Based on courses taught at the University of Dublin, Carnegie Mellon University, and mostly at Simon Fraser University, this book presents the special theory of relativity from a mathematical point of view. It begins with the axioms of the Minkowski vector space and the flat spacetime manifold. Then it discusses the kinematics of special relativity in terms of Lorentz tranformations, and treats the group structure of Lorentz transformations. Extending the discussion to spinors, the author shows how a unimodular mapping of spinor (vector) space can induce a proper, orthochronous Lorentz mapping on the Minkowski vector space. The second part begins with a discussion of relativistic particle mechanics from both the Lagrangian and Hamiltonian points of view. The book then turns to the relativistic (classical) field theory, including a proof of Noether's theorem and discussions of the Klein-Gordon, electromagnetic, Dirac, and non-abelian gauge fields. The final chapter deals with recent work on classical fields in an eight-dimensional covariant phase space.
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ISBN-13
978-1-4612-0893-8 (9781461208938)
DOI
10.1007/978-1-4612-0893-8
Schweitzer Klassifikation
1. Four-Dimensional Vector Spaces and Linear Mappings.- 1.1. Minkowski Vector Space V4.- 1.2. Lorentz Mappings of V4.- 1.3. The Minkowski Tensors.- 2. Flat Minkowski Space-Time Manifold M4 and Tensor Fields.- 2.1. A Four-Dimensional Differentiable Manifold.- 2.2. Minkowski Space-Time M4 and the Separation Function.- 2.3. Flat Submanifolds of Minkowski Space-Time M4.- 2.4. Minkowski Tensor Fields on M4.- 3. The Lorentz Transformation.- 3.1. Applications of the Lorentz Transformation.- 3.2. The Lorentz Group ?4.- 3.3. Real Representations of the Lorentz Group ?4.- 3.4. The Lie Group ?+4+.- 4. Pauli Matrices, Spinors, Dirac Matrices, and Dirac Bispinors.- 4.1. Pauli Matrices, Rotations, and Lorentz Transformations.- 4.2. Spinors and Spinor-Tensors.- 4.3. Dirac Matrices and Dirac Bispinors.- 5. The Special Relativistic Mechanics.- 5.1. The Prerelativistic Particle Mechanics.- 5.2. Prerelativistic Particle Mechanics in Space and Time E3 × ?.- 5.3. The Relativistic Equation of Motion of a Particle.- 5.4. The Relativistic Lagrangian and Hamiltonian Mechanics of a Particle.- 6. The Special Relativistic Classical Field Theory.- 6.1. Variational Formalism for Relativistic Classical Fields.- 6.2. The Klein-Gordon Scalar Field.- 6.3. The Electromagnetic Tensor Field.- 6.4. Nonabelian Gauge Fields.- 6.5. The Dirac Bispinor Field.- 6.6. Interaction of the Dirac Field with Gauge Fields.- 7. The Extended (or Covariant) Phase Space and Classical Fields.- 7.1. Classical Fields.- 7.2. The Generalized Klein-Gordon Equation.- 7.3. Spin-½ Fields in the Extended Phase Space.- Answers and Hints to Selected Exercises.- Index of Symbols.
