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This book is an exposition of the algebra and calculus of differentialforms, of the Clifford and Spin-Clifford bundle formalisms, and of vistas to aformulation of important concepts of differential geometry indispensable for anin-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 objectsof the same mathematical nature, namely sections of an appropriate Cliffordbundle. This approach reveals unity in diversity and suggests relationshipsthat are hidden in the standard formalisms and opens new paths for research.
This thoroughly revised second edition also adds three new chapters: on the Cliffordbundle approach to the Riemannian or semi-Riemannian differential geometry ofbranes; on Komar currents in the context of the General Relativity theory; andan analysis of the similarities and main differences between Dirac, Majoranaand ELKO spinor fields.
The exercises with solutions, thecomprehensive list of mathematical symbols, and the list of acronyms andabbreviations are provided for self-study for students as well as for classes.
From the reviews of the firstedition:
"The text is written in a veryreadable manner and is complemented with plenty of worked-out exercises whichare in the style of extended examples. ... their book could also serve as atextbook for graduate students in physics or mathematics." (AlbertoMolgado, Mathematical Reviews, 2008 k)
List of contents
Preface.- Introduction.- Multivector and Extensor Calculus.- The Hidden Geometrical Nature of Spinors.- Some Differential Geometry.- Clifford Bundle Approach to the Differential Geometry of Branes.- Some Issues in Relativistic Spacetime Theories.- Clifford and Dirac-Hestenes Spinor Fields.- A Clifford Algebra Lagrangian Formalism in Minkowski Spacetime.- Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes.- The DHE on a RCST and the Meaning of Active Local Lorentz Invariance.- On the Nature of the Gravitational Field.- On the Many Faces of Einstein Equations.- Maxwell, Dirac and Seiberg-Witten Equations.- Superparticles and Superfields.- Maxwell, Einstein, Dirac and Navier-Stokes Equations.- Magnetic Like Particles and Elko Spinor Fields.-Appendices A1-5.- Acronyms and Abbreviations.- List of Symbols.- Index.
About the author
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.
Summary
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)
Additional text
From the reviews of the 1st edition:
"The main intention of the present book is to familiarise the reader with the algebra and calculus within the Clifford bundle formalism … . 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. ... From my personal point of view, the authors elegantly succeed in their ambitions and, in my opinion, their book could also serve as a textbook for graduate students in physics or mathematics." (Alberto Molgado, Mathematical Reviews, 2008 k)
Report
From the reviews of the 1st edition:
"The main intention of the present book is to familiarise the reader with the algebra and calculus within the Clifford bundle formalism ... . 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. ... From my personal point of view, the authors elegantly succeed in their ambitions and, in my opinion, their book could also serve as a textbook for graduate students in physics or mathematics." (Alberto Molgado, Mathematical Reviews, 2008 k)
