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This book, intended to commemorate the work of Paul Dirac, highlights new developments in the main directions of Clifford analysis. Just as complex analysis is based on the algebra of the complex numbers, Clifford analysis is based on the geometric Clifford algebras. Many methods and theorems from complex analysis generalize to higher dimensions in various ways. However, many new features emerge in the process, and much of this work is still in its infancy. Some of the leading mathematicians working in this field have contributed to this book in conjunction with “Clifford Analysis and Related Topics: a conference in honor of Paul A.M. Dirac,” which was held at Florida State University, Tallahassee, on December 15-17, 2014. The content reflects talks given at the conference, as well as contributions from mathematicians who were invited but were unable to attend. Hence much of the mathematics presented here is not only highly topical, but also cannot be found elsewhere in print. Given its scope, the book will be of interest to mathematicians and physicists working in these areas, as well as students seeking to catch up on the latest developments.
The contributions to this volume are devoted to a discussion of state-of-the-art research and treatment of problems of a wide spectrum of areas in complex analysis ranging from pure to applied and interdisciplinary mathematical research. Topics covered include: holomorphic approximation, hypercomplex analysis, special functions of complex variables, automorphic groups, zeros of the Riemann zeta function, Gaussian multiplicative chaos, non-constant frequency decompositions, minimal kernels, one-component inner functions, power moment problems, complex dynamics, biholomorphic cryptosystems, fermionic and bosonic operators. The book will appeal to graduate students and research mathematicians as well as to physicists, engineers, and scientists, whose work is related to the topics covered.
Quaternionic and Clifford analysis are an extension of complex analysis into higher dimensions. The unique starting point of Wolfgang Sprößig’s work was the application of quaternionic analysis to elliptic differential equations and boundary value problems. Over the years, Clifford analysis has become a broad-based theory with a variety of applications both inside and outside of mathematics, such as higher-dimensional function theory, algebraic structures, generalized polynomials, applications of elliptic boundary value problems, wavelets, image processing, numerical and discrete analysis. The aim of this volume is to provide an essential overview of modern topics in Clifford analysis, presented by specialists in the field, and to honor the valued contributions to Clifford analysis made by Wolfgang Sprößig throughout his career.
This book contains a selection of papers presented at the session "Quaternionic and Clifford Analysis" at the 10th ISAAC Congress held in Macau in August 2015. The covered topics represent the state-of-the-art as well as new trends in hypercomplex analysis and its applications.
Foundations of the relativistic quantum mechanics and field theory of arbitrary spin are presented. New relativistic wave equations without redundant components for the particle-antiparticle doublets of arbitrary spin are considered. The comparison with known arbitrary spin equations of Bhabha, Bargman-Wigner and with Pauli-Fierz, Rarita-Schwinger equations (for the spin s=3/2) demonstrates the advantages of the presented approach. The special procedure of synthesis of higher spin relativistic wave equations is suggested. New equations are considered on three levels of (i) relativistic canonical quantum mechanics, (ii) canonical Foldy-Wouthuysen type field theory, and (iii) manifestly covariant field theory. The derivation of field equations based on the start from the relativistic canonical quantum mechanics is given. The corresponding transition operator, which is the extended Foldy-Wouthuysen transformation, is suggested and described. This model of relativistic quantum mechanics is described here on the level of von Neumann's consideration of non-relativistic case. The Lagrange approach for the spinor field in the Foldy-Wouthuysen representation is analyzed. The proof of the Fermi-Bose duality property of a few main equations of field theory, which before were known to have only single Fermi (or single Bose) property, is given. Hidden Bose properties (symmetry, solutions, and conservation laws) of the Dirac equation are proved. Both cases of non-zero and zero mass are considered. New useful mathematical objects, which are the pure matrix representations of the 64-dimensional Clifford and 28-dimensional SO(8) algebras over the field of real numbers, are put into consideration. The application of such algebras to the Dirac and Dirac-like equations properties analysis is demonstrated. Fermi and Bose SO(4) symmetries of the relativistic hydrogen atom are found. New symmetries and solutions of the Maxwell equations are considered. The Maxwell equations in the form, having maximal symmetry, are suggested and described. The application of such field-strength equations to the atomic microworld phenomena is demonstrated. On the basis of such Maxwell system the relativistic hydrogen atom spectrum and quantum properties of this atom are described. The Sommerfeld-Dirac fine structure formula, Plank constant and the Bohr postulates are derived in the frameworks of classical electrodynamics. The limits and boarders of classical physics applications in inneratomic microworld are discussed. In order to determine the place of our approach among other investigations the 26 variants of the Dirac equation derivation are considered.
Symmetries play a fundamental role in physics. Non-Abelian gauge symmetries are the symmetries behind theories for massless spin-1 particles, while the reparametrization symmetry is behind Einstein's gravity theory for massless spin-2 particles. In supersymmetric theories these particles can be connected also to massless fermionic particles. Does Nature stop at spin-2 or can there also be massless higher spin theories. In the past strong indications have been given that such theories do not exist. However, in recent times ways to evade those constraints have been found and higher spin gauge theories have been constructed. With the advent of the AdS/CFT duality correspondence even stronger indications have been given that higher spin gauge theories play an important role in fundamental physics.All these issues were discussed at a recent international workshop in Singapore where the leading scientists in the field participated. This volume presents an up-to-date, detailed overview of the theories including its historic background, as well as the latest accomplishments in understanding the foundational properties of higher spin physics.
In its traditional form, Clifford analysis provides the function theory for solutions of the Dirac equation. From the beginning, however, the theory was used and applied to problems in other fields of mathematics, numerical analysis, and mathematical physics. recently, the theory has enlarged its scope considerably by incorporating geometrical methods from global analysis on manifolds and methods from representation theory. New, interesting branches of the theory are based on conformally invariant, first-order systems other than the Dirac equation, or systems that are invariant with respect to a group other than the conformal group. This book represents an up-to-date review of Clifford analysis in its present form, its applications, and directions for future research. Readership: Mathematicians and theoretical physicists interested in Clifford analysis itself, or in its applications to other fields.
Provides an overview of the eigenspinors of the charge conjugation operator and mass one dimension fermions.
William Kingdon Clifford published the paper defining his "geometric algebras" in 1878, the year before his death. Clifford algebra is a generalisation to n-dimensional space of quaternions, which Hamilton used to represent scalars and vectors in real three-space: it is also a development of Grassmann's algebra, incorporating in the fundamental relations inner products defined in terms of the metric of the space. It is a strange fact that the Gibbs Heaviside vector techniques came to dominate in scientific and technical literature, while quaternions and Clifford algebras, the true associative algebras of inner-product spaces, were regarded for nearly a century simply as interesting mathematical curiosities. During this period, Pauli, Dirac and Majorana used the algebras which bear their names to describe properties of elementary particles, their spin in particular. It seems likely that none of these eminent mathematical physicists realised that they were using Clifford algebras. A few research workers such as Fueter realised the power of this algebraic scheme, but the subject only began to be appreciated more widely after the publication of Chevalley's book, 'The Algebraic Theory of Spinors' in 1954, and of Marcel Riesz' Maryland Lectures in 1959. Some of the contributors to this volume, Georges Deschamps, Erik Folke Bolinder, Albert Crumeyrolle and David Hestenes were working in this field around that time, and in their turn have persuaded others of the importance of the subject.
Accompanying DVD-ROM contains the electronic proceedings of the summer school on mathematical general relativity and global properties of solutions of Einstein's equations held at Cargèse, Corsica, France, July 20-Aug. 10, 2002.