Physics of the Lorentz group /

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This book explains the Lorentz group in a language familiar to physicists, namely in terms of two-by-two matrices. While the three-dimensional rotation group is one of the standard mathematical tools in physics, the Lorentz group applicable to the four-dimensional Minkowski space is still very stran...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autores principales: Baçkal, Sibel (Autor), Kim, Y. S. (Autor), Noz, Marilyn E. (Autor)
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Bristol [England] (Temple Circus, Temple Way, Bristol BS1 6HG, UK) : IOP Publishing, [2021]
Edición:Second edition.
Colección:IOP (Series). Release 21.
IOP ebooks. 2021 collection.
Temas:
Lorentz groups.
Rotation groups.
Mathematical physics.
SCIENCE / Physics / Mathematical & Computational.
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 1. Lorentz group and its representations
  • 1.1. Generators of the Lorentz group
  • 1.2. Two-by-two representation of the Lorentz group
  • 1.3. Conformal representation of the Lorentz group
  • 1.4. Representations of the Poincaré group
  • 1.5. Representations of the Lorentz group based on harmonic oscillators
  • 1.6. Wigner functions for the Lorentz group
  • 2. Wigner's little groups for internal space-time symmetries
  • 2.1. Euler decomposition of Wigner's little group
  • 2.2. O(3)-like little group for massive particles
  • 2.3. E(2)-like little group for massless particles
  • 2.4. O(2, 1)-like little group for imaginary-mass particles
  • 2.5. Further properties of Wigner's little groups
  • 2.6. Little groups in the light-cone coordinate system
  • 3. Group contractions
  • 3.1. Contraction with squeeze transformations
  • 3.2. Contractions of the O(3) rotation group
  • 3.3. Contraction of the O(2, 1) Lorentz group
  • 3.4. Contraction of the Lorentz group
  • 3.5. Tangential spheres
  • 4. Two-by-two representations of Wigner's little groups
  • 4.1. Transformation properties of the energy-momentum four-vector
  • 4.2. Two-by-two representations of Wigner's little groups
  • 4.3. Lorentz completion of the little groups
  • 4.4. Bargmann and Wigner decompositions
  • 4.5. Conjugate transformations
  • 4.6. One little group with three branches
  • 4.7. Classical damped harmonic oscillator
  • 5. Relativistic spinors and polarization of photons and neutrinos
  • 5.1. Two-component spinors
  • 5.2. Massive and massless particles
  • 5.3. Dirac spinors and massless particles
  • 5.4. Polarization of massless neutrinos
  • 5.5. Scalars, vectors, tensors, and the polarization of photons
  • 6. Lorentz-covariant harmonic oscillators
  • 6.1. Dirac's plan to construct Lorentz-covariant quantum mechanics
  • 6.2. Dirac's forms of relativistic dynamics
  • 6.3. Running waves and standing waves
  • 6.4. Little groups for relativistic extended particles
  • 6.5. Further properties of covariant oscillator wave functions
  • 6.6. Lorentz contraction of harmonic oscillators
  • 6.7. Feynman's rest of the Universe
  • 7. Quarks and partons in the Lorentz-covariant world
  • 7.1. Lorentz-covariant quark model
  • 7.2. Feynman's parton picture
  • 7.3. Proton structure function
  • 7.4. Proton form factor and Lorentz coherence
  • 7.5. Coherence in energy-momentum space
  • 7.6. Hadronic temperature and boiling quarks
  • 8. Wigner functions and their symmetries
  • 8.1. Symmetries and the uncertainty principle in the Wigner phase space
  • 8.2. Four-dimensional phase space
  • 8.3. Canonical transformations
  • 8.4. SL(4, r) symmetry
  • 8.5. Dirac matrices for O(3, 3)
  • 8.6. O(3, 3) symmetry
  • 9. Coupled harmonic oscillators and squeezed states of light
  • 9.1. Coupled oscillators
  • 9.2. Lorentz-covariant oscillators
  • 9.3. Squeezed states of light
  • 9.4. Further notes on squeezed states
  • 9.5. O(3, 2) symmetry from Dirac's coupled oscillators
  • 9.6. Canonical and non-canonical transformations from the coupled oscillators
  • 9.7. Entropy and the expanding Wigner phase space
  • 10. Special relativity from quantum mechanics?
  • 10.1. Definition of the problem
  • 10.2. Symmetries of the single oscillator
  • 10.3. Symmetries from two oscillators
  • 10.4. Contraction of O(3, 2) to the inhomogeneous Lorentz group
  • 11. Lorentz group in ray optics
  • 11.1. The group of ABCD matrices applied to ray optics
  • 11.2. Equi-diagonalization of the ABCD matrix
  • 11.3. Decomposition of the ABCD matrix
  • 11.4. Laser cavities
  • 11.5. Composition of lens and translation matrices
  • 11.6. Optical beam propagation through multilayers
  • 11.7. Camera optics
  • 12. Polarization optics
  • 12.1. Jones vectors
  • 12.2. Squeeze transformation and phase shift
  • 12.3. Rotation of the polarization axes
  • 12.4. The SL(2, c) group content of polarization optics
  • 12.5. Optical activities
  • 12.6. Correspondence to space-time symmetries
  • 12.7. More optical filters from E(2)-like groups
  • 13. Poincaré sphere
  • 13.1. Decoherence in polarization optics
  • 13.2. Coherency matrix
  • 13.3. Poincaré sphere
  • 13.4. Two concentric Poincaré spheres
  • 13.5. Symmetries derivable from the Poincaré sphere
  • 13.6. O(3, 2) symmetry for energy couplings
  • 13.7. Entropy problem
  • Appendix A. Physics as art of synthesis
  • A.1. Illustration of Hume, Kant, and Hegel
  • A.2. Kant and Einstein
  • A.3. Kantianism and Taoism
  • A.4. Einstein and Hegel.

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