Thyssen P. Shattered symmetry: group theory from the eightfold way to the periodic table / P.Thyssen, A.Ceulemans. - Oxford: Oxford university press, 2017. - xxiii, 498 p.: ill., tab. - Bibliogr. in the noten. - Ind.: p.491-498. - ISBN 978-0-19-061139-2 Шифр: (И/В31-Т51) 02  Место хранения:   02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents

List of Figures ................................................ XV List of Tables ................................................. XV Preface ....................................................... xxi Part One Space Symmetries 1 primer on symmetry ......................................... 3 1.1. L The tragic life of Evariste Galois ....................... 3 1.1.1. Entrance exams ...................................... 5 1.1.2. Publish or perish ................................... 5 1.1.3. Galois' mathematical testament 1.2. The concept of symmetry .................................... 7 1.2.1. Symmetry defined .................................... 8 1.2.2. The symmetries of a triangle ....................... 11 1.2.3. Quantifying symmetry ............................... 12 1.2.4. Discrete and continuous symmetries ................. 13 1.2.5. Multiplying symmetries ............................. 13 2. The elements of group theory .............................. 15 2.1. Mathematical definition ................................... 16 2.2. The abstract and the concrete ............................. 18 2.3. Abelian groups ............................................ 20 2.4. Examples of groups ........................................ 21 2.5. Subgroups ................................................. 24 2.6. Symmetry breaking ......................................... 24 2.7. Isomorphisms and homomorphisms ............................ 24 2.8. Historical interlude ...................................... 27 2.8.1. Evariste Galois .................................... 28 2.8.2. The French school .................................. 29 2.8.3. Sir Arthur Cayley .................................. 29 3. The axial rotation group .................................. 31 3.1. Active versus passive view of symmetry .................... 32 3.2. Rotation operators ........................................ 34 3.3. The axial rotation group .................................. 34 3.4. Transformations of coordinates ............................ 35 3.5. Transformations of coordinate functions ................... 37 3.6. Matrix representations .................................... 42 3.6.1. Matrix representation of coordinate operators R .... 42 3.6.2. Matrix representation of function operators R ...... 43 3.7. The orthogonal group O(2) ................................. 44 3.7.1. Symmetry and invariance ............................ 44 3.7.2. Proper and improper rotation matrices .............. 46 3.7.3. Orthogonal groups: O(2) and SO(2) .................. 47 4. The SO(2) group ........................................... 50 4.1. Infinite continuous groups ................................ 53 4.1.1. The nature of infinite continuous groups ........... 53 4.1.2. Parameters of continuous groups .................... 54 4.1.3. Examples of continuous groups ...................... 54 4.1.4. The composition functions .......................... 57 4.2. Lie groups ................................................ 58 4.2.1. Definition ......................................... 59 4.2.2. Parameter space .................................... 59 4.2.3. Connectedness and compactness ...................... 60 4.3. The infinitesimal generator ............................... 61 4.3.1. Matrix form of the S0(2) generator ................. 62 4.3.2. Operator form of the S0(2) generator ............... 64 4.4. Angular momentum .......................................... 65 4.4.1. Classical mechanical picture ....................... 65 4.4.2. Quantum mechanical picture ......................... 66 4.5. SO(2) symmetry and aromatic molecules ..................... 67 4.5.1. The particle on a ring model ....................... 67 4.5.2. The shell perspective .............................. 71 4.5.3. Aromatic molecules ................................. 71 5. The SO(3) group ........................................... 75 5.1. The spherical rotation group .............................. 78 5.2. The orthogonal group in three dimensions .................. 80 5.2.1. Rotation matrices .................................. 80 5.2.2. The orthogonal group O(3) .......................... 82 5.2.3. The special orthogonal group SO(3) ................. 82 5.3. Rotations and SO(3) ....................................... 83 5.3.1. Orthogonality and skew-symmetry .................... 83 5.3.2. The matrix representing an infinitesimal rotation .. 84 5.3.3. The exponential map ................................ 86 5.3.4. The Euler parameterization ......................... 87 5.4. The (3) Lie algebra .................................... 88 5.4.1. The (3) generators .............................. 88 5.4.2. Operator form of the SO(3) generators .............. 90 5.5. Rotations in quantum mechanics ............................ 91 5.5.1. Angular momentum as the generator of rotations ..... 91 5.5.2. The rotation operator .............................. 92 5.6. Angular momentum .......................................... 92 5,6.1. The angular momentum algebra ....................... 92 5.6.2. Casimir invariants ................................. 92 5.6.3. The eigenvalue problem ............................. 93 5.6.4, Dirac's ladder operator method ..................... 94 5.7. Application: Particle on a sphere ........................ 100 5.7.1. Spherical components of the Hamiltonian 5.7.2. The flooded planet model and BuckminsterfuUerene .. 103 5.8. Epilogue ................................................. 107 6. Scholium I ............................................... 108 6.1. Symmetry in quantum mechanics ............................ 111 6.1.1. State vector transformations ...................... 111 6.1.2. Unitarity of symmetry operators ................... 112 6.1.3. Transformation of observables ..................... 113 6.1.4. Symmetry transformations of the Hamiltonian ....... 114 6.1.5. Symmetry group of the Hamiltonian ................. 114 6.1,6, Symmetry and degeneracy ........................... 115 6.2. Lie groups and Lie algebras .............................. 116 6.2.1. Lie generators .................................... 117 6.2.2. Lie algebras ...................................... 117 6.2.3. Hermiticity and Lie generators .................... 120 6.3. Symmetry and conservation laws ........................... 121 6.3.1. Noether's theorem ................................. 121 6.3.2. Conserved quantities in quantum mechanics ......... 122 6.4. The Cartan-Weyl method ................................... 123 6.4.1. The threefold path toward a Cartan-Weyl basis ..... 124 6.4.2. Review of the angular momentum algebra ............ 127 6.5. The three pillars of group theory ........................ 130 Part Two Dynamic Symmetries 7. The SU(3) group .......................................... 135 7.1. Historical prelude: The particle zoo ..................... 135 7.1.1. Probing the inner structure of the atom (1897- 1947) ............................................. 135 7.1.2. Yukawa's pion (1947-1949) ......................... 136 7.1.3. The growing particle jungle (1950-1960) ........... 138 7.1.4, Quantum numbers and conservation laws ............. 139 7.1.5, Strangeness (1953) ................................ 141 7.1.6. The Mendeleev of elementary particle physics ...... 141 7.2. The one-dimensional harmonic oscillator .................. 143 7.2.1. Classical mechanical treatment .................... 143 7.2.2. Quantum mechanical treatment ...................... 146 7.3. The three-dimensional harmonic oscillator ................ 153 7.4. The unitary groups U(3) and SU(3) ........................ 158 7.4.1. The unitary group U(3) ............................ 158 7.4.2. Subgroups of the unitary group U(3) ............... 159 7.4.3. The generators of U(3) ............................ 160 7.4.4. The generators of SU(3) ........................... 163 7.4.5. The (3) Lie algebra ............................ 166 7.4.6. The Cartan subalgebra of (3) ................... 169 7.4.7. The , , and subalgebras of (3) ........... 174 7.5. The eightfold way (1961) ................................. 175 7.5.1. An octet of particles ............................. 177 7.5.2. Different SU(3) representations ................... 179 7.5.3. Broken symmetries ................................. 181 7.5.4. In Mendeleev's footsteps .......................... 182 7.6. Three quarks for Muster Mark (1964) ...................... 184 7.6.1. Aces and Quarks ................................... 185 7.6.2. Mesons ............................................ 190 7.6.3. Baryons ........................................... 191 7.6.4. Symmetry from the quark perspective ............... 195 8. SU(2) and electron spin .................................. 200 8.1. From SU(3) to SU(2) ...................................... 200 8.2. Spinors and half-integer angular momentum ................ 201 8.3. The relationship between SU(2) and SO(3) ................. 203 8.4. The spinning electron .................................... 207 9. The SO(4) group .......................................... 208 9.1. The hydrogen atom ........................................ 209 9.1.1. A quartet of quantum numbers ...................... 209 9.1.2. "Accidental" degeneracies and the Fock (n) rule ... 210 9.3. Classical Kepler problem ................................. 212 9.3.1. General formulation ............................... 214 9.3.2. Hamiltonian formulation ........................... 216 9.3.3. Constants of the motion ........................... 217 9.3.4. The Laplace-Runge-Lenz vector ..................... 218 9.3.5. Equations of motion ............................... 223 9.3.6. History of the Kepler problem ..................... 226 9.3.7. History of the LRL vector (I) ..................... 228 9.4. Quantum mechanics of the hydrogen atom ................... 229 9.4.1. Operators for conserved quantities ................ 230 9.4.2. Conservation laws ................................. 231 9.5. The special orthogonal group SO(4) ....................... 232 9.5.1. The generators of SO(4) ........................... 232 9.5.2. The (4) Lie algebra ............................ 234 9.5.3. The Cartan subalgebra of so(4) .................... 239 9.6. The origin of accidental degeneracies .................... 244 9.6.1. Energy levels of the hydrogen atom ................ 244 9.6.2. History of the LRL vector (II) .................... 247 9.6.3. Hodographs in momentum space ...................... 249 9.6.4. Stereographic projections in hyperspace ........... 252 9.6.5. Peeking at the fourth dimension ................... 257 10. Scholium II .............................................. 260 10.1. Oscillator roots of the hydrogen problem ................ 261 10.1.1. Dimensional considerations ....................... 261 10.1.2. The SU(4) oscillator ............................. 262 10.1.3. The Kustaanheimo-Stiefel transformation .......... 263 10.2. The transformation of the hydrogen Hamьtonian ........... 266 10.3. Shattering SU(4) symmetry ............................... 268 Part Three Spectrum-Generating Symmetries 11. The SO(2,l) group ....................................... 273 11.1. The road to noninvariance groups ........................ 275 11.1.1. Historical prelude .............................. 275 11.1.2. Invariance groups ............................... 277 11.1.3. Noninvariance groups ............................ 278 11.2. The pseudo-orthogonal group SO(2,l) ..................... 279 11.2.1. From SO(3) to SO(2,l) ........................... 279 11.2.2. Theso(2,l) Lie algebra .......................... 281 11.2.3. The Cartan-Weyl basis of (2,1) ............... 281 11.2.4. S0(3) revisited ................................. 286 11.3. Hydrogenic realization of the SO(2,l) group ............. 287 11.3.1. The radial Schrцdinger equation ................. 288 11.3.2. The spectrum-generating SO(2,1) algebra ......... 289 11.5. Dirac's harmonic oscillator revisited ................... 293 12. The SO(4,2) group ....................................... 297 12.1. The pseudo-orthogonal group SO(4,2) ..................... 299 12.1.1. From SO(6) to SO(4,2) ........................... 299 12.1.2. Hydrogenic realization of the SO(4,2) Lie algebra ......................................... 300 12.2. The Cartan-Weyl basis ................................... 304 12.2.1. Cartan subalgebra and Cartan generators ......... 304 12,2.2. Casimir invariants .............................. 306 12.2.3. Weyl generators ................................. 307 12.2.4. SO(4,2) root diagram ............................ 310 12.2.5. SO(4,2) weight diagrams ......................... 311 12.3. Quantum alchemy ......................................... 313 12.3.1. Raising and lowering m .......................... 313 12.3,2, Raising and lowering n .......................... 315 12.3.3. Raising and lowering l .......................... 316 13. The periodic table ...................................... 323 13.1. Chemical periodicity .................................... 324 13.2. Quantum mechanics of atomic systems ..................... 327 13.2.1. One-electron systems ............................ 327 13.2.2. Many-electron systems ........................... 328 13.3. Quantum mechanics of the periodic system ................ 329 13.3.1. The Fock (n) rule ............................... 330 13.3.2. The hydrogenic (n, 1) rule ...................... 331 13.3.3. The Madelung (n+l, n) rule ...................... 334 13.4. The Madelung (n+1, n) rule .............................. 335 13.4.1. The left-step periodic table .................... 336 13.4.2. Period doubling ................................. 340 13.4.3. Ab initio derivation of the (n+1, n) rule ....... 342 13.5. Misapplying the Madelung rule ........................... 343 13.5.1. The n+1 blunder ................................. 343 13.5.2. The concept of an element ....................... 343 13.5.3. Two interpretations of the (n+1, n) rule ........ 346 13.6. Group theory and the periodic system .................... 352 13.6.1. APA versus EPA .................................. 352 13.6.2. Group theoretical classification of the elements ........................................ 353 13.6.3. System and system states ........................ 358 13.6.4. Symmetry and symmetry breaking .................. 359 13.6.5. Mass formulas ................................... 360 13.7. Literature study ........................................ 360 13.7.1. Historical prelude .............................. 360 13.7.2. Demkov and Ostrovsky (St. Petersburg, Russia) ... 372 13.8. Conclusion .............................................. 380 14. SO(4,2) and the rules of atomic chess ................... 382 14.1. From the eightfold way to the periodic table ............ 382 14.1.1. The atomic chessboard ........................... 383 14.1.2. Correlation diagram ............................. 384 14.2. The rules of atomic chess ............................... 386 14.2.1. The king and queen .............................. 386 14.2.2. The rook ........................................ 387 14.2.3. The knight ...................................... 389 14.2.4. The bishop ...................................... 392 14.3. The origin of period doubling ........................... 395 14.4. The quest for the chiral bishop ......................... 396 14.4.1. The Madelung sequences .......................... 396 14.4.2. The Regge sequences ............................. 397 14.5. The algebra of chiral bishops ........................... 398 14.5.1. Standard embedding of SO(3,1) in SO(3,2) ........ 399 14.5.2. The S operator .................................. 401 14.5.3. Construction of Regge and Madelung operators .... 403 14.5.4. Regge-like algebras ............................. 405 14.5.5. Madelung-like algebras .......................... 406 14.5.6. Casimir operators ............................... 408 14.6. Nonlinear Lie algebras .................................. 411 14.7. Conclusion .............................................. 412 Epilogue ...................................................... 415 A. Vector algebra ........................................... 417 A.l. Conceptual definition .................................... 417 A.1.1. Notation .......................................... 417 A.2. Representation ........................................... 417 A.3. Vector operations ........................................ 418 A.4. The inner product ........................................ 419 A.5. The outer product ........................................ 420 A.6. Higher dimensional vectors ............................... 421 B. Matrix algebra ........................................... 423 B.l. Conceptual definition .................................... 423 Б.2. Special matrices ......................................... 423 B.3. Matrix operations ........................................ 424 B.3.1. Equality .......................................... 424 B.3.2. Matrix transposition .............................. 424 B.3.3. Matrix addition and subtraction ................... 425 B.3.4. Scalar multiplication ............................. 425 B.4. Matrix products .......................................... 425 B.4.1. Definition ........................................ 425 B.4.2. Properties of matrix products ..................... 426 B.4.3. The inner product in matrix notation .............. 426 B.4.4. The transpose of a matrix product ................. 427 B.5. Trace and determinant of a square matrix ................. 428 B.5.1. The trace of a square matrix ...................... 428 B.5.2. The determinant of a square matrix ................ 428 B.6. Complex matrices ......................................... 430 B.7. Orthogonal matrices ...................................... 431 B.8. Unitary matrices ......................................... 431 C. Taylor and Maclaurin series .............................. 433 D. Quantum mechanics in a nutshell .......................... 436 D.I. Wave mechanics: the intuitive approach ................... 436 D.2. Quantum mechanics: the formal structure .................. 439 D.2.1. Operators and eigenfunctions ...................... 439 D.2.2. The bra-ket formalism ............................. 440 D.2.3. Hermitian and unitary operators ................... 441 D.2.4. The spectral decomposition postulate .............. 443 D.3. Operator commutations and uncertainty relations .......... 444 D.4. Time dependence .......................................... 445 E. Determinant of an orthogonal niatrix ..................... 447 F. Lie bracket .............................................. 449 G. Laplacian in radial, and angular momentum ................ 453 H. Quantum states of the SU(2) oscillator ................... 455 I. Commutation relations .................................... 458 I.l. Canonical commutation relations .......................... 458 I.1.1. Commutation relations for [i, j] ................. 458 I.1.2. Commutation relations for [i,j] and [i,j] ...... 459 I.2. Angular momentum commutation relations ................... 460 I.2.1. Commutation relations for i,j ................... 460 I.2.2. Commutation relations for i,j ................... 460 I.2.3. Commutation relations for i,j ................... 461 I.3. Auxiliary commutators .................................... 462 I.3.1. Commutation relations for [i,j/r] ............... 462 I.3.2. Commutation relations for i,j/r ................. 463 I.4. Reformulation of the LRL vector .......................... 465 I.5. Commutators with the LRL operator ........................ 466 I.5.1. Commutation relations for i,j .................. 466 I.5.2. Commutation relations for i,j .................. 468 I.6. More auxiliary commutators ............................... 471 I.6.1. Commutation relations for i, ................... 471 I.6.2. Commutation relations for i2,j/r ................. 472 I.7. Commutators with the Hamiltonian ......................... 473 I.7.1. Commutation relations for i, ................... 473 I.7.2. Commutation relations for i, .................. 474 I.8. Angular momentum commutation relations ................... 475 J. Identities of the Laplace-Runge-Lenz vector .............. 477 J.1. The scalar product • ................................ 477 J.2. The scalar product 2 ................................... 479 К. Indicial notation ........................................ 482 K.1. The Kronecker delta δij .................................. 482 K.2. The Levi-Civita tensor εijk ............................... 482 K.2.1. Contracted epsilon identities ..................... 482 K.2.2. Indicial notation for cross products .............. 484 L. S-transform of the so(4,2) algebra ....................... 485 Index ......................................................... 491