Spinors in Physics

Invented by Dirac in creating his relativistic quantum theory of the electron, spinors are important in quantum theory, relativity, nuclear physics, atomicand molecular physics, and condensed matter physics. Essentially, they are the mathematical entities that correspond to electrons in the same way that ordinary wave functions correspond to classical particles (including photons). Because of their relations to the rotation group SO(n) and the unitary group SU(n), the discussion should be of interest to applied mathematicians as well as physicists.

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I Spinors in Three-Dimensional Space.- 1 Two-Component Spinor Geometry.- 1.1 Definition of a Spinor.- 1.1.1 Stereographic Projection.- 1.1.2 Vectors Associated with a Spinor.- 1.1.3 The Definition of a Spinor.- 1.2 Geometrical Properties.- 1.2.1 Plane Symmetries.- 1.2.2 Rotations.- 1.2.3 The Olinde-Rodrigues Parameters.- 1.2.4 Rotations Defined in Terms of the Euler Angles.- 1.3 Infinitesimal Properties of Rotations.- 1.3.1 The Infinitesimal Rotation Matrix.- 1.3.2 The Pauli Matrices.- 1.3.3 Properties of the Pauli Matrices.- 1.4 Algebraic Properties of Spinors.- 1.4.1 Operations on Spinors.- 1.4.2 Properties of Operations on Spinors.- 1.4.3 The Basis of the Vector Space of Spinors.- 1.4.4 Hermitian Vector Spaces.- 1.4.5 Properties of the Hermitian Product.- 1.4.6 The Use of an Antisymmetric Metric Tensor.- 1.5 Solved Problems.- 2 Spinors and SU (2) Group Representations.- 2.1 Lie Groups.- 2.1.1 Examples of Continuous Groups.- 2.1.2 Analytic Definition of Continuous Groups.- 2.1.3 Linear Representations.- 2.1.4 Infinitesimal Generators.- 2.1.5 Infinitesimal Matrices.- 2.1.6 Exponential Mapping.- 2.1.7 The Nomenclature of Continuous Linear Groups.- 2.2 Unimodular Unitary Groups.- 2.2.1 The Unitary Group U (2).- 2.2.2 The Unitary Unimodular Group SU (2).- 2.2.3 Three-Dimensional Representations.- 2.2.4 Representations of the Groups SU (2).- 2.2.5 Irreducible Representations of SU (2).- 2.3 Solved Problems.- 3 Spinor Representation of SO (3).- 3.1 The Rotation Group SO (3).- 3.1.1 Rotations About a Point.- 3.1.2 The Infinitesimal Matrices of the Group.- 3.1.3 Rotations About a Given Axis.- 3.1.4 The Exponential Matrix of a Rotation About a Given Axis.- 3.2 Irreducible Representations of SO (3).- 3.2.1 The Structure Equations.- 3.2.2 The Infinitesimal Matrices of the Representations of the Group SO (3).- 3.2.3 Eigenvectors and Eigenvalues of the Infinitesimal Matrices of the Representations.- 3.2.4 Irreducible Representations.- 3.2.5 The Infinitesimal Matrices of an Irreducible Representation in the Canonical Basis.- 3.2.6 The Characters of the Rotation Matrices of a Representation.- 3.3 Spherical Harmonics.- 3.3.1 The Infinitesimal Operators in Spherical Coordinates.- 3.3.2 Spherical Harmonics.- 3.4 Spinor Representations.- 3.4.1 The Two-Dimensional Irreducible Representation.- 3.4.2 The Three-Dimensional Irreducible Representation.- 3.4.3 (2 j + 1)-Dimensional Irreducible Representations.- 3.5 Solved Problems.- 4 Pauli Spinors.- 4.1 Spin and Spinors.- 4.2 The Linearized Schrödinger Equations.- 4.2.1 The Free Particle.- 4.2.2 Particle in an Electromagnetic Field.- 4.2.3 The Spinors in Pauli's Equation.- 4.3 Spinor and Vector Fields.- 4.3.1 The Transformation of a Vector Field by a Rotation.- 4.3.2 The Rotation of a Spinor Field.- 4.4 Solved Problems.- II Spinors in Four-Dimensional Space.- 5 The Lorentz Group.- 5.1 The Generalized Lorentz Group.- 5.1.1 Rotations and Reflections.- 5.1.2 Orthochronous and Anti-Orthochronous Transformations.- 5.1.3 Sheets of the Generalized Lorentz Group.- 5.2 The Four-Dimensional Rotation Group.- 5.2.1 Four-Dimensional Orthogonal Transformations.- 5.2.2 Matrix Representations of the Group SO (4).- 5.2.3 Infinitesimal Matrices.- 5.2.4 Irreducible Representations.- 5.3 Solved Problems.- 6 Representations of the Lorentz Groups.- 6.1 Irreducible Representations.- 6.1.1 Relations Between the Groups SO (3, 1)?andSO(4).- 6.1.2 Infinitesimal Matrices.- 6.1.3 Irreducible Representations.- 6.2 The Group SL(2,?).- 6.2.1 Two-Component Spinors.- 6.2.2 Higher-Order Spinors.- 6.2.3 Representations of the GroupsSL(2,?).- 6.2.4 Irreducible Representations.- 6.3 Spinor Representations of the Lorentz Group.- 6.3.1 Four-Dimensional Irreducible Representations.- 6.3.2 Two-Dimensional Representations.- 6.3.3 The Direct Product of Irreducible Representations.- 6.4 Solved Problems.- 7 Dirac Spinors.- 7.1 The Dirac Equation.- 7.1.1 The Classical Relativistic Wave Equation.- 7.1.2 The Dirac Equation for a Free Particle.- 7.1.3 A Particle in an Electromagnetic Field.- 7.2 Relativistic Invariance of the Dirac Equation.- 7.2.1 The Relativistic Invariance Condition.- 7.2.2 The Type of Representation for the Wave Function.- 7.2.3 The Link Between a Spinor and a Four-Vector.- 7.2.4 Dirac's Equation in the Spinor Representation.- 7.2.5 The Symmetric Form of the Dirac Equation.- 7.3 Solved Problems.- 8 Clifford and Lie Algebras.- 8.1 Lie Algebras.- 8.1.1 The Definition of an Algebra.- 8.1.2 Lie Algebras.- 8.1.3 Isomorphic Lie Algebras.- 8.2 Representations of Lie Algebras.- 8.2.1 Definition.- 8.2.2 Representations of a Lie Group and of Its Lie Algebra.- 8.2.3 Connected Groups.- 8.2.4 Reducible and Irreducible Representations.- 8.3 Clifford Algebras.- 8.3.1 Definition.- 8.3.2 Examples of Clifford Algebras.- 8.3.3 Clifford and Lie Algebras.- 8.3.4 Spinor Groups.- 8.4 Solved Problems.- Appendix: Groups and Their Representations.- A.1 The Definition of a Group.- A.1.1 Examples of Groups.- A.1.2 The Axioms Defining a Group.- A.1.3 Elementary Properties of Groups.- A.2 Linear Operators.- A.2.1 The Operator Representing an Element of a Group.- A.2.2 The Operators Acting on the Vectors of Geometric Space.- A.2.3 The Operators Acting on Wave Functions.- A.2.4 Operators Representing a Group.- A.3 Matrix Representations.- A.3.1 The Rotation Matrix Acting on the Vectors of a Three-Dimensional Space.- A.3.2 The Matrix of an Operator Acting on Functions.- A.3.3 The Matrices Representing the Elements of a Group.- A.4 Matrix Representations.- A.4.1 The Definition of a Matrix Representation.- A.4.2 The Fundamental Property of the Matrices of a Representation.- A.4.3 Representation by Regular Matrices.- A.4.4 Equivalent Representations.- A.5 Reducible and Irreducible Representations.- A.5.1 The Direct Sum of Two Vector Spaces.- A.5.2 The Direct Sum of Two Representations.- A.5.3 Irreducible Representations.- A.6 The Direct Product of Representations.- A.6.1 The Direct Product of Two Matrices.- A.6.2 Properties of Tensor Products of Matrices.- A.6.3 The Direct Product of Two Representations.- References.