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

Franck Laloë is a researcher at the Kastler-Brossel laboratory of the Ecole Normale Supérieure in Paris. His first assignment was with the University of Paris VI before he was appointed to the CNRS, the French National Research Center. His research is focused on optical pumping, statistical mechanics of quantum gases, musical acoustics and the foundations of quantum mechanics.