dorsal/arxiv
View SchemaNoncommutative geometry and its relation to stochastic calculus and symplectic mechanics
| Authors | A. Dimakis, C. Tzanakis |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9606011 |
| URL | https://arxiv.org/abs/q-alg/9606011 |
Abstract
In the context of a noncommutative differential calculus on the algebra of real valued functions of an $n$-dimensional manifold $M$, a commutative and associative product of 1-forms is naturally defined. Ordinary differential calculus corresponds to this product being trivially zero. We consider the minimal generalization in which the algebra of 1-forms with this product is nilpotent of degree 3. Basic tensor analysis and differential geometry are developped in this context and applied to the formulation of symplectic geometry and Hamiltonian dynamics. It is shown that the corresponding Liouville equation can take the form of a generalized Fokker-Planck equation, well known in statistical mechanics of open systems, as well as in stochastic calculus. Specifically it may be the same with that obtained via Ito stochastic calculus, when solving a linear stochastic differential equation with a Wiener process as the stochastic term. The close connection between noncommutative differential structures of this kind, with semimartingale stochastic processes on manifolds thus suggested, is further explored.
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"abstract": "In the context of a noncommutative differential calculus on the algebra of\nreal valued functions of an $n$-dimensional manifold $M$, a commutative and\nassociative product of 1-forms is naturally defined. Ordinary differential\ncalculus corresponds to this product being trivially zero. We consider the\nminimal generalization in which the algebra of 1-forms with this product is\nnilpotent of degree 3. Basic tensor analysis and differential geometry are\ndevelopped in this context and applied to the formulation of symplectic\ngeometry and Hamiltonian dynamics. It is shown that the corresponding Liouville\nequation can take the form of a generalized Fokker-Planck equation, well known\nin statistical mechanics of open systems, as well as in stochastic calculus.\nSpecifically it may be the same with that obtained via Ito stochastic calculus,\nwhen solving a linear stochastic differential equation with a Wiener process as\nthe stochastic term. The close connection between noncommutative differential\nstructures of this kind, with semimartingale stochastic processes on manifolds\nthus suggested, is further explored.",
"arxiv_id": "q-alg/9606011",
"authors": [
"A. Dimakis",
"C. Tzanakis"
],
"categories": [
"q-alg",
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],
"title": "Noncommutative geometry and its relation to stochastic calculus and symplectic mechanics",
"url": "https://arxiv.org/abs/q-alg/9606011"
},
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