Quantum Exotic PDE's

TL;DR

This paper develops a new algebraic and geometric framework for quantum PDEs, introducing quantum hypercomplex manifolds, quantum logic structures, and proving existence theorems, culminating in a quantum Poincaré conjecture.

Contribution

It introduces quantum hypercomplex manifolds and associates Heyting algebras to quantum PDEs, extending geometric and logical structures in quantum PDE theory.

Findings

Association of integral Heyting algebra to quantum PDEs

Existence theorems for solutions in quantum hypercomplex manifolds

A smooth quantum Poincaré conjecture

Abstract

Following the previous works on the A. Pr\'astaro's formulation of algebraic topology of quantum (super) PDE's, it is proved that a canonical Heyting algebra ({\em integral Heyting algebra}) can be associated to any quantum PDE. This is directly related to the structure of its global solutions. This allows us to recognize a new inside in the concept of quantum logic for microworlds. Furthermore, the Prastaro's geometric theory of quantum PDE's is applied to the new category of {\em quantum hypercomplex manifolds}, related to the well-known Cayley-Dickson construction for algebras. Theorems of existence for local and global solutions are obtained for (singular) PDE's in this new category of noncommutative manifolds. Finally the extension of the concept of exotic PDE's, recently introduced by A.Pr\'astaro, has been extended to quantum PDE's. Then a smooth quantum version of the quantum (generalized) Poincar\'e conjecture is given too. These results extend ones for quantum (generalized) Poincar\'e conjecture, previously given by A. Pr\'astaro.