Quantized Partial Differential Equations
World 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.
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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