Quantum Mechanics via Lie Algebras

Arnold Neumaier and Dennis Westra, University of Vienna, Austria.

• PrefaceI An invitation to quantum mechanics1 Motivation1.1 Classical mechanics1.2 Relativity theory1.3 Statistical mechanics and thermodynamics1.4 Hamiltonian mechanics1.5 Quantum mechanics1.6 Quantum field theory1.7 The Schrödinger picture1.8 The Heisenberg picture1.9 Outline of the book2 The simplest quantum system2.1 Matrices, relativity and quantum theory2.2 Continuous motions and matrix groups2.3 Infinitesimal motions and matrix Lie algebras2.4 Uniform motions and the matrix exponential2.5 Volume preservation and special linear groups2.6 The vector product, quaternions, and SL(2,C)2.7 The Hamiltonian form of a Lie algebra 2.8 Atomic energy levels and unitary groups 2.9 Qubits and Bloch sphere 2.10 Polarized light and beam transformations 2.11 Spin and spin coherent states 2.12 Particles and detection probabilities 2.13 Photons on demand 2.14 Unitary representations of SU(2) 3 The symmetries of the universe 3.1 Rotations and SO(n) 3.2 3-dimensional rotations and SO(3) 3.3 Rotations and quaternions 3.4 Rotations and SU(2) 3.5 Angular velocity 3.6 Rigid motions and Euclidean groups 3.7 Connected subgroups of SL(2,R) 3.8 Connected subgroups of SL(3,R) 3.9 Classical mechanics and Heisenberg groups 3.10 Angular momentum, isospin, quarks 3.11 Connected subgroups of SL(4,R) 3.12 The Galilean group 3.13 The Lorentz groups O(1, 3), SO(1, 3), SO(1, 3)0 3.14 The Poincare group ISO(1, 3) 3.15 A Lorentz invariant measure 3.16 Kepler's laws, the hydrogen atom, and SO(4) 3.17 The periodic systemand the conformal group SO(2, 4) 3.18 The interacting boson model and U(6) 3.19 Casimirs 3.20 Unitary representations of the Poincaré group 3.21 Some representations of the Poincaré group 3.22 Elementary particles 3.23 The position operator 4 From the theoretical physics FAQ 4.1 To be done 4.2 Postulates for the formal core of quantum mechanics 4.3 Lie groups and Lie algebras 4.4 The Galilei group as contraction of the Poincare group 4.5 Representations of the Poincare group 4.6 Forms of relativistic dynamics 4.7 Is there a multiparticle relativistic quantum mechanics? 4.8 What is a photon? 4.9 Particle positions and the position operator 4.10 Localization and position operators 4.11 SO(3) = SU(2)/Z2 5 Classical oscillating systems 5.1 Systems of damped oscillators 5.2 The classical anharmonic oscillator 5.3 Harmonic oscillators and linear field equations 5.4 Alpha rays 5.5 Beta rays 5.6 Light rays and gamma rays 6 Spectral analysis 6.1 The quantum spectrum 6.2 Probing the spectrum of a system 6.3 The early history of quantum mechanics 6.4 The spectrum of many-particle systems 6.5 Black body radiation6.6 Derivation of Planckés law6.7 Stefan´s law and Wien´s displacement lawII Statistical mechanics7 Phenomenological thermodynamics 7.1 Standard thermodynamical systems 7.2 The laws of thermodynamics 7.3 Consequences of the first law 7.4 Consequences of the second law 7.5 The approach to equilibrium 7.6 Description levels 8 Quantities, states, and statistics 8.1 Quantities 8.2 Gibbs states 8.3 Kubo product and generating functional 8.4 Limit resolution and uncertainty 9 The laws of thermodynamics 9.1 The zeroth law: Thermal states 9.2 The equation of state 9.3 The first law: Energy balance 9.4 The second law: Extremal principles 9.5 The third law: Quantization 10 Models, statistics, and measurements 10.1 Description levels 10.2 Local, microlocal, and quantum equilibrium 10.3 Statistics and probability 10.4 Classical measurements 10.5 Quantum probability 10.6 Entropy and information theory 10.7 Subjective probabilityIII Lie algebras and Poisson algebras11 Lie algebras11.1 Basic definitions 11.2 Lie algebras from derivations 11.3 Linear groups and their Lie algebras 11.4 Classical Lie groups and their Lie algebras 11.5 Heisenberg algebras and Heisenberg groups 11.6 Lie-algebras 12 Mechanics in Poisson algebras 12.1 Poisson algebras 12.2 Rotating rigid bodies 12.3 Rotations and angular momentum 12.4 Classical rigid body dynamics 12.5 Lie-Poisson algebras 12.6 Classical symplectic mechanics 12.7 Molecular mechanics 12.8 An outlook to quantum field theory 13 Representation and classification 13.1 Poisson representations 13.2 Linear representations 13.3 Finite-dimensional representations 13.4 Representations of Lie groups 13.5 Finite-dimensional semisimple Lie algebras 13.6 Automorphisms and coadjoint orbits IV Nonequilibrium thermodynamics 14 Markov Processes 14.1 Activities 14.2 Processes 14.3 Forward morphisms and quantum dynamical semigroups 14.4 Forward derivations 14.5 Single-time, autonomous Markov processes 15 Diffusion processes 15.1 Stochastic differential equations 15.2 Closed diffusion processes 15.3 Ornstein-Uhlenbeck processes 15.4 Linear processes with memory 15.5 Dissipative Hamiltonian Systems 16 Collective Processes 16.1 The master equation 16.2 Canonical form and thermodynamic limit 16.3 Stirred chemical reactions 16.4 Linear response theory 16.5 Open system 16.6 Some philosophical afterthoughts V Mechanics and differential geometry 17 Fields, forms, and derivatives 17.1 Scalar fields and vector fields 17.2 Multilinear forms 17.3 Exterior calculus 17.4 Manifolds as differential geometries 17.5 Manifolds as topological spaces 17.6 Noncommutative geometry 17.7 Lie groups as manifolds 18 Conservative mechanics on manifolds 18.1 Poisson algebras from closed 2-forms 18.2 Conservative Hamiltonian dynamics 18.3 Constrained Hamiltonian dynamics 18.4 Lagrangian mechanics 19 Hamiltonian quantum mechanics19.1 Quantum dynamics as symplectic motion 19.2 Quantum-classical dynamics 19.3 Deformation quantization19.4 The Wigner transformVI Representations and spectroscopy20 Harmonic oscillators and coherent states20.1 The classical harmonic oscillator20.2 Quantizing the harmonic oscillator20.3 Representations of the Heisenberg algebra 20.4 Bras and Kets20.5 Boson Fock space20.6 Bargmann.Fock representation20.7 Coherent states for the harmonic oscillator 20.8 Monochromatic beams and coherent states 21 Spin and fermions 21.1 Fermion Fock space 21.2 Extension to many degrees of freedom 21.3 Exterior algebra representation 21.4 Spin and metaplectic representation 22 Highest weight representations 22.1 Triangular decompositions 22.2 Triangulated Lie algebras of rank and degree one 22.3 Unitary representations of SU(2) and SO(3) 22.4 Some unitary highest weight representations 23 Spectroscopy and spectra 23.1 Introduction and historical background 23.2 Spectra of systems of particles 23.3 Examples of spectra23.4 Dynamical symmetries23.5 The hydrogen atom23.6 Chains of subalgebrasReferences