QUANTIZED PARTIAL DIFFERENTIAL EQUATIONS

This book contains three chapters and two addenda. Quantized PDE's.I. In this first part we consider quantum (super) manifolds as topological spaces locally identified with open sets of some locally convex topological vector spaces built starting from suitable topological algebras $A$, \textit{quantum (super)algebras}. The noncommutative character of such quantum (super)manifolds is given by the underlying noncommutative algebras $A$. In fact, here $A$ plays the role of ''fundamental algebra of numbers'', like ${\mathbb K}={\mathbb R},{\mathbb C}$ does for usual commutative manifolds. Therefore, quantum (super)manifolds are the natural generalizations of manifolds , when one substitutes commutative numbers with noncommutative ones. Commutative manifolds are contained into quantum (super)manifolds, as quantum (super)algebras $A$ are required to contain ${\mathbb K}$. This aspect is also reflected by the fact that the class of differentiability $Q^k_w$ for pseudogroup structures defining quantum (super)manifolds, contains the usual $C^k$ differentiability for manifolds. In fact, the class of differentiability of such topological manifolds is defined by requiring weak differentiability and $Z$-linearity of the derivatives, where $Z$ is the centre of the underlying quantum (super)algebras. We give (co)homological characterizations of quantum (super)algebras and quantum (super)manifolds, by applaying to these noncommutative topological manifolds standard methods of algebraic topology. In particular, we calculate also (co)bordism groups in quantum (super)manifolds. Quantized PDE's.II. Here we give a geometric theory of PDE's in the category of quantum (super)manifolds. This theory is the natural extension of the geometric theory of PDE's in the category of commutative (super)manifolds. Emphasis is put on some new algebraic topological techniques that allow us to calculate the integral (co)bordism groups of quantum (super)PDE's, hence to characterize global properties of solutions of quantum (super)PDE's. Many applicatio ns to important equations of quantum field theory are considered also. Quantized PDE's.III. Here we consider a process that allows us to associate to a (super)PDE, defined in the category of (super)commutative manifolds, a quantum (super)PDE. This process is the \textit{covariant quantization}. We describe it in some steps. In fact, we first define quantizations of PDE's in the framework of the mathematical logic, by means of evaluations of the logic of a PDE $E_k$, that is the Boolean algebra of subsets of the classic limit $\Omega(E_k)_c$ of the \textit{quantum situs} $\Omega(E_k)$ of $E_k$, into \textit{quantum logics} $A\subset{\mathfrak L}({\mathfrak H})$, that are algebras of (self-adjoint) operators on a locally convex topological vector (Hilbert) space ${\mathfrak H}$, in such a way to define (pre-)spectral measures on $\Omega(E_k)_c$: $\Omega(E_k)_c\SRA {\mathfrak L}({\mathfrak H})$. We show that these quantizations can be obtained by means of a geometric process called \textit{covariant quantization}, (or \textit{canonical quantization}), of PDE's, that is, roughly speaking the covariant quantization observed by a physical frame. In fact, in a purely geometric context, we prove that any physical observable deforms the classical PDE, $E_k\subset J{\it D}^k(W)$, around its solutions. In this way we can associate to the Lie filtered (super)algebra of the (super)classical observables, ${\mathfrak B}$, of $E_k$, a filtered quantum (super)algebra $\hat{\mathfrak B}$, defined by means of distributive kernels, $\tilde G_q$, propagators, canonically associated to $E_k$. We characterize also the propagators of PDE's by means oftheir integral bordism groups. The final step is the relation between the formal properties $E_\infty\cdots\to E_{k+1}\to E_k\to\cdots$ of the classical equation $E_k$, with quantum ones $\hat E_\infty\cdots\to \hat E_{k+1}\to \hat E_k\to\cdots$. These are obtained in the category of QPDE's, where the quantum (super)algebra $\hat{\mathfrak B}$, so obtained as covariant quantization of $E_k$, identifies a quantum (super)PDE. Addendum I. In refs.[38, 41] are calculated, for the first time, the integral bordism groups of the $3D$ nonisothermal Navier-Stokes equation $(NS)$. A direct consequence of these results is the proof of existence of global (smooth) solutions for $(NS)$. Here we go in some further results emphasizing surgery techniques that allow us to better understand this geometric proof of existence of (smooth) global solutions for any (smooth) boundary condition. Addendum II. In the framework of the geometry of PDE's, we classify variational equations of any order with respect to their formal properties. A variational sequence is introduced for constrained variational PDE's that extends previous ones for variational calculus on fiber bundles. Such extended variational sequence allows us to locally and globally solve variational problems, constrained by PDE's of any order, $E_k\subset J^k_n(W)$, by means of some cohomological properties of $E_k$. Moreover, we relate constrained variational PDE's to the integral (co)bordism groups for PDE's. In this way we are able to characterize the structure properties of global solutions of constrained variational PDE's and to relate them to the structure of global solutions for the corresponding constraint equations. Contents: Quantized PDE's.I. Noncommutative Manifolds: Algebraic topology; Quantum algebras; Quantum manifolds; Quantum supermanifolds. Quantized PDE's.II. Noncommutative PDE's: Quantum PDE's; The quantum Navier-Stokes equation; Quantum super PDE's; The quantum super Yang-Mills equations. Quantized PDE's.III. Quantizations of commutative PDE's: Integral (co)bordism groups in PDE's; Algebraic geometry of PDE's; Spectral measures of PDE's; Quantizations of PDE's; Covariant and canonical quantizations of PDE's. Addendum I: Bordism groups and the $(NS)$-problem. Addendum II: Bordism groups and variational PDE's. References. Index.