The Many Faces of Maxwell, Dirac and Einstein Equations: A Clifford Bundle Approach

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This book is an exposition of the algebra and calculus of differential forms, of the Clifford and Spin-Clifford bundle formalisms, and of vistas to a formulation of important concepts of differential geometry indispensable for an in-depth understanding of space-time physics.
The formalism discloses the hidden geometrical nature of spinor fields. Maxwell, Dirac and Einstein fields are shown to have representatives by objects of the same mathematical nature, namely sections of an appropriate Clifford bundle. This approach reveals unity in diversity and suggests relationships that are hidden in the standard formalisms and opens new paths for research.
This thoroughly revised second edition also adds three new chapters: on the Clifford bundle approach to the Riemannian or semi-Riemannian differential geometry of branes; on Komar currents in the context of the General Relativity theory; and an analysis of the similarities and main differences between Dirac, Majorana and ELKO spinor fields.
The exercises with solutions, the comprehensive list of mathematical symbols, and the list of acronyms and abbreviations are provided for self-study for students as well as for classes.

From the reviews of the first edition:

“The text is written in a very readable manner and is complemented with plenty of worked-out exercises which are in the style of extended examples. ... their book could also serve as a textbook for graduate students in physics or mathematics." (Alberto Molgado, Mathematical Reviews, 2008 k)


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1 Introduction
2 Multivector and Extensor Calculus
3 The Hidden Geometrical Nature of Spinors
4 Some Differential Geometry
5 Clifford Bundle Approach to the Differential Geometry of Branes
6 Some Issues in Relativistic Spacetime Theories
7 Clifford and DiracHestenes Spinor Fields
8 A Clifford Algebra Lagrangian Formalism in Minkowski Spacetime
12 On the Many Faces of Einstein Equations
13 Maxwell Dirac and SeibergWitten Equations
14 Superparticles and Superfields
15 Maxwell Einstein Dirac and NavierStokes Equations
16 Magnetic Like Particles and Elko Spinor Fields
A Principal Bundles Vector Bundles and Connections
Acronyms and Abbreviations
List of Symbols

9 Conservation Laws on RiemannCartan and Lorentzian Spacetimes
10 The DHE on a RCST and the Meaning of Active Local Lorentz Invariance
11 On the Nature of the Gravitational Field

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About the author (2016)

Waldyr Alves Rodrigues Jr, full professor of Mathematical Physics, at the State University of Campinas, São Paulo, Brazil, received his BSc from the University of São Paulo, and his PhD from the University of Torino, Italy. He has held various positions at Brazilian and European universities, one of them being director of the Institute of Mathematics, Statistics and Scientific Computation, State University of Campinas (IMECC-UNICAMP) from 1994 to1998. He has more than 40 years of teaching experience in more than 70 courses at graduate and post-graduate levels. As a researcher he received many grants, and various prizes - especially in 2012 he won the Paul Sabatier honor medal in 2012 for his contributions to applications of Clifford algebras to mathematical physics. He is the editor-in-chief of the journal “Advances in Applied Clifford Algebras”.

Edmundo Capelas de Oliveira is a member of the Mathematical Physics Group at the State University of Campinas, Sao Paulo, Brazil. He received his PhD in Physics in 1982, and became a full professor in Applied Mathematics in 2015. His research topics involve projective relativity, differential equations, complex analysis and fractional calculus. He is author of several successful textbooks (in Portuguese), teaches undergraduate as well as graduate courses, and held positions as Library Coordinator, Graduate Studies Coordinator of the Department and Associate Director of the Institute.

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