Quantum Theory, Groups and Representations: An ...

1 Quantum theory , Groups and Representations: an introduction (Final draft version)Peter WoitDepartment of Mathematics, Columbia 2017 Peter WoitAll rights Acknowledgements .. xvi1 introduction and introduction .. Basic principles of Quantum mechanics .. axioms of Quantum mechanics .. of measurement theory .. Unitary group representations .. Groups .. representations .. group representations .. Representations and Quantum mechanics .. Groups and symmetries .. For further reading .. 112 The GroupU(1)and its Some representation theory .. The groupU(1) and its representations .. The charge operator .. Conservation of charge andU(1) symmetry.

2 Summary .. For further reading .. 233 Two-state Systems andSU(2) The two-state Quantum system .. Pauli matrices: observables of the two-state quantumsystem .. of Pauli matrices: unitary transformationsof the two-state system .. Commutation relations for Pauli matrices .. Dynamics of a two-state system .. For further reading .. 33ii4 Linear Algebra Review, Unitary and Orthogonal Vector spaces and linear maps .. Dual vector spaces .. Change of basis .. Inner products .. Adjoint operators .. Orthogonal and unitary transformations .. Groups .. Groups .

3 Eigenvalues and eigenvectors .. For further reading .. 465 Lie Algebras and Lie Algebra Lie algebras .. Lie algebras of the orthogonal and unitary Groups .. algebra of the orthogonal group .. algebra of the unitary group .. A summary .. Lie algebra representations .. Complexification .. For further reading .. 616 The Rotation and Spin Groups in 3 and 4 The rotation group in three dimensions .. Spin Groups in three and four dimensions .. and spin Groups in four dimensions .. and spin Groups in three dimensions .. spin group andSU(2) .. A summary.

4 For further reading .. 747 Rotations and the Spin12 Particle in a Magnetic The spinor representation .. The spin12particle in a magnetic field .. The Heisenberg picture .. Complex projective space .. The Bloch sphere .. For further reading .. 888 Representations ofSU(2)andSO(3) Representations ofSU(2): classification .. decomposition .. algebra representations: raising and lowering operators Representations ofSU(2): construction .. Representations ofSO(3) and spherical harmonics .. The Casimir operator .. For further reading .. 1089 Tensor Products, Entanglement, and Addition of Tensor products.

5 Composite Quantum systems and tensor products .. Indecomposable vectors and entanglement .. Tensor products of representations .. products ofSU(2) representations .. of representations .. examples .. Bilinear forms and tensor products .. Symmetric and antisymmetric multilinear forms .. For further reading .. 12010 Momentum and the Free The groupRand its representations .. Translations in time and space .. Energy and the groupRof time translations .. Momentum and the groupR3of space translations .. The energy-momentum relation and the Schr odinger equation fora free particle .. For further reading.

6 12811 Fourier Analysis and the Free Periodic boundary conditions and the groupU(1) .. The groupRand the Fourier transform .. Distributions .. Linear transformations and distributions .. Solutions of the Schr odinger equation in momentum space .. For further reading .. 14212 Position and the Free The position operator .. Momentum space representation .. Dirac notation .. Heisenberg uncertainty .. The propagator in position space .. Propagators in frequency-momentum space .. Green s functions and solutions to the Schr odinger equations .. For further reading .. 15613 The Heisenberg group and the Schr odinger The Heisenberg Lie algebra.

7 The Heisenberg group .. The Schr odinger representation .. For further reading .. 16314 The Poisson Bracket and Symplectic Classical mechanics and the Poisson bracket .. The Poisson bracket and the Heisenberg Lie algebra .. Symplectic geometry .. For further reading .. 17115 Hamiltonian Vector Fields and the Moment Vector fields and the exponential map .. Hamiltonian vector fields and canonical transformations .. group actions onMand the moment map .. Examples of Hamiltonian group actions .. The dual of a Lie algebra and symplectic geometry .. For further reading .. 18516 Quadratic Polynomials and the Symplectic The symplectic group .

8 The symplectic group ford= 1 .. The symplectic group for arbitraryd.. The symplectic group and automorphisms of the Heisenberg The adjoint representation and inner automorphisms .. The symplectic group as automorphism group .. The case of arbitrary d .. For further reading .. 19717 Canonical quantization .. The Groenewold-van Hove no-go theorem .. Canonical quantization inddimensions .. Quantization and symmetries .. More general notions of quantization .. For further reading .. 20418 Semi-direct An example: the Euclidean group .. Semi-direct product Groups .. Semi-direct product Lie algebras .. For further reading.

9 21019 The Quantum Free Particle as a representation of the Eu-clidean The Quantum free particle and representations ofE(2) .. The case ofE(3) .. Other representations ofE(3) .. For further reading .. 221v20 Representations of Semi-direct Intertwining operators and the metaplectic representation .. Constructing intertwining operators .. Explicit calculations .. TheSO(2) action by rotations of the plane ford= 2 .. AnSO(2) action on thed= 1 phase space .. The Fourier transform as an intertwining operator .. AnRaction on thed= 1 phase space .. Representations ofNoK,Ncommutative .. For further reading .. 23321 Central Potentials and the Hydrogen Quantum particle in a central potential.

10 (4) symmetry and the Coulomb potential .. The hydrogen atom .. For further reading .. 24422 The Harmonic The harmonic oscillator with one degree of freedom .. Creation and annihilation operators .. The Bargmann-Fock representation .. Quantization by annihilation and creation operators .. For further reading .. 25423 Coherent States and the Propagator for the Harmonic Coherent states and the Heisenberg group action .. Coherent states and the Bargmann-Fock state space .. The Heisenberg group action on operators .. The harmonic oscillator propagator .. The propagator in the Bargmann-Fock representation .. The coherent state propagator.