by Y Ohnuki (Nagoya, Japan), translated by S Kitakado (Nagoya, Japan), & T Sugiyama (Nagoya, Japan)

This book is devoted to an extensive and systematic study on unitary representations of the Poincaré group. The Poincaré group plays an important role in understanding the relativistic picture of particles in quantum mechanics. Complete knowledge of every free particle states and their behaviour can be obtained once all the unitary irreducible representations of the Poincaré group are found. It is a surprising fact that a simple framework such as the Poincaré group, when unified with quantum theory, fixes our possible picture of particles severely and without exception. In this connection, the theory of unitary representations of the Poincaré group provides a fundamental concept of relativistic quantum mechanics and field theory.

• Introduction:
  • Transformation and Invariance
  • Poincaré Group and Free Particles
• Lorentz Group:
  • Double-Valued Representations
  • Spinor Representations
  • Infinitesimal Transformations
• Irreducible Representations of the Poincaré Group:
  • Translational Transformations
  • Lorentz Transformations
  • Little Groups
  • Irreducible Representations
• Unitary Representations of Little Groups:
  • Rotation Group
  • Two-Dimensional Euclidean Group
  • Lorentz Group
  • Three-Dimensional Lorentz Group
  • Classifications of Free Particles
• Wigner Rotations:
  • Particles with Finite Mass
  • Particles with Zero Mass
  • Particles with Imaginary Mass
  • Angular Momenta of Massless Particles
• Covariant Formalism I — Massive Particles:
  • Particles with Spin O
  • Dirac Particles
  • Particles with Higher Spin
  • Generalized Bargmann-Wigner Equations
  • γ Matrices
  • Discrete Transformations
  • Other Covariant Formalisms
• Covariant Formalism II — Massless Particles:
  • Particles with Discrete Spin
  • Discrete Transformations
  • Covariant Inner Products
  • Particles with Continuous Spin
• Quantized Fields:
  • Quantum Theory of Matter Waves
  • Harmonic Oscillators
  • Scalar Fields
  • Spin and Statistics
  • Poincaré Group and Free Fields