Quantized Partial Differential EquationsWorld Scientific, 2004 - 485 sider This book presents, for the first time, a systematic formulation of the geometric theory of noncommutative PDE's which is suitable enough to be used for a mathematical description of quantum dynamics and quantum field theory. A geometric theory of supersymmetric quantum PDE's is also considered, in order to describe quantum supergravity. Covariant and canonical quantizations of (super) PDE's are shown to be founded on the geometric theory of PDE's and to produce quantum (super) PDE's by means of functors from the category of commutative (super) PDE's to the category of quantum (super) PDE's. Global properties of solutions to (super) (commutative) PDE's are 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 integral called canonical quantization Cauchy characteristic numbers cohomology commutative diagram compact consider corresponding covariant defined Definition denote differential forms epimorphism Example fact fiber bundle finite following commutative diagram following exact formally integrable fullquantum function functor Furthermore geometric hence homology homology theory homomorphism Homz Homz(A ideal identified induced integral bordism groups integral manifold integral quantum isomorphism K-algebra linear manifold of dimension map f measure Mes(Q morphism Navier-Stokes equation Noetherian PDE's PDE’s Proposition QPDE QSPDE quantum algebra quantum coordinates quantum manifold quantum super quantum superalgebra quantum supermanifold Radon measure regular solution resp respect ring satisfies short exact sequence singular ſº space-like space-time Spec(A spectral structure submanifold subset subspace superalgebra supermanifold Theorem topological space vector bundle vector field vector space zero