The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra
Lichnerowicz's algebra of differential geometric operators acting on symmetric tensors can be obtained from generalized geodesic motion of an observer carrying a complex tangent vector. This relation is based upon quantizing the classical evolution equations, and identifying wavefunctions with...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2007-09-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Subjects: | |
| Online Access: | http://www.emis.de/journals/SIGMA/2007/089/ |
| _version_ | 1818250397634002944 |
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| author | Karl Hallowell Andrew Waldron |
| author_facet | Karl Hallowell Andrew Waldron |
| author_sort | Karl Hallowell |
| collection | DOAJ |
| description | Lichnerowicz's algebra of differential geometric operators acting on symmetric tensors can be obtained from generalized geodesic motion of an observer carrying a complex tangent vector. This relation is based upon quantizing the classical evolution equations, and identifying wavefunctions with sections of the symmetric tensor bundle and Noether charges with geometric operators. In general curved spaces these operators obey a deformation of the Fourier-Jacobi Lie algebra of sp(2,R). These results have already been generalized by the authors to arbitrary tensor and spinor bundles using supersymmetric quantum mechanical models and have also been applied to the theory of higher spin particles. These Proceedings review these results in their simplest, symmetric tensor setting. New results on a novel and extremely useful reformulation of the rank 2 deformation of the Fourier-Jacobi Lie algebra in terms of an associative algebra are also presented. This new algebra was originally motivated by studies of operator orderings in enveloping algebras. It provides a new method that is superior in many respects to common techniques such as Weyl or normal ordering. |
| first_indexed | 2024-12-12T15:51:45Z |
| format | Article |
| id | doaj.art-c736c737dfb948fdbe2316be9e0046f0 |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-12-12T15:51:45Z |
| publishDate | 2007-09-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-c736c737dfb948fdbe2316be9e0046f02022-12-22T00:19:37ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592007-09-013089The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi AlgebraKarl HallowellAndrew WaldronLichnerowicz's algebra of differential geometric operators acting on symmetric tensors can be obtained from generalized geodesic motion of an observer carrying a complex tangent vector. This relation is based upon quantizing the classical evolution equations, and identifying wavefunctions with sections of the symmetric tensor bundle and Noether charges with geometric operators. In general curved spaces these operators obey a deformation of the Fourier-Jacobi Lie algebra of sp(2,R). These results have already been generalized by the authors to arbitrary tensor and spinor bundles using supersymmetric quantum mechanical models and have also been applied to the theory of higher spin particles. These Proceedings review these results in their simplest, symmetric tensor setting. New results on a novel and extremely useful reformulation of the rank 2 deformation of the Fourier-Jacobi Lie algebra in terms of an associative algebra are also presented. This new algebra was originally motivated by studies of operator orderings in enveloping algebras. It provides a new method that is superior in many respects to common techniques such as Weyl or normal ordering.http://www.emis.de/journals/SIGMA/2007/089/symmetric tensorsFourier-Jacobi algebrashigher spinsoperator orderings |
| spellingShingle | Karl Hallowell Andrew Waldron The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra Symmetry, Integrability and Geometry: Methods and Applications symmetric tensors Fourier-Jacobi algebras higher spins operator orderings |
| title | The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra |
| title_full | The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra |
| title_fullStr | The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra |
| title_full_unstemmed | The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra |
| title_short | The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra |
| title_sort | symmetric tensor lichnerowicz algebra and a novel associative fourier jacobi algebra |
| topic | symmetric tensors Fourier-Jacobi algebras higher spins operator orderings |
| url | http://www.emis.de/journals/SIGMA/2007/089/ |
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