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Classical and quantum mechanics via Lie algebras Arnold Neumaier Dennis Westra University of Vienna, Austria April 14, 2011 arXiv:0810.1019v2 [quant-ph] 14 Apr 2011 This is the draft of a book. The manuscript has not yet full book quality. Please refer to the book once it is published. Until then, we’d appreciate suggestions for improvements; please send them to [email protected] copyright c by Arnold Neumaier and Dennis Westra Contents Preface ix I An invitation to quantum mechanics 1 1 Motivation 3 1.1 Classicalmechanics................................ 3 1.2 Relativitytheory ................................. 5 1.3 Statistical mechanics and thermodynamics . ... 7 1.4 Hamiltonianmechanics.............................. 8 1.5 Quantummechanics ............................... 10 1.6 Quantumfieldtheory............................... 13 1.7 TheSchr¨odingerpicture . .. .. 13 1.8 TheHeisenbergpicture.............................. 15 1.9 Outlineofthebook................................ 17 2 The simplest quantum system 25 2.1 Matrices, relativity and quantum theory . .. 26 2.2 Continuousmotionsandmatrixgroups . 28 2.3 Infinitesimal motions and matrix Lie algebras . 29 2.4 Uniform motions and the matrix exponential . .. 31 2.5 Volume preservation and special linear groups . ... 33 2.6 The vector product, quaternions, and SL(2, C)................. 35 i ii CONTENTS 2.7 The Hamiltonian form of a Lie algebra . 38 2.8 Atomic energy levels and unitary groups . 39 2.9 QubitsandBlochsphere ............................. 40 2.10 Polarized light and beam transformations . ... 41 2.11 Spinandspincoherentstates . 44 2.12 Particles and detection probabilities . .. 47 2.13Photonsondemand................................ 48 2.14 Unitary representations of SU(2) ........................ 53 3 The symmetries of the universe 57 3.1 Rotations and SO(n)............................... 57 3.2 3-dimensional rotations and SO(3) ....................... 60 3.3 Rotationsandquaternions . .. .. .. 63 3.4 Rotations and SU(2)............................... 65 3.5 Angularvelocity.................................. 66 3.6 Rigid motions and Euclidean groups . 68 3.7 Connected subgroups of SL(2, R) ........................ 70 3.8 Connected subgroups of SL(3, R) ........................ 70 3.9 Classical mechanics and Heisenberg groups . ... 70 3.10 Angularmomentum,isospin,quarks. .. 70 3.11 Connected subgroups of SL(4, R) ........................ 71 3.12TheGalileangroup ................................ 71 3.13 The Lorentz groups O(1, 3), SO(1, 3), SO(1, 3)0 ................ 72 3.14 The Poincare group ISO(1, 3).......................... 75 3.15 ALorentzinvariantmeasure . 77 3.16 Kepler’s laws, the hydrogen atom, and SO(4) ................. 78 3.17 The periodic systemand the conformal group SO(2, 4)............. 78 3.18 The interacting boson model and U(6) ..................... 78 CONTENTS iii 3.19Casimirs...................................... 78 3.20 Unitary representations of the Poincar´egroup . ....... 79 3.21 Some representations of the Poincare group. ...... 80 3.22Elementaryparticles ............................... 80 3.23 Thepositionoperator............................... 81 4 From the theoretical physics FAQ 83 4.1 Tobedone .................................... 83 4.2 Postulates for the formal core of quantum mechanics . ...... 84 4.3 LiegroupsandLiealgebras ........................... 90 4.4 The Galilei group as contraction of the Poincare group . .... 92 4.5 RepresentationsofthePoincaregroup . ... 92 4.6 Forms of relativistic dynamics . 94 4.7 Is there a multiparticle relativistic quantum mechanics? . .... 98 4.8 Whatisaphoton? ................................ 99 4.9 Particle positions and the position operator . 105 4.10 Localization and position operators . 111 4.11 SO(3) = SU(2)/Z2 ................................114 5 Classical oscillating systems 119 5.1 Systemsofdampedoscillators . 119 5.2 The classical anharmonic oscillator . 123 5.3 Harmonic oscillators and linear field equations . 126 5.4 Alpharays.....................................128 5.5 Betarays .....................................130 5.6 Lightraysandgammarays ...........................133 6 Spectral analysis 137 6.1 Thequantumspectrum..............................137 iv CONTENTS 6.2 Probingthespectrumofasystem . 141 6.3 Theearlyhistoryofquantummechanics . 144 6.4 Thespectrumofmany-particlesystems . .148 6.5 Blackbodyradiation ...............................150 6.6 Derivation of Planck’s law . 152 6.7 Stefan’s law and Wien’s displacement law . 156 II Statistical mechanics 159 7 Phenomenological thermodynamics 161 7.1 Standardthermodynamicalsystems . .161 7.2 Thelawsofthermodynamics. .. .. .. .167 7.3 Consequencesofthefirstlaw. 169 7.4 Consequencesofthesecondlaw . 171 7.5 The approach to equilibrium . 175 7.6 Description levels . 176 8 Quantities, states, and statistics 179 8.1 Quantities .....................................180 8.2 Gibbsstates....................................187 8.3 Kuboproductandgeneratingfunctional . .190 8.4 Limit resolution and uncertainty . 195 9 The laws of thermodynamics 201 9.1 Thezerothlaw:Thermalstates . 201 9.2 Theequationofstate...............................205 9.3 Thefirstlaw:Energybalance . .. .. .. .210 9.4 The second law: Extremal principles . 214 9.5 The third law: Quantization . 215 CONTENTS v 10 Models, statistics, and measurements 219 10.1 Description levels . 219 10.2 Local, microlocal, and quantum equilibrium . 226 10.3 Statistics and probability . 230 10.4 Classicalmeasurements. 233 10.5Quantumprobability ...............................236 10.6 Entropyandinformationtheory . 240 10.7 Subjective probability . 244 III Lie algebras and Poisson algebras 247 11 Lie algebras 249 11.1Basicdefinitions..................................250 11.2 Lie algebras from derivations . 253 11.3 Linear groups and their Lie algebras . 255 11.4 Classical Lie groups and their Lie algebras . 258 11.5 Heisenberg algebras and Heisenberg groups . .263 11.6 Lie -algebras ...................................268 ∗ 12 Mechanics in Poisson algebras 273 12.1Poissonalgebras..................................273 12.2 Rotating rigid bodies . 276 12.3 Rotationsandangularmomentum. 277 12.4 Classical rigid body dynamics . 280 12.5Lie–Poissonalgebras ...............................282 12.6 Classical symplectic mechanics . 286 12.7Molecularmechanics ...............................288 12.8 Anoutlooktoquantumfieldtheory . 290 vi CONTENTS 13 Representation and classification 293 13.1 Poissonrepresentations. 293 13.2 Linearrepresentations . .. .. .. 295 13.3 Finite-dimensional representations . .297 13.4 RepresentationsofLiegroups . 298 13.5 Finite-dimensional semisimple Lie algebras . 300 13.6 Automorphismsandcoadjointorbits . 304 IV Nonequilibrium thermodynamics 305 14 Markov Processes 307 14.1Activities .....................................308 14.2Processes .....................................312 14.3 Forward morphisms and quantum dynamical semigroups . .315 14.4Forwardderivations................................ 316 14.5 Single-time, autonomous Markov processes . .319 14.6tobedone.....................................321 15 Diffusion processes 323 15.1 Stochastic differential equations . .323 15.2 Closeddiffusionprocesses . 327 15.3 Ornstein-Uhlenbeckprocesses . .331 15.4 Linearprocesseswithmemory . 333 15.5 Dissipative Hamiltonian Systems . 335 16 Collective Processes 339 16.1Themasterequation ...............................339 16.2 Canonical form and thermodynamic limit . 345 16.3 Stirredchemicalreactions . 350 CONTENTS vii 16.4 Linearresponsetheory . .. .. 352 16.5Opensystem ...................................354 16.6 Somephilosophicalafterthoughts . .356 V Mechanics and differential geometry 357 17 Fields, forms, and derivatives 359 17.1 Scalarfieldsandvectorfields. 360 17.2 Multilinear forms . 363 17.3Exteriorcalculus .................................367 17.4 Manifolds as differential geometries . 370 17.5 Manifolds as topological spaces . 371 17.6 Noncommutativegeometry. 375 17.7Liegroupsasmanifolds..............................376 18 Conservative mechanics on manifolds 379 18.1 Poissonalgebrasfromclosed2-forms . .379 18.2 Conservative Hamiltonian dynamics . 382 18.3 Constrained Hamiltonian dynamics . 385 18.4 Lagrangianmechanics ..............................388 19 Hamiltonian quantum mechanics 393 19.1 Quantum dynamics as symplectic motion . 393 19.2 Quantum-classicaldynamics . 395 19.3 Deformationquantization . 398 19.4TheWignertransform ..............................401 VI Representations and spectroscopy 411 20 Harmonic oscillators and coherent states 413 viii CONTENTS 20.1 The classical harmonic oscillator . 414 20.2 Quantizing the harmonic oscillator . 415 20.3 Representations oftheHeisenberg algebra . .418 20.4BrasandKets...................................420 20.5BosonFockspace .................................423 20.6 Bargmann–Fockrepresentation . .425 20.7 Coherent states for the harmonic oscillator . .426 20.8 Monochromatic beamsandcoherent states . .432 21 Spin and fermions 435 21.1FermionFockspace................................435 21.2 Extensiontomanydegreesoffreedom . .438 21.3 Exterioralgebrarepresentation . .441 21.4 Spin and metaplectic representation . .443 22 Highest weight representations 447 22.1 Triangular decompositions . 447 22.2 Triangulated Lie algebras of rank and degree one . .450 22.3 Unitary representations of SU(2) and SO(3)..................451 22.4 Some unitary highest weight representations . .453 23 Spectroscopy and spectra 455 23.1 Introductionandhistoricalbackground
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