Introduction to Continuous Symmetries: From Space-Time to Quantum Mechanics

ISBN-13: 9783527414161 / Angielski / Twarda / 2023 / 500 str.

Franck Laloe

Spis treści

I Symmetry TransformationsA Fundamental SymmetriesB Symmetries in Classical Mechanics C Symmetries in Quantum MechanicsA_I Euler's and Lagrange's Views in Classical Mechanics1 Euler's Point of View2 Lagrange's Point of ViewII Notions on Group TheoryA General Properties of GroupsB Linear Representations of a GroupA_II Residual Classes of a Subgroup; Quotient Group1 Residual Classes on the Left2 Quotient GroupIII Introduction to Continuous Groups and Lie GroupsA General Properties B ExamplesC Galileo and Poincaré GroupsA_III Adjoint Representation, Killing Form, Casimir Operator1 Representation Adjoint to the Lie Algebra2 Killing Form; Scalar Product and Change of Basis in L3 Totally Antisymmetric Structure Constants4 Casimir OperatorIV Representations Induced in the State SpaceA Conditions Imposed on Transformations in the State SpaceB Wigner's TheoremC Transformations of ObservablesD Linear Representations in the State SpaceE Phase Factors and Projective RepresentationsA_IV Finite-Dimensional Unitary Projective Representations of Related Lie Groups1 Case Where G is Simply Connected2 Case Where G is P-ConnectedB_IV Uhlhorn-Wigner Theorem1 Real Space2 Complex SpaceV Representations of the Galileo and Poincaré Groups: Mass, Spin and EnergyA Galileo GroupB Poincaré GroupA_V Some Properties of the Operators S and W_21 Operator S2 Eigenvalues of the Operator W_2B_V Geometric Displacement Group1 Reminders: Classical Properties of Displacements2 Associated Operators in the State SpaceC_V Clean Lorentz Group1 Link with the Group SL(2,C)2 Small Group Associated with a Four-Vector3 Operator W_2D_V Space Reflections (Parity)1 Action in Real Space2 Associated Operator in the State Space3 Retention of ParityVI Construction of State Spaces and Wave EquationsA Galileo Group, Schrödinger EquationB Poincaré Group, Klein-Gordon and Dirac EquationsA_VI Lagrangians of Wave Equations1 Lagrangian of a Field2 Schrödinger's Equation3 Klein-Gordon Equation4 Dirac's EquationVII Irreducible Representations of the Group of Rotations, SpinorsA Irreducible Unitary Representations of the Group of RotationsB Spin 1/2 Particles; SpinorsC Composition of the Kinetic MomentsA_VII Homorphism Between SU(2) and Rotation Matrices1 Transformation of a Vector P Induced by an SU(2) Matrix2 The Transformation is a Rotation3 Homomorphism4 Link to the Reasoning of Chapter VII5 Link with Bivalent RepresentationsVIII Transformation of Observables by RotationA Vector Operators B Tensor OperatorsC Wigner-Eckart TheoremD Decomposition of the Density Matrix on Tensor OperatorsA_VIII Basic Reminders on Classical Tensors1 Vectors2 Tensors3 Properties4 Tensoriality Criterion5 Symmetric and Antisymmetric Tensors6 Special Tensors7 Irreducible TensorsB_VIII Second Order Tensor Operators1 Tensor Product of Two Vector Operators2 Cartesian Components of the Tensor in the General CaseC_VIII Multipolar Moments1 Electrical Multipole Moments2 Magnetic Multipole Moments3 Multipole Moments of a Quantum System for a Given Kinetic Moment Multiplicity JIX 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 Number1 Partial or Total Antisymmetrization of a State Vector2 Correspondence Between the States of Two Physical Systems3 Physical ConsequencesB_IX Operators Changing the Symmetry of a State Vector by Permutation1 Fermions2 BosonsX Symmetry BreakingA Magnetism, Breaking of the Rotation SymmetryB Some Other ExamplesAPPENDIXI The Reversal of Time1 Time Reversal in Classical Mechanics2 Antilinear and Antiunitary Operators in Quantum Mechanics3 Time Reversal and Antilinearity4 Explicit Form of the Time Reversal Operator5 Applications

O autorze

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.