2002 Fiscal Year Final Research Report Summary
Application of Deformation Quantization theory to Geometry and Mathematical Physics
| Project/Area Number |
13640088
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Tokyo University of Science |
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Principal Investigator |
YOSHIOKA Akira Tokyo University of Science, Mathematics, Professor, 理学部, 教授 (40200935)
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| Co-Investigator(Kenkyū-buntansha) |
MAEDA Yoshiaki Keio University, Mathematical science, Professor, 理工学部, 教授 (40101076)
HAR Tamio Tokyo University of Science, Engineering, Assistant Professor, 工学部, 助教授 (10120205)
OMORI Hideki Tokyo University of Science, Mathematics, Professor, 理工学部, 教授 (20087018)
MIYAZAKI Naoya Keio University, Economics, Assistant Professor, 経済学部, 助教授 (50315826)
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| Project Period (FY) |
2001 – 2002
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| Keywords | Deformation Quantization / star products / non-commutative geometry / Hamiltonian mechanics / simplistic geometry / quantization |
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| Research Abstract |
In this research program we obtain the following results.
1. In the complex 2n Euclidian space we introduce the Modyal products, the normal product and the anti-normal *oduct. We set up a function space, for which these three products are convergent and have meaning. We deonte by F(p) the space of all entire functions of order p (p is nonnegative). Then for p equal to or less than 2, the space F(p) becomes a noncommutative topological algebra with repect to each product
2. By means of differential equations, we construct exponential functions of quadratic functions for these products respectively. Thank to the formula of the expoenetial functions, we investigate their singularities with resect to the time variable. We define the Laplace transform by means of the exponential and we construct several transcendental elements of the noncommutative algebra
3. We consider the Weyl algebra W of 2n generators over the complex number. The Weyl ordring, the normal ordering and the anti-normal ordering on the Weyl algebra give isomorphisms of W to the space of all polynomials on C, P(C, 2n). These isomorphisms naturally induces noncommutative, associative products on P(C, 2n). We remarked these produ*
Composing these isomorphisms gines intertwiners between the algebras on P(C, 2n) and the Freshet space F(p). Patching these algebra together by these intertwiner we construct a certain noncommutative manifold
We show the noncommutative manifold is not a manifold in the ordinary sense by has the garb structure
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