Discrete dynamics and supergeometry
Abstract We formulate a geometric measurement theory of dynamical classical systems possessing both continuous and discrete degrees of freedom. The approach is covariant with respect to choices of clocks and naturally incorporates laboratories. The latter are embedded symplectic submanifolds of an o...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2024-09-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP09(2024)164 |
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| _version_ | 1846137385593602048 |
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| author | Subhobrata Chatterjee Andrew Waldron Cem Yetişmişoğlu |
| author_facet | Subhobrata Chatterjee Andrew Waldron Cem Yetişmişoğlu |
| author_sort | Subhobrata Chatterjee |
| collection | DOAJ |
| description | Abstract We formulate a geometric measurement theory of dynamical classical systems possessing both continuous and discrete degrees of freedom. The approach is covariant with respect to choices of clocks and naturally incorporates laboratories. The latter are embedded symplectic submanifolds of an odd-dimensional symplectic structure. When suitably defined, symplectic geometry in odd dimensions is exactly the structure needed for covariance. A fundamentally probabilistic viewpoint allows classical supergeometries to describe discrete dynamics. We solve the problem of how to construct probabilistic measures on supermanifolds given a (possibly odd dimensional) supersymplectic structure. This relies on a superanalog of the Hodge star for differential forms and a description of probabilities by convex cones. We also show how stochastic processes such as Markov chains can be described by supergeometry. |
| format | Article |
| id | doaj-art-d4c8faecce6f43a1b9e10b184525b2c8 |
| institution | Kabale University |
| issn | 1029-8479 |
| language | English |
| publishDate | 2024-09-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj-art-d4c8faecce6f43a1b9e10b184525b2c82024-12-08T12:14:52ZengSpringerOpenJournal of High Energy Physics1029-84792024-09-012024914610.1007/JHEP09(2024)164Discrete dynamics and supergeometrySubhobrata Chatterjee0Andrew Waldron1Cem Yetişmişoğlu2Center for Quantum Mathematics and Physics (QMAP) and Department of Physics and Astronomy, University of CaliforniaCenter for Quantum Mathematics and Physics (QMAP) and Department of Mathematics, University of CaliforniaDepartment of Mathematics, İstanbul Technical UniversityAbstract We formulate a geometric measurement theory of dynamical classical systems possessing both continuous and discrete degrees of freedom. The approach is covariant with respect to choices of clocks and naturally incorporates laboratories. The latter are embedded symplectic submanifolds of an odd-dimensional symplectic structure. When suitably defined, symplectic geometry in odd dimensions is exactly the structure needed for covariance. A fundamentally probabilistic viewpoint allows classical supergeometries to describe discrete dynamics. We solve the problem of how to construct probabilistic measures on supermanifolds given a (possibly odd dimensional) supersymplectic structure. This relies on a superanalog of the Hodge star for differential forms and a description of probabilities by convex cones. We also show how stochastic processes such as Markov chains can be described by supergeometry.https://doi.org/10.1007/JHEP09(2024)164Differential and Algebraic GeometryRandom SystemsStochastic ProcessesSuperspaces |
| spellingShingle | Subhobrata Chatterjee Andrew Waldron Cem Yetişmişoğlu Discrete dynamics and supergeometry Journal of High Energy Physics Differential and Algebraic Geometry Random Systems Stochastic Processes Superspaces |
| title | Discrete dynamics and supergeometry |
| title_full | Discrete dynamics and supergeometry |
| title_fullStr | Discrete dynamics and supergeometry |
| title_full_unstemmed | Discrete dynamics and supergeometry |
| title_short | Discrete dynamics and supergeometry |
| title_sort | discrete dynamics and supergeometry |
| topic | Differential and Algebraic Geometry Random Systems Stochastic Processes Superspaces |
| url | https://doi.org/10.1007/JHEP09(2024)164 |
| work_keys_str_mv | AT subhobratachatterjee discretedynamicsandsupergeometry AT andrewwaldron discretedynamicsandsupergeometry AT cemyetismisoglu discretedynamicsandsupergeometry |