Introduction to Continuous Symmetries

Franck Laloë is a researcher at the Kastler-Brossel Laboratory of the Ecole Normale Supérieure in Paris, France. His research is focused on optical pumping, the statistical mechanics of quantum gases, musical acoustics, and the foundations of quantum mechanics.

• I Symmetry transformations 1A Basic symmetries 1B Symmetries in classical mechanics 5C Symmetries in quantum mechanics 26AI Eulerian and Lagrangian points of view in classical mechanics 311 Eulerian point of view 322 Lagrangian point of view 34BI Noether’s theorem for a classical field 381 Lagrangian density and Lagrange equations for continuous variables 382 Symmetry transformations and current conservation 403 Generalization, relativistic notation 414 Local conservation of energy 42II Some ideas about group theory 45A General properties of groups 46B Linear representations of a group 56AII Left coset of a subgroup; quotient group 651 Left cosets 652 Quotient group 66III Introduction to continuous groups and Lie groups 69A General properties 70B Examples 85C Galilean and Poincaré groups 98AIII Adjoint representation, Killing form, Casimir operator 1091 Adjoint representation of a Lie algebra 1092 Killing form ; scalaire product and change of basis in L 1113 Completely antisymmetric structure constants 1134 Casimir operator 114IV Induced representations in the state space 117A Conditions imposed on the transformations in the state space 119B Wigner’s theorem 121C Transformations of observables 126D Linear representations in the state space 128E Phase factors and projective representations 133AIV Unitary projective representations, with finite dimension, of connected Lie groups. Bargmann’s theorem 1411 Case where G is simply connected 1422 Case where G is p-connected 145BIV Uhlhorn-Wigner theorem 1491 Real space 1492 Complex space 153V Representations of Galilean and Poincaré groups: mass, spin, and energy 157A Representations in the state space 158B Galilean group 159C Poincaré group 173AV Proper Lorentz group and SL(2C) group 1911 Link to the SL(2, C) group 1912 Little group associated with a four-vector 1983 W2 operator 202BV Commutation relations of spin components, Pauli–Lubanski four-vector 2051 Operator S 2052 Pauli–Lubanski pseudovector 2073 Energy-momentum eigensubspace with any eigenvalues 210CV Group of geometric displacements 2131 Brief review: classical properties of displacements 2142 Associated operators in the state space 223DV Space reflection (parity) 2331 Action in real space 2332 Associated operator in the state space 2353 Parity conservation 237VI Construction of state spaces and wave equations 241A Galilean group, the Schrödinger equation 242B Poincaré group, Klein–Gordon, Dirac, and Weyl equations 254AVI Relativistic invariance of Dirac equation and non-relativistic limit 2731 Relativistic invariance 2732 Non-relativistic limit of the Dirac equation 276BVI Finite Poincaré transformations and Dirac state space 2811 Displacement group 2812 Lorentz transformations 2833 State space and Dirac operators 287CVI Lagrangians and conservation laws for wave equations 2931 Complex fields 2932 Schrödinger equation 2953 Klein–Gordon equation 2974 Dirac equation 300VII Rotation group, angular momenta, spinors 303A General properties of rotation operators 304B Spin 1/2 particule; spinors 323C Addition of angular momenta 329AVII Rotation of a spin 1/2 and SU(2) matrices 3391 Modification of a spin 1/2 polarization induced by an SU(2) matrix 3402 The transformation is a rotation 3413 Homomorphism 3424 Link with the chapter VII discussion 3445 Link with double-valued representations 346BVII Addition of more than two angular momenta 3471 Zero total angular momentum; 3-j coefficients 3472 6-j Wigner coefficients 351VIII Transformation of observables under rotation 355A Scalar and vector operators 358B Tensor operators 363C Wigner–Eckart theorem 379D Applications and examples 384AVIII Short review of classical tensors 3971 Vectors 3972 Tensors 3983 Properties 4014 Criterium for a tensor 4035 Symmetric and antisymmetric tensors 4036 Specific tensors 4047 Irreducible tensors 405BVIII Second-order tensor operators 4091 Tensor product of two vector operators 4092 Cartesian components of the tensor in the general case 411CVIII Multipole moments 4151 Electric multipole moments 4162 Magnetic multipole moments 4283 Multipole moments of a quantum system with a given angular momentum J 434DVIII Density matrix expansion on tensor operators 4391 Liouville space 4392 Rotation transformation 4413 Basis of the T[K]Q operators 4424 Rotational invariance in a system’s evolution 444IX Internal symmetries, SU(2) and SU(3) groups 449A System of distinguishable but equivalent particles 451B SU(2) group and isospin symmetry 466C SU(3) symmetry 472AIX The nature of a particle is equivalent to an internal quantum number 4971 Partial or complete symmetrization, or antisymmetrization, of a state vector 4972 Correspondence between the states of two physical systems 4993 Physical consequences 501BIX Operators changing the symmetry of a state vector by permu-tation 5031 Fermions 5032 Bosons 506X Symmetry breaking 507A Magnetism, breaking of rotational symmetry 508B A few other examples 515Appendix 521Time reversal 5211 Time reversal in classical mechanics 5222 Antilinear and antiunitary operators in quantum mechanics 5273 Time reversal and antilinearity 5344 Explicit form of the time reversal operator 5425 Applications 546