Symmetries in Quantum Mechanics: From Angular Momentum to Supersymmetry (Graduate Student Series in Physics)

SYMMETRIES IN QUANTUM MECHANICS: FROM ANGULAR MOMENTUM TO SUPERSYMMETRY......Page 1
Title Page......Page 2
Contents......Page 4
Preface......Page 10
1.1 Notation......Page 12
1.2 Some basic concepts in quantum mechanics......Page 15
1.3.1 Groups: finite, infinite, continuous, Abelian, non-Abelian; subgroup of a group, cosets......Page 18
1.3.2 Isomorphism, automorphism, homomorphism......Page 19
1.3.3 Lie groups and Lie algebras......Page 20
1.3.4 Representations: faithful, irreducible, reducible, completely reducible (decomposable), indecomposable, adjoint, fundamental......Page 21
1.3.5 Relation between Lie algebras and Lie groups, Casimir operators, rank of a group......Page 22
1.3.7 Semidirect sum of Lie algebras and semidirect product of Lie groups (inhomogeneous Lie algebras and groups)......Page 23
1.3.8 The Haar measure......Page 24
1.4 Remark about the introduction of angular momentum......Page 25
2.1.1 General considerations......Page 27
2.1.2 Formal definition of symmetry; ray correspondence......Page 31
2.2 Wigner's theorem: the existence of unitary or anti-unitary representations......Page 32
2.3.1 General considerations......Page 39
2.3.2 Continuous matrix groups; decomposition into pieces......Page 40
2.3.3 The Lie algebra (Lie ring, infinitesimal ring)......Page 41
2.3.4 Canonical coordinates......Page 43
2.3.5 The structure of the group and its infinitesimal ring......Page 47
2.3.6 Summary: continuous matrix groups and their Lie algebra......Page 49
2.3.7 Group representations......Page 50
2.4.1 Continuous groups connected to the identity; Noether's theorem......Page 53
2.4.3 Super-selection rules......Page 56
2.4.4 Complete symmetry group, complete sets of commuting observables, complete sets of states......Page 60
2.4.5 Summary of the chapter......Page 63
3.1.1 Interpretation......Page 64
3.1.2 Parameters describing a rotation......Page 65
3.1.3 Representation of a rotation......Page 66
3.2 Sequences of rotations......Page 67
3.2.1 Considering the 'abstract' rotations R......Page 68
3.2.2 Considering the 3 × 3 rotation matrices M(R)......Page 70
3.3.1 The rotation matrix M[sub(p)](η)......Page 74
3.3.2 The generators of the rotation group......Page 76
3.3.3 The local group......Page 78
3.3.4 Canonical parameters of the first and the second kind......Page 79
3.4 The unitary representation U(R) induced by the three-dimensional rotation R......Page 80
4.1.1 The physical significance of J......Page 83
4.2 Commutation relations for angular momenta......Page 86
4.3 Direct sum and direct product......Page 91
4.4 Angular momenta of interacting systems......Page 96
4.5 Irreducible representations; Schur's lemma......Page 98
4.6 Eigenstates of angular momentum......Page 102
4.7 Orbital angular momentum......Page 110
4.7.1 Angular momentum operators in polar coordinates......Page 111
4.7.2 Construction of the eigenfunctions......Page 113
4.7.3 Orbital angular momenta have only integer eigenvalues......Page 115
4.7.4 Spherical harmonics......Page 116
4.7.7 Particular cases......Page 119
4.8 Spin-1/2 eigenstates and operators......Page 122
4.9 Double-valued representations; the covering group SU(2)......Page 125
4.10 Construction of the general j, m-state from spin-1/2 states......Page 127
5.1 The general problem......Page 132
5.2 Complete sets of mutually commuting (angular momentum) observables......Page 133
5.3.1 Notation......Page 139
5.3.2 Definition and some properties of the Clebsch–Gordan coefficients......Page 142
5.3.3 Orthogonality of the Clebsch–Gordan coefficients......Page 144
5.3.4 Sketch of the calculation of the Clebsch–Gordan coefficients; phase convention and reality......Page 145
5.3.5 Calculation of ......Page 149
5.3.6 Obvious symmetry relations for CGCs......Page 151
5.3.7 Wigner's 3j-symbol and Racah's V(j[sub(1)]j[sub(2)]j[sub(3)]|m[sub(1)]m[sub(2)]m[sub(3)])-symbol......Page 158
5.3.8 Racah's formula for the CGCs......Page 160
5.3.9 Regge's symmetry of CGCs......Page 165
5.3.10 Collection of formulae for the CGCs; a table of special values......Page 167
5.4.1 General remarks; statement of the problem......Page 170
5.4.2 The 6j-symbol and the Racah coefficients......Page 173
5.4.3 Collection of formulae for recoupling coefficients......Page 175
5.6 Numerical tables and important references on addition of angular momenta......Page 179
6.1 Active and passive interpretation; definition of D[sup((j))][sub(m'm)]; the invariant  subspaces H[sub(j)]......Page 181
6.2.1 The spin-1/2 case......Page 184
6.2.2 The general case......Page 186
6.3.1 Relation to the Clebsch–Gordan coefficients......Page 188
6.3.2 Significance of the relation to the CGCs......Page 190
6.3.3.1 Transformation of eigenfunctions under rotations......Page 194
6.3.3.2 The D[sup((l))][sub(m0)] are spherical harmonics......Page 195
6.3.3.3 The addition theorem and the composition rule of spherical harmonics; integral over three spherical harmonics......Page 197
6.3.4 Orthogonality relations and integrals over D-matrices......Page 199
6.3.5 A projection formula......Page 201
6.3.6 Completeness relation for the D matrices......Page 202
6.3.7 Symmetry properties of the D matrices......Page 205
7.1 Bosonic operators......Page 207
7.2 Realization of su(2) Lie algebra and the rotation matrix in terms of bosonic operators......Page 210
7.3 A short note about the new field of quantum groups......Page 215
8.1 Introduction......Page 218
8.2 Definition and properties......Page 220
8.3 Tensor product; irreducible combination of irreducible tensors; scalar product......Page 222
8.4 Invariants and covariant equations......Page 224
8.5 Spinor and vector spherical harmonics......Page 226
8.6 Angular momenta as spherical tensor operators......Page 230
8.7 The Wigner–Eckart theorem......Page 231
8.8.1 The trace of T(kq)......Page 233
8.9 Projection theorem for irreducible tensor operators of rank 1......Page 234
9.1 Introduction......Page 238
9.2 Properties of rotations in two-dimensional space and fractional statistics......Page 239
9.3 Particle–flux system: example of anyon......Page 243
9.4 Possible role of anyons in physics......Page 248
10.1 Introduction......Page 250
10.2 The generators of the inhomogeneous Lorentz group (Poincaré group)......Page 251
10.2.1 Translations; four-momentum......Page 252
10.2.2.1 Introducing a new notation adapted to space-time......Page 253
10.2.2.2 Extension from space to space-time......Page 255
10.2.2.3 Physical significance of the new generators......Page 257
10.3.1 Commutation relations of the J[sup(μν)] with each other......Page 259
10.4 A complete set of commuting observables......Page 260
10.4.1 The spin four-vector w[sup(μ)] and the spin tensor S[sup(μν)]......Page 261
10.4.2 Commutation relations for w[sup(μ)] and S[sup(μν)]......Page 263
10.4.3 Construction of a complete set of commuting observables; helicity......Page 266
10.4.4 Zero-mass particles......Page 271
10.5.1 Construction of one-particle helicity states of arbitrary p......Page 274
10.5.3 Eigenstates of the total angular momentum......Page 276
10.5.4 The S-matrix; cross-sections......Page 278
10.5.5 Evaluation of cross-section formulae......Page 280
10.5.6.1 Parity......Page 281
10.5.6.2 Time reversal......Page 283
10.5.6.3 Identical particles......Page 284
11.1 What is supersymmetry?......Page 286
11.2 SUSY quantum mechanics......Page 290
11.3 Factorization and the hierarchy of Harniltonians......Page 292
11.4 Broken supersymmetry......Page 296
Appendix A. Remarks on symmetric and self-adjoint operators......Page 298
Appendix B. The distinction between finite and infinite numbers of degrees of freedom in quantum mechanics......Page 302
Bibliography......Page 304
Index......Page 308
Back Cover......Page 316