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Supervisor's Foreword; Abstract; Preface; The original contributions of this thesis are based on the following publications:G. Barnich and B. Oblak, "Holographic positive energy theorems in three-dimensional gravity," Class. Quant. Grav. 31 (2014) 152001, 1403.3835.G. Barnich and B. Oblak, "Notes on the BMS group in three dimensions: I. Induced representations," JHEP 06 (2014) 129, 1403.5803.G. Barnich and B. Oblak, "Notes on the BMS group in three dimensions: II. Coadjoint representation," JHEP 03 (2015) 033, 1502.00010.B. Oblak, "Char; Acknowledgements; Professional Community
Family and FriendsPhysics and Art; Financial and Logistic Support; Contents; 1 Introduction; 1.1 Asymptotic BMS Symmetry; 1.2 Global BMS and Extended BMS; 1.3 Holography; 1.4 BMS Particles and Soft Gravitons; 1.5 Plan of the Thesis; References; Part I Quantum Symmetries; 2 Quantum Mechanics and Central Extensions; 2.1 Symmetries and Projective Representations; 2.1.1 Quantum Mechanics; 2.1.2 Symmetry Representation Theorem; 2.1.3 Projective Representations; 2.1.4 Central Extensions; 2.1.5 Topological Central Extensions; 2.1.6 Classifying Projective Representations; 2.2 Lie Algebra Cohomology
2.2.1 Cohomology2.2.2 Central Extensions; 2.3 Group Cohomology; 2.3.1 Cohomology; 2.3.2 Central Extensions; References; 3 Induced Representations ; 3.1 Wavefunctions and Measures; 3.1.1 Measures; 3.1.2 Hilbert Spaces of Wavefunctions; 3.1.3 Equivalent Measures and Radon
Nikodym Derivatives; 3.2 Quasi-regular Representations; 3.2.1 Quasi-invariant Measures on Homogeneous Spaces; 3.2.2 The Simplest Induced Representations; 3.2.3 Radon
Nikodym Is a Cocycle*; 3.3 Defining Induced Representations; 3.3.1 Standard Boosts; 3.3.2 Induced Representations; 3.3.3 Properties of Induced Representations
3.3.4 Plane Waves3.4 Characters; 3.4.1 Characters Are Partition Functions; 3.4.2 The Frobenius Formula; 3.4.3 Characters and Fixed Points; 3.5 Systems of Imprimitivity*; 3.5.1 Projections and Imprimitivity; 3.5.2 Imprimitivity Theorem; References; 4 Semi-direct Products; 4.1 Representations and Particles; 4.1.1 Semi-direct Products; 4.1.2 Momenta; 4.1.3 Orbits and Little Groups; 4.1.4 Particles; 4.1.5 Exhaustivity Theorem; 4.2 Poincaré Particles; 4.2.1 Poincaré Groups; 4.2.2 Orbits and Little Groups; 4.2.3 Particles; 4.2.4 Massive Characters; 4.2.5 Massless Characters
4.2.6 Wigner Rotations and Entanglement*4.3 Poincaré Particles in Three Dimensions; 4.3.1 Poincaré Group in Three Dimensions; 4.3.2 Particles in Three Dimensions; 4.3.3 Characters; 4.4 Galilean Particles*; 4.4.1 Bargmann Groups; 4.4.2 Orbits and Little Groups; 4.4.3 Particles; 4.4.4 Characters; References; 5 Coadjoint Orbits and Geometric Quantization; 5.1 Symmetric Phase Spaces; 5.1.1 Lie Groups; 5.1.2 Adjoint and Coadjoint Representations; 5.1.3 Poisson Structures; 5.1.4 Symplectic Structures; 5.1.5 Kirillov
Kostant Structures; 5.1.6 Momentum Maps; 5.2 Geometric Quantization
Family and FriendsPhysics and Art; Financial and Logistic Support; Contents; 1 Introduction; 1.1 Asymptotic BMS Symmetry; 1.2 Global BMS and Extended BMS; 1.3 Holography; 1.4 BMS Particles and Soft Gravitons; 1.5 Plan of the Thesis; References; Part I Quantum Symmetries; 2 Quantum Mechanics and Central Extensions; 2.1 Symmetries and Projective Representations; 2.1.1 Quantum Mechanics; 2.1.2 Symmetry Representation Theorem; 2.1.3 Projective Representations; 2.1.4 Central Extensions; 2.1.5 Topological Central Extensions; 2.1.6 Classifying Projective Representations; 2.2 Lie Algebra Cohomology
2.2.1 Cohomology2.2.2 Central Extensions; 2.3 Group Cohomology; 2.3.1 Cohomology; 2.3.2 Central Extensions; References; 3 Induced Representations ; 3.1 Wavefunctions and Measures; 3.1.1 Measures; 3.1.2 Hilbert Spaces of Wavefunctions; 3.1.3 Equivalent Measures and Radon
Nikodym Derivatives; 3.2 Quasi-regular Representations; 3.2.1 Quasi-invariant Measures on Homogeneous Spaces; 3.2.2 The Simplest Induced Representations; 3.2.3 Radon
Nikodym Is a Cocycle*; 3.3 Defining Induced Representations; 3.3.1 Standard Boosts; 3.3.2 Induced Representations; 3.3.3 Properties of Induced Representations
3.3.4 Plane Waves3.4 Characters; 3.4.1 Characters Are Partition Functions; 3.4.2 The Frobenius Formula; 3.4.3 Characters and Fixed Points; 3.5 Systems of Imprimitivity*; 3.5.1 Projections and Imprimitivity; 3.5.2 Imprimitivity Theorem; References; 4 Semi-direct Products; 4.1 Representations and Particles; 4.1.1 Semi-direct Products; 4.1.2 Momenta; 4.1.3 Orbits and Little Groups; 4.1.4 Particles; 4.1.5 Exhaustivity Theorem; 4.2 Poincaré Particles; 4.2.1 Poincaré Groups; 4.2.2 Orbits and Little Groups; 4.2.3 Particles; 4.2.4 Massive Characters; 4.2.5 Massless Characters
4.2.6 Wigner Rotations and Entanglement*4.3 Poincaré Particles in Three Dimensions; 4.3.1 Poincaré Group in Three Dimensions; 4.3.2 Particles in Three Dimensions; 4.3.3 Characters; 4.4 Galilean Particles*; 4.4.1 Bargmann Groups; 4.4.2 Orbits and Little Groups; 4.4.3 Particles; 4.4.4 Characters; References; 5 Coadjoint Orbits and Geometric Quantization; 5.1 Symmetric Phase Spaces; 5.1.1 Lie Groups; 5.1.2 Adjoint and Coadjoint Representations; 5.1.3 Poisson Structures; 5.1.4 Symplectic Structures; 5.1.5 Kirillov
Kostant Structures; 5.1.6 Momentum Maps; 5.2 Geometric Quantization