Theory and Applications of the Poincaré Group

ISBN-13: 9789401085267 / Angielski / Miękka / 2011 / 331 str.

Young Suh Kim; M. Noz

Opis

Special relativity and quantum mechanics, formulated early in the twentieth century, are the two most important scientific languages and are likely to remain so for many years to come. In the 1920's, when quantum mechanics was developed, the most pressing theoretical problem was how to make it consistent with special relativity. In the 1980's, this is still the most pressing problem. The only difference is that the situation is more urgent now than before, because of the significant quantity of experimental data which need to be explained in terms of both quantum mechanics and special relativity. In unifying the concepts and algorithms of quantum mechanics and special relativity, it is important to realize that the underlying scientific language for both disciplines is that of group theory. The role of group theory in quantum mechanics is well known. The same is true for special relativity. Therefore, the most effective approach to the problem of unifying these two important theories is to develop a group theory which can accommodate both special relativity and quantum mechanics. As is well known, Eugene P. Wigner is one of the pioneers in developing group theoretical approaches to relativistic quantum mechanics. His 1939 paper on the inhomogeneous Lorentz group laid the foundation for this important research line. It is generally agreed that this paper was somewhat ahead of its time in 1939, and that contemporary physicists must continue to make real efforts to appreciate fully the content of this classic work.

Spis treści

I: Elements of Group Theory.- 1. Definition of a Group.- 2. Subgroups, Cosets, and Invariant Subgroups.- 3. Equivalence Classes, Orbits, and Little Groups.- 4. Representations and Representation Spaces.- 5. Properties of Matrices.- 6. Schur’s Lemma.- 7. Exercises and Problems.- II: Lie Groups and Lie Algebras.- 1. Basic Concepts of Lie Groups.- 2. Basic Theorems Concerning Lie Groups.- 3. Properties of Lie Algebras.- 4. Properties of Lie Groups.- 5. Further Theorems of Lie Groups.- 6. Exercises and Problems.- III: Theory of the Poincaré Group.- 1. Group of Lorentz Transformations.- 2. Orbits and Little Groups of the Proper Lorentz Group.- 3. Representations of the Poincaré Group.- 4. Lorentz Transformations of Wave Functions.- 5. Lorentz Transformations of Free Fields.- 6. Discrete Symmetry Operations.- 7. Exercises and Problems.- IV: Theory of Spinors.- 1. SL(2, c) as the Covering Group of the Lorentz Group.- 2. Subgroups of SL(2, c).- 3. SU (2).- 4. 5L(2, c) Spinors and Four-Vectors.- 5. Symmetries of the Dirac Equation.- 6. Exercises and Problems.- V: Covariant Harmonic Oscillator Formalism.- 1. Covariant Harmonic Oscillator Differential Equations.- 2. Normalizable Solutions of the Relativistic Oscillator Equation.- 3. Irreducible Unitary Representations of the Poincaré Group.- 4. Transformation Properties of Harmonic Oscillator Wave Functions.- 5. Harmonic Oscillators in the Four-Dimensional Euclidean Space.- 6. Moving O(4) Coordinate System.- 7. Exercises and Problems.- VI: Dirac’s Form of Relativistic Quantum Mechanics.- 1. C-Number Time-Energy Uncertainty Relation.- 2. Dirac’s Form of Relativistic Theory of “Atom ”.- 3. Dirac’s Light-Cone Coordinate System.- 4. Harmonic Oscillators in the Light-Cone Coordinate System.- 5. Lorentz-Invariant Uncertainty Relations.- 6. Exercises and Problems.- VII: Massless Particles.- 1. What is the E(2) Group?.- 2. E(2)-like Little Group for Photons.- 3. Transformation Properties of Photon Polarization Vectors.- 4. Unitary Transformation of Photon Polarization Vectors.- 5. Massless Particles with Spin 1/2.- 6. Harmonic Oscillator Wave Functions for Massless Composite Particles.- 7. Exercises and Problems.- VIII: Group Contractions.- 1. SE(2) Group as a Contraction of SO(3).- 2. E(2)-like Little Group as an Infinite-momentum/zero-mass Limit of the O(3)-like Little Group for Massive Particles.- 3. Large-momentum/zero-mass Limit of the Dirac Equation.- 4. Finite-dimensional Non-unitary Representations of the SE(2) Group.- 5. Polarization Vectors for Massless Particles with Integer Spin.- 6. Lorentz and Galilei Transformations.- 7. Group Contractions and Unitary Representations of SE(2).- 8. Exercises and Problems.- IX: SO(2, 1) and SU(1, 1).- 1. Geometry of SL(2, r) and Sp(2).- 2. Finite-dimensional Representations of SO(2, 1).- 3. Complex Angular Momentum.- 4. Unitary Representations of SU(1, 1).- 5. Exercises and Problems.- X: Homogeneous Lorentz Group.- 1. Statement of the Problem.- 2. Finite-dimensional Representations of the Homogeneous Lorentz Group.- 3. Transformation Properties of Electric and Magnetic Fields.- 4. Pseudo-unitary Representations for Dirac Spinors.- 5. Harmonic Oscillator Wave Functions in the Lorentz Coordinate System.- 6. Further Properties of the Homogeneous Lorentz Group.- 7. Concluding Remarks.- XI: Hadronic Mass Spectra.- 1. Quark Model.- 2. Three-particle Symmetry Classifications According to the Method of Dirac.- 3. Construction of Symmetrized Wave Functions.- 4. Symmetrized Products of Symmetrized Wave Functions.- 5. Spin Wave Functions for the Three-Quark System.- 6. Three-quark Unitary Spin and SU(6) Wave Functions.- 7. Three-body Spatial Wave Functions.- 8. Totally Symmetric Baryonic Wave Functions.- 9. Baryonic Mass Spectra.- 10. Mesons.- 11. Exercises and Problems.- XII: Lorentz-Dirac Deformation in High-Energy Physics.- 1. Lorentz-Dirac Deformation of Hadronic Wave Functions.- 2. Form Factors of Nucléons.- 3. Calculation of the Form Factors.- 4. Scaling Phenomenon and the Parton Picture.- 5. Covariant Harmonic Oscillators and the Parton Picture.- 6. Calculation of the Parton Distribution Function for the Proton.- 7. Jet Phenomenon.- 8. Exercises and Problems.- References.