Quantized Partial Differential EquationsWorld Scientific, 2004 - 485 sider This book presents, for the first time, a systematic formulation ofthe geometric theory of noncommutative PDE''s which is suitable enoughto be used for a mathematical description of quantum dynamics andquantum field theory. A geometric theory of supersymmetric quantumPDE''s is also considered, in order to describe quantumsupergravity. Covariant and canonical quantizations of (super) PDE''sare shown to be founded on the geometric theory of PDE''s and toproduce quantum (super) PDE''s by means of functors from the categoryof commutative (super) PDE''s to the category of quantum (super)PDE''s. Global properties of solutions to (super) (commutative) PDE''sare obtained by means of their integral bordism groups. |
Innhold
Noncommutative Manifolds | 1 |
Noncommutative PDEs | 103 |
Quantizations of Commutative PDEs | 193 |
Bordism groups and the NSproblem | 377 |
Bordism groups and variational PDEs | 435 |
| 461 | |
| 473 | |
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3-dimensional A-module admissible associated assume bordism groups boundary called canonical Cartan characteristic closed compact complex connection consider contained coordinates corresponding covariant defined Definition denote derivative differential dimension distribution element equation equivalent exact sequence Example exists extension fact fiber bundle finite flow formally function Furthermore geometric given global hence homology homomorphism homotopy Homz ideal identified induced integral manifold isomorphism Lemma linear locally means measure natural Note obtained operator particular PDE's problem Proof properties Proposition quantization quantum algebra quantum manifold quantum supermanifold regular relation Remark represented resp respect ring satisfies sequence singular smooth solutions space structure submanifold subset super taking Theorem theory topological unique variational vector field vertical write zero