Analysis over Cayley-Dickson numbers and its applications. Hypercomplex holomorphic functions, meromorphic functions, partial differential equations, operational calculus
The book deals with function theory over so-called Cayley-Dickson numbers, which are non-associative generalizations of real quaternions. Cayley-Dickson algebras contain complex numbers, quaternions, octonions, sedenions (sedonions) and so on. The octonion algebra is not longer associative, but at least alternative. Sedenions and higher dimensional algebras in this row do not longer have to be alternative. It should be mentioned that all Clifford algebras are associative, i.e. they can be represented by matrices over the fields of real numbers, complex numbers or quaternions. This is in Cayley-Dickson algebras not longer the case. Nevertheless, Cayley-Dickson algebras have a rich structure, and hence in some way this also applies to the analysis over them. In some sense, a similar theory to complex analysis can be successfully developed. This analysis is called in this book super-analysis. Quantum mechanics, quantum field theory and super-gravitational theories motivate to study such a general topic.NEWLINENEWLINEChapter 1 introduces the notion of super-differentiability over domains in Cayley-Dickson algebras, a corresponding function theory is developed. There is a natural connection between super-differentiability and Dirac spinors in quantum field theory.NEWLINENEWLINEThe second chapter is devoted to the theory of meromorphic functions with Cayley-Dickson variables. Elements of a residue theory are presented. Residues are alway operators here. Algorithms for their computations are described and demonstrated by examples.NEWLINENEWLINE In the third chapter some types of partial differential equations over Cayley-Dickson algebras are considered; a method adapted from ordinary differential equations is used. Examples are taken over to octonions. Also, some non-linear differential equations can be investigated.NEWLINENEWLINEIn Chapter 4 a non-commutative integration method for the solution of partial differential equations in mathematical physics, optics and quantum field theory is lined out. The Klein-Gordon operator, the Helmholtz operator and the wave operator belong to the operators considered here. New decomposition theorems are obtained as well. Fundamental solutions of linear first and second order partial differential equations with so-called variable \(z\)-differentiable Cayley-Dickson algebra coefficients are successful constructed.NEWLINENEWLINEChapter 5 is devoted to a transform analysis over Cayley-Dickson algebras. In particular, a non-commutative multidimensional Laplace transform is described. Corresponding examples for such non-commutative and non-associative transforms are given. This approach can help to solve partial differential equations. A moving boundary value problem and partial differential equations with discontinuous coefficients are studied by using non-commutative analysis.NEWLINENEWLINEThe book contains very tightly presented material on non-commutative and non-associative analysis with lots of examples and applications. Many ideas and suggestions for the use of such a type of analysis are made available. It is really surprising that so many theorems first developed in complex analysis have a counterpart in super-analysis over Cayley-Dickson algebras. Quantum groups and Hopf algebras seem to be further fields of applications.
scientific article; zbMATH DE number 6090399