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This book explores the Lipschitz spinorial groups (versor, pinor, spinor and rotor groups) of a real non-degenerate orthogonal geometry (or orthogonal geometry, for short) and how they relate to the group of isometries of that geometry. After a concise mathematical introduction, it offers an axiomatic presentation of the geometric algebra of an orthogonal geometry. Once it has established the language of geometric algebra (linear grading of the algebra; geometric, exterior and interior products; involutions), it defines the spinorial groups, demonstrates their relation to the isometry groups, and illustrates their suppleness (geometric covariance) with a variety of examples. Lastly, the book provides pointers to major applications, an extensive bibliography and an alphabetic index. Combining the characteristics of a self-contained research monograph and a state-of-the-art survey, this book is a valuable foundation reference resource on applications for both undergraduate and graduate students.

Spinor analysis. --- Clifford algebras. --- Geometric algebras --- Algebras, Linear --- Calculus of spinors --- Spinor calculus --- Spinors, Theory of --- Algebra --- Wave mechanics --- Calculus of tensors --- Vector analysis --- Geometry. --- Group theory. --- Mathematical physics. --- Group Theory and Generalizations. --- Mathematical Methods in Physics. --- Physical mathematics --- Physics --- Groups, Theory of --- Substitutions (Mathematics) --- Mathematics --- Euclid's Elements --- Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics

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This book contains a systematic exposition of the theory of spinors in finite-dimensional Euclidean and Riemannian spaces. The applications of spinors in field theory and relativistic mechanics of continuous media are considered. The main mathematical part is connected with the study of invariant algebraic and geometric relations between spinors and tensors. The theory of spinors and the methods of the tensor representation of spinors and spinor equations are thoroughly expounded in four-dimensional and three-dimensional spaces. Very useful and important relations are derived that express the derivatives of the spinor fields in terms of the derivatives of various tensor fields. The problems associated with an invariant description of spinors as objects that do not depend on the choice of a coordinate system are addressed in detail. As an application, the author considers an invariant tensor formulation of certain classes of differential spinor equations containing, in particular, the most important spinor equations of field theory and quantum mechanics. Exact solutions of the Einstein–Dirac equations, nonlinear Heisenberg’s spinor equations, and equations for relativistic spin fluids are given. The book presents a large body of factual material and is suited for use as a handbook. It is intended for specialists in theoretical physics, as well as for students and post-graduate students of physical and mathematical specialties.

Spinor analysis. --- Calculus of spinors --- Spinor calculus --- Spinors, Theory of --- Algebra --- Wave mechanics --- Calculus of tensors --- Vector analysis --- Physics. --- Elementary particles (Physics). --- Quantum field theory. --- Cosmology. --- Mathematical physics. --- Algebraic geometry. --- Mathematical Methods in Physics. --- Elementary Particles, Quantum Field Theory. --- Mathematical Physics. --- Algebraic Geometry. --- Algebraic geometry --- Geometry --- Physical mathematics --- Physics --- Astronomy --- Deism --- Metaphysics --- Relativistic quantum field theory --- Field theory (Physics) --- Quantum theory --- Relativity (Physics) --- Elementary particles (Physics) --- High energy physics --- Nuclear particles --- Nucleons --- Nuclear physics --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Mathematics