An algebraic formulation of nonassociative quantum mechanics
Fuente:
arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910439773831168 |
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| author | Schupp, Peter Szabo, Richard J. |
| author_facet | Schupp, Peter Szabo, Richard J. |
| contents | We develop a version of quantum mechanics that can handle nonassociative algebras of observables and which reduces to standard quantum theory in the traditional associative setting. Our algebraic approach is naturally probabilistic and is based on using the universal enveloping algebra of a general nonassociative algebra to introduce a generalized notion of associative composition product. We formulate properties of states together with notions of trace, and use them to develop GNS constructions. We describe Heisenberg and Schrödinger pictures of completely positive dynamics, and we illustrate our formalism on the explicit examples of finite-dimensional matrix Jordan algebras as well as the octonion algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_03647 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An algebraic formulation of nonassociative quantum mechanics Schupp, Peter Szabo, Richard J. Quantum Physics High Energy Physics - Theory Mathematical Physics Rings and Algebras We develop a version of quantum mechanics that can handle nonassociative algebras of observables and which reduces to standard quantum theory in the traditional associative setting. Our algebraic approach is naturally probabilistic and is based on using the universal enveloping algebra of a general nonassociative algebra to introduce a generalized notion of associative composition product. We formulate properties of states together with notions of trace, and use them to develop GNS constructions. We describe Heisenberg and Schrödinger pictures of completely positive dynamics, and we illustrate our formalism on the explicit examples of finite-dimensional matrix Jordan algebras as well as the octonion algebra. |
| title | An algebraic formulation of nonassociative quantum mechanics |
| topic | Quantum Physics High Energy Physics - Theory Mathematical Physics Rings and Algebras |
| url | https://arxiv.org/abs/2311.03647 |