Book contents
- Frontmatter
- Contents
- Preface to the Second Edition
- Preface to the Revised Printing
- Preface to the First Edition
- I Manifolds, Tensors, and Exterior Forms
- II Geometry and Topology
- III Lie Groups, Bundles, and Chern Forms
- 15 Lie Groups
- 16 Vector Bundles in Geometry and Physics
- 17 Fiber Bundles, Gauss–Bonnet, and Topological Quantization
- 18 Connections and Associated Bundles
- 19 The Dirac Equation
- 20 Yang–Mills Fields
- 21 Betti Numbers and Covering Spaces
- 22 Chern Forms and Homotopy Groups
- Appendix A Forms in Continuum Mechanics
- Appendix B Harmonic Chains and Kirchhoff's Circuit Laws
- Appendix C Symmetries, Quarks, and Meson Masses
- Appendix D Representations and Hyperelastic Bodies
- Appendix E Orbits and Morse–Bott Theory in Compact Lie Groups
- References
- Index
19 - The Dirac Equation
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface to the Second Edition
- Preface to the Revised Printing
- Preface to the First Edition
- I Manifolds, Tensors, and Exterior Forms
- II Geometry and Topology
- III Lie Groups, Bundles, and Chern Forms
- 15 Lie Groups
- 16 Vector Bundles in Geometry and Physics
- 17 Fiber Bundles, Gauss–Bonnet, and Topological Quantization
- 18 Connections and Associated Bundles
- 19 The Dirac Equation
- 20 Yang–Mills Fields
- 21 Betti Numbers and Covering Spaces
- 22 Chern Forms and Homotopy Groups
- Appendix A Forms in Continuum Mechanics
- Appendix B Harmonic Chains and Kirchhoff's Circuit Laws
- Appendix C Symmetries, Quarks, and Meson Masses
- Appendix D Representations and Hyperelastic Bodies
- Appendix E Orbits and Morse–Bott Theory in Compact Lie Groups
- References
- Index
Summary
Spin is what makes the world go 'round.
The Groups SO(3) and SU(2)
How does SU(2) act on its Lie algebra?
For physical and mathematical motivation for this section (which involves nonrelativistic quantum mechanics) we refer the reader to some remarks of Feynman and of Weyl. Specifically, Feynman, in his section entitled “Degeneracy,” shows that a process involving a specific choice of direction in space requires that the process be described not by a single wave function ψ but rather by a multicomponent column vector of wave functions Ψ = (ψ1, …, ψN)T. He then indicates, roughly speaking, that since the physics cannot depend on the choice of cartesian coordinates (x1, x2, x3) of space, the N-tuples must transform under some representation ρ: SO(3) → U(N) of the rotation group SO(3) of space. This is not quite accurate; since eiγΨ represents the same wave function (when γ is a constant), ρ is only a “ray” representation, ρ(g)ρ(h) = eiγ(g,h) ρ(gh) for a function γ(g, h). Weyl shows that this can be made into a genuine representation, except that it is (perhaps) double-valued. We shall show in this section that there is a natural 2 : 1 homomorphism π of the special unitary group SU(2) onto SO(3), thus yielding a (perhaps double valued) representation of SU(2) into U(N).
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- Information
- The Geometry of PhysicsAn Introduction, pp. 491 - 522Publisher: Cambridge University PressPrint publication year: 2003