of such a vector field. This interpretation is in fact nothing but the Lie group-Lie group algebra correspondence, for a geometric space that can be interpreted as the Cayley transform of the usual, Hermitian operator space. In order to clarify the geometric nature of this setting, we realize the Cayley transform as a member of a natural octahedral group that can be associated to any triple of pairwise transversal elements. |
author2 |
Institut Élie Cartan de Lorraine (IECL) Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS) |
format |
Report |
author |
Bertram, Wolfgang |
author_facet |
Bertram, Wolfgang |
author_sort |
Bertram, Wolfgang |
title |
An essay on the completion of quantum theory. II: unitary time evolution |
title_short |
An essay on the completion of quantum theory. II: unitary time evolution |
title_full |
An essay on the completion of quantum theory. II: unitary time evolution |
title_fullStr |
An essay on the completion of quantum theory. II: unitary time evolution |
title_full_unstemmed |
An essay on the completion of quantum theory. II: unitary time evolution |
title_sort |
essay on the completion of quantum theory. ii: unitary time evolution |
publisher |
HAL CCSD |
publishDate |
2018 |
url |
https://hal.archives-ouvertes.fr/hal-01822440 https://hal.archives-ouvertes.fr/hal-01822440/document https://hal.archives-ouvertes.fr/hal-01822440/file/WB-TB-2.pdf |
long_lat |
ENVELOPE(-60.833,-60.833,-64.417,-64.417) |
geographic |
Cayley North Pole South Pole |
geographic_facet |
Cayley North Pole South Pole |
genre |
North Pole South pole |
genre_facet |
North Pole South pole |
op_source |
https://hal.archives-ouvertes.fr/hal-01822440 2018 |
op_relation |
info:eu-repo/semantics/altIdentifier/arxiv/1807.04650 hal-01822440 https://hal.archives-ouvertes.fr/hal-01822440 https://hal.archives-ouvertes.fr/hal-01822440/document https://hal.archives-ouvertes.fr/hal-01822440/file/WB-TB-2.pdf ARXIV: 1807.04650 |
op_rights |
info:eu-repo/semantics/OpenAccess |
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1766140664942166016 |