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The Theory of Groups and Quantum Mechanics


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AUTHOR'S PREFACES TRANSLATOR'S PREFACE INTRODUCTION I. UNITARY GEOMETRY I. The n-dimensional Vector Space 2. Linear Correspondences. Matrix Calculus 3. The Dual Vector Space 4. Unitary Geometry and Hermitian Forms 5. Transformation to Principal Axes 6. Infinitesimal Unitary Transformations 7. Remarks on 8-dimensional Space II. QUANTUM THEORY 1. Physical Foundations 2. The de Broglie Waves of a Particle 3. Schrodinger's Wave Equation. The Harmonic Oscillator 4. Spherical Harmonics 5. Electron in Spherically Symmetric Field. Discretional Quantization 6. Collision Phenomena 7. The Conceptual Structure of Quantum Mechanics 8. The Dynamical Law. Transition Probabilities 9. Peturbation Theory 10. The Problem of Several Bodies. Product Space 11. Commutation Rules. Canonical Transformations 12. Motion of a Particle in an Electro-magnetic Field. Zeeman Effect and Stark Effect 13. Atom in Interatction with Radiation III. GROUPS AND THEIR REPRESENTATIONS 1. Transformation Groups 2. Abstract Groups and their Realization 3. Sub-groups an Conjugate Classes 4. Representation of Groups by Linear Transformations 5. Formal Processes. Clebsch-Gordan Series 6. The Jordan-Holder Theorem and its Analogues 7. Unitary Representations 8. Rotation and Lorentz Groups 9. Character of a Representation 10. Schur's Lemma and Burnside's Theorem 11. Orthogonality Properties of Group Characters 12. Extension to Closed Continuous Groups 13. The Algebra of a Group 14. Invariants and Covariants 15. Remarks on Lie's Theory of Continuous Groups of Transformations 16. Representation by Rotations of Ray Space IV. APPLICATION OF THE THEORY OF GROUPS TO QUANTUM MECHANICS A. The Rotation Group 1. The Representation Induced in System Space by the Rotation Group 2. Simple States and Term Analysis. Examples 3. Selection and Intensity Rules 4. "The Spinning Electron, Multiplet Structure and Anomalous Zeeman Effect" B. The Lorentz Group 5. Relativistically Invariant Equations of Motion of an Electron 6. Energy and Momentum. Remarks on the Interchange of Past and Future 7. Electron in Spherically Symmetric Field 8. Selection Rules. Fine Structure C. The Permutation Group 9. Resonance between Equivalent Individuals 10. The Pauli Exclusion Principle and the Structure of the Periodic Table 11. The Problem of Several Bodies and the Quantization of the Wave Equation 12. Quantization of the Maxwell-Dirac Field Equations 13. The Energy and Momentum Laws of Quantum Physics. Relativistic Invariance D. Quantum Kinematics 14. Quantum Kinematics as an Abelian Group of Rotations 15. Derivation of the Wave Equation from the Commutation Rules V. THE SYMMETRIC PERMUTATION GROUP AND THE ALGEBRA OF SYMMETRIC TRANSFORMATIONS A. General Theory 1. The Group induced in Tensor Space and the Algebra of Symmetric Transformations 2. Symmetry Classes of Tensors 3. Invariant Sub-spaces in Group Space 4. Invariant Sub-spaces in Tensor Space 5. Fields and Algebras 6. Representations of Algebras 7. Constructive Reduction of an Algebra into Simple Matric Algebras B. Extension of the Theory and Physical Applications 8. The Characters of the Symmetric Group and Equivalence Degeneracy in Quantum Mechanics 9. Relation between the Characters of the Symmetric Permutation and Affine Groups 10. Direct Product. Subgroups 11. Perturbation Theory for the Construction of Molecules 12. The Symmetry Problem of Quantum Theory C. Explicit Algebraic Construction 13. Young's Symmetry Operators 14. "Irreducibility, Linear Independence, Inequivalence and Completeness" 15. Spin and Valence. Group-theoretic Classification of Atomic Spectra 16. Determination of the Primitive Characters of u and p 17. Calculation of Volume on u 18. Branching Laws APPENDIX I. PROOF OF AN INEQUALITY 2. A COMPOSITION PROPERTY OF GROUP CHARACTERS 3. A THEOREM CONCERNING NON-DEGENERATE ANTI-SYMMETRIC BI-LINEAR FORMS BIBLIOGRAPHY LIST OF OPERATIONAL SYMBOLS LIST OF LETTERS HAVING A FIXED SIGNIFICANCE INDEX

About the Author

Along with his fundamental contributions to most branches of mathematics, Hermann Weyl (1885-1955) took a serious interest in theoretical physics. In addition to teaching in Zurich, Gottingen, and Princeton, Weyl worked with Einstein on relativity theory at the Institute for Advanced Studies. Hermann Weyl: The Search for Beautiful Truths One of the most influential mathematicians of the twentieth century, Hermann Weyl (1885-1955) was associated with three major institutions during his working years: the ETH Zurich (Swiss Federal Institute of Technology), the University of Gottingen, and the Institute for Advanced Study in Princeton. In the last decade of Weyl's life (he died in Princeton in 1955), Dover reprinted two of his major works, The Theory of Groups and Quantum Mechanics and Space, Time, Matter. Two others, The Continuum and The Concept of a Riemann Surface were added to the Dover list in recent years. In the Author's Own Words: "My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful." "We are not very pleased when we are forced to accept mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim and of the road; we want to understand the idea of the proof, the deeper context." "A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details." - Hermann Weyl Critical Acclaim for Space, Time, Matter: "A classic of physics ... the first systematic presentation of Einstein's theory of relativity." - British Journal for Philosophy and Science

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