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.
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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. Schurs 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: Diracs Form of Relativistic Quantum Mechanics.-
1. C-Number Time-Energy Uncertainty Relation.-
2. Diracs Form of
Relativistic Theory of Atom .-
3. Diracs Light-Cone Coordinate System.-
4.
Harmonic Oscillators in the Light-Cone Coordinate System.-
5.
Lorentz-InvariantUncertainty 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.
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