The Theory of Groups and Quantum Mechanics

This landmark among mathematics texts applies group theory to quantum mechanics, first covering unitary geometry, quantum theory, groups and their representations, then applications themselves-rotation, Lorentz, permutation groups, symmetric permutation groups, and the algebra of symmetric transformations. Unabridged republication of the English (1931) edition.

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 Zürich, Göttingen, 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

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. Schrödinger'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-Hölder 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