Analysis of Dirac Systems and Computational Algebra

The subject of Clifford algebras has become an increasingly rich area of research with a significant number of important applications not only to mathematical physics but to numerical analysis, harmonic analysis, and computer science.

The main treatment is devoted to the analysis of systems of linear partial differential equations with constant coefficients, focusing attention on null solutions of Dirac systems. In addition to their usual significance in physics, such solutions are important mathematically as an extension of the function theory of several complex variables. The term "computational" in the title emphasizes two main features of the book, namely, the heuristic use of computers to discover results in some particular cases, and the application of Gröbner bases as a primary theoretical tool.

Knowledge from different fields of mathematics such as commutative algebra, Gröbner bases, sheaf theory, cohomology, topological vector spaces, and generalized functions (distributions and hyperfunctions) is required of the reader. However, all the necessary classical material is initially presented.

The book may be used by graduate students and researchers interested in (hyper)complex analysis, Clifford analysis, systems of partial differential equations with constant coefficients, and mathematical physics.

1 Background Material.- 1.1 Algebraic tools.- 1.2 Analytical tools.- 1.3 Elements of hyperfunction theory.- 1.4 Appendix: category theory.- 2 Computational Algebraic Analysis.- 2.1 A primer of algebraic analysis.- 2.2 The Ehrenpreis-Palamodov Fundamental Principle.- 2.3 The Fundamental Principle for hyperfunctions.- 2.4 Using computational algebra software.- 3 The Cauchy-Fueter System and its Variations.- 3.1 Regular functions of one quaternionic variable.- 3.2 Quaternionic hyperfunctions in one variable.- 3.3 Several quaternionic variables: analytic approach.- 3.4 Several quaternionic variables: an algebraic approach.- 3.5 The Moisil-Theodorescu system.- 4 Special First Order Systems in Clifford Analysis.- 4.1 Introduction to Clifford algebras.- 4.2 Introduction to Clifford analysis.- 4.3 The Dirac complex for two, three and four operators.- 4.4 Special systems in Clifford analysis.- 5 Some First Order Linear Operators in Physics.- 5.1 Physics and algebra of Maxwell and Proca fields.- 5.2 Variations on Maxwell system in the space of biquaternions.- 5.3 Properties of DZ-regular functions.- 5.4 The Dirac equation and the linearization problem.- 5.5 Octonionic Dirac equation.- 6 Open Problems and Avenues for Further Research.- 6.1 The Cauchy-Fueter system.- 6.2 The Dirac system.- 6.3 Miscellanea.