Introduction to Continuous Symmetries

I Symmetry Transformations
A Fundamental Symmetries
B Symmetries in Classical Mechanics C Symmetries in Quantum Mechanics
A_I Euler's and Lagrange's Views in Classical Mechanics
1 Euler's Point of View
2 Lagrange's Point of View

II Notions on Group Theory
A General Properties of Groups
B Linear Representations of a Group
A_II Residual Classes of a Subgroup; Quotient Group
1 Residual Classes on the Left
2 Quotient Group

III Introduction to Continuous Groups and Lie Groups
A General Properties B Examples
C Galileo and Poincaré Groups
A_III Adjoint Representation, Killing Form, Casimir Operator
1 Representation Adjoint to the Lie Algebra
2 Killing Form; Scalar Product and Change of Basis in L
3 Totally Antisymmetric Structure Constants
4 Casimir Operator

IV Representations Induced in the State Space
A Conditions Imposed on Transformations in the State Space
B Wigner's Theorem
C Transformations of Observables
D Linear Representations in the State Space
E Phase Factors and Projective Representations
A_IV Finite-Dimensional Unitary Projective Representations of Related Lie Groups
1 Case Where G is Simply Connected
2 Case Where G is P-Connected
B_IV Uhlhorn-Wigner Theorem
1 Real Space
2 Complex Space

V Representations of the Galileo and Poincaré Groups: Mass, Spin and Energy
A Galileo Group
B Poincaré Group
A_V Some Properties of the Operators S and W_2
1 Operator S
2 Eigenvalues of the Operator W_2
B_V Geometric Displacement Group
1 Reminders: Classical Properties of Displacements
2 Associated Operators in the State Space
C_V Clean Lorentz Group
1 Link with the Group SL(2,C)
2 Small Group Associated with a Four-Vector
3 Operator W_2
D_V Space Reflections (Parity)
1 Action in Real Space
2 Associated Operator in the State Space
3 Retention of Parity

VI Construction of State Spaces and Wave Equations
A Galileo Group, Schrödinger Equation
B Poincaré Group, Klein-Gordon and Dirac Equations
A_VI Lagrangians of Wave Equations
1 Lagrangian of a Field
2 Schrödinger's Equation
3 Klein-Gordon Equation
4 Dirac's Equation

VII Irreducible Representations of the Group of Rotations, Spinors
A Irreducible Unitary Representations of the Group of Rotations
B Spin 1/2 Particles; Spinors
C Composition of the Kinetic Moments
A_VII Homorphism Between SU(2) and Rotation Matrices
1 Transformation of a Vector P Induced by an SU(2) Matrix
2 The Transformation is a Rotation
3 Homomorphism
4 Link to the Reasoning of Chapter VII
5 Link with Bivalent Representations

VIII Transformation of Observables by Rotation
A Vector Operators B Tensor Operators
C Wigner-Eckart Theorem
D Decomposition of the Density Matrix on Tensor Operators
A_VIII Basic Reminders on Classical Tensors
1 Vectors
2 Tensors
3 Properties
4 Tensoriality Criterion
5 Symmetric and Antisymmetric Tensors
6 Special Tensors
7 Irreducible Tensors
B_VIII Second Order Tensor Operators
1 Tensor Product of Two Vector Operators
2 Cartesian Components of the Tensor in the General Case
C_VIII Multipolar Moments
1 Electrical Multipole Moments
2 Magnetic Multipole Moments
3 Multipole Moments of a Quantum System for a Given Kinetic Moment Multiplicity J

IX Groups SU(2) and SU(3)
A System of Discernible but Equivalent Particles
B SU(2) Group and Isospin Symmetry
C Symmetry SU(3)
A_IX the Nature of a Particle Is Equivalent to an Internal Quantum Number
1 Partial or Total Antisymmetrization of a State Vector
2 Correspondence Between the States of Two Physical Systems
3 Physical Consequences
B_IX Operators Changing the Symmetry of a State Vector by Permutation
1 Fermions
2 Bosons

X Symmetry Breaking
A Magnetism, Breaking of the Rotation Symmetry
B Some Other Examples

APPENDIX
I The Reversal of Time
1 Time Reversal in Classical Mechanics
2 Antilinear and Antiunitary Operators in Quantum Mechanics
3 Time Reversal and Antilinearity
4 Explicit Form of the Time Reversal Operator
5 Applications