Quantum theory and symmetries with Lie theory... Vol. 2

Preface of Volume 2......Page 6
 Acknowledgements......Page 7
Contents......Page 8
List of Participants......Page 11
 Summary......Page 15
Plenary Talks......Page 16
 1 Introduction......Page 17
 2 Integrable Spin Chain and Hexagon Form Factors......Page 18
 3 Hexagonalisation and Fishnet Integrals......Page 23
 4 Conclusion......Page 28
 References......Page 29
 1 Introduction......Page 31
 2 A Few Examples in 3d and 4d......Page 33
 3 Two-Point Correlators and Effective Actions......Page 35
 4 The Local and Non-local Fronsdal Equations......Page 37
  5.2 Tadpoles and Seagulls......Page 40
 6 Comments......Page 42
 References......Page 43
Algebraic Structures in Exceptional Geometry......Page 45
 References......Page 52
Duality in the Sachdev-Ye-Kitaev Model......Page 56
 1 Introduction......Page 57
 2 Overview of SYK......Page 58
 3 Bulk Space-Time......Page 61
 4 3D Realization and c=1......Page 68
 References......Page 73
 1 Motivations and Contents......Page 75
  2.1 Basic Superspaces......Page 77
  2.2 Basic Superfields......Page 78
  2.3 mathscrN=(1,0) Superfield Actions......Page 79
 3 Higher-Dimension mathscrN=(1,0) and mathscrN=(1,1) Invariants......Page 80
 4 mathscrN=(1,1) Harmonic Superspace......Page 81
 5 Invariants in mathscrN=(1,1) Superspace......Page 83
 6 Quantum mathscrN=(1,0) and mathscrN=(1,1) SYM......Page 84
 7 Summary and Outlook......Page 85
 References......Page 86
 1 Introduction......Page 88
  2.1 Physical and Mirror Channels......Page 90
  2.2 Thermal Partition Function......Page 91
  2.3 The Partition Function as a Sum over Mode Numbers......Page 92
  2.4 From Mode Numbers to Rapidities......Page 94
  2.5 Graph Expansion of the Gaudin Determinant......Page 95
  2.6 Performing the Sum over Trees......Page 98
 3 The Energy of an Excited State......Page 99
 4 One-Point Functions at Finite Volume/Temperature......Page 101
  4.1 The One-Point Function in Terms of Connected Diagonal Form Factors......Page 102
  4.2 LeClair-Mussardo Series from the Tree Expansion......Page 103
  4.3 The Case of a Conserved Charge......Page 105
 5 Conclusion......Page 107
 References......Page 108
Wheeler–DeWitt Quantization of Gravity Models of Unified Dark Energy and Dark Matter......Page 110
 1 Introduction......Page 111
  2.1 Hidden Noether Symmetry and Unification of Dark Energy and Dark Matter......Page 112
  2.2 Quintessential Inflation and Unified Dark Energy  and Dark Matter......Page 113
 3 Wheeler–De Witt Minisuperspace Quantization......Page 118
 4 Conclusions......Page 122
 References......Page 123
 1 Introduction......Page 125
  2.1 Generalised Form Factors of Wilson Loops......Page 128
  2.2 Dual Variables......Page 129
  2.3 Duality Relation......Page 130
  2.4 Duality Relation at MHV Level......Page 131
  2.5 Duality Beyond MHV......Page 134
  3.1 calN=4 Super-Yang-Mills in LHC Superspace......Page 137
  3.2 Chiral Wilson Loop in LHC Superspace......Page 138
 4 Diagrammatic Approach to the Duality......Page 139
  4.1 MHV Example......Page 140
 5 Concluding Remarks......Page 141
 References......Page 144
 1 Introduction......Page 146
  1.1 Thermalization......Page 147
  1.3 Localized Systems......Page 148
 2 Many-Body Interacting Model by Using Coproducts......Page 149
  2.1 Coproducts......Page 150
 3 Eigenstates and Eigenvalues......Page 151
  3.1 Level 1 States......Page 152
  3.2 Level p States......Page 153
 4 Entanglement Entropy......Page 155
 5 Discussion......Page 156
 References......Page 157
 1 Introduction......Page 158
 2 Cooperative Game Theory: The Whole is Greater  than the Sum of Its Parts......Page 159
 3 Generalized Bases of Mixed States with a Resolution  of the Identity......Page 161
  5.1 Location Indices, Comonotonic Operators  and Comonotonicity Intervals......Page 164
 6 Discussion......Page 166
 References......Page 167
Holography and String Theory......Page 168
 1 Introduction......Page 169
 2 Constant Roll Inflation......Page 170
 3 Parameter Space of the New Solutions......Page 172
 4 Stability Under Scalar Perturbations......Page 173
 5 Scalar Spectral Index......Page 175
 6 Other Non-slow Roll Regimes......Page 177
 References......Page 178
 1 Introduction......Page 180
 2 The Dp/Dq Brane Holographic Set-Up......Page 182
 3 Holographic Calculation of the Condensate Susceptibility......Page 183
 4 Field Theory Comparison......Page 184
 5 Conclusion......Page 185
 References......Page 186
 1 Introduction and Summary......Page 187
 2 Non-perturbative Scalars in the Wilson Loop......Page 188
 3 Fermion Contribution to the Wilson Loop......Page 190
  3.1 Emergence of a Bound State......Page 191
  3.2 Mesons Bound States, TBA and Beyond......Page 194
 References......Page 196
 1 Introduction......Page 198
 2 Four-Point String Amplitudes......Page 200
  3.1 Linear Relations in Hard Limit......Page 203
  3.2 Recurrence Relations in Regge Limit......Page 205
 4 Symmetry of Four-Point Amplitudes at General Energy......Page 206
 References......Page 208
 1 Introduction......Page 210
 2 Basics of Thermo Field Dynamics......Page 211
 3 The Fisher Information Metric for Closed Strings  in Plane Wave Geometry......Page 213
 4 Reconstruction of Probability Density Functions  from Fisher Metric......Page 216
 References......Page 218
Gravity and Cosmology......Page 220
 1 Introduction......Page 221
  2.2 Duality Structures......Page 223
  2.3 Electromagnetic Structures......Page 224
  2.4 Scalar-Duality and Scalar-Electromagnetic Structures......Page 225
  2.5 Pulled-Back Electromagnetic Structures......Page 226
  2.6 The Mathematical Formulation of Generalized ESM Theories......Page 227
  2.7 Sheaves of Scalar-Electromagnetic Configurations  and Solutions......Page 228
 3 Scalar-Electromagnetic Dualities and Symmetries......Page 229
  4.1 Integral Duality Structures and Integral Electromagnetic Structures......Page 231
  4.2 The Twisted Dirac Quantization Condition......Page 233
 References......Page 234
 1 Introduction......Page 236
  2.1 Einstein–Scalar Theories with 2-Dimensional Scalar Manifolds......Page 237
  2.2 Cosmological Models Defined by (Σ,mathcalG, V)......Page 238
  3.1 Lifted Trajectories and Tilings......Page 239
  3.3 Well-Behaved Scalar Potentials......Page 240
 4 Examples of Trajectories for the Hyperbolic Triply Punctured Sphere......Page 241
 References......Page 244
 1 Introduction......Page 245
 2 The BRST Formulation of Supergravity......Page 246
 3 The Full Topological Structure of Supergravity......Page 247
 4 The Cohomological Equations of Localization......Page 249
 References......Page 252
 1 The Two-Measures Model......Page 253
 2 The Darkon Model in FLRW Metric......Page 254
 3 The Two-Measures Theory – Including the Inflaton......Page 257
 4 Conclusions......Page 261
 References......Page 262
 1 Introduction......Page 263
 2 Basic Equations and Boundary Conditions......Page 264
 3 Numerical Results......Page 265
 References......Page 268
Conformal and Gauge Theories......Page 270
 1 Introduction......Page 271
 2 From Conformal Ward–Takahashi Identities  to Intertwining Relations......Page 272
 3 Timelike Conformal Killing Vectors Associated  with the Subgroup SO(1,1)subsetSO(2,d)......Page 274
 4 Intertwining Relations in the SO(1,1) Basis......Page 279
 References......Page 281
 1 Introduction......Page 283
  2.1 The Theory......Page 284
  2.2 Mixing of the Fields......Page 289
 3 N=2 Superconformal Models......Page 292
 References......Page 295
 1 Introduction......Page 296
  2.1 Solution: General Expressions......Page 298
 3 Extremely Localized Input: Fundamental Solution......Page 299
  3.1 Numerical Analysis: V-Like Shape of |S(ξ,τ)|2......Page 300
 4 Gaussian Input......Page 301
 5 Salpeter Equation and Optics......Page 302
  5.1 Bessel Input: Band-Limited Initial Condition......Page 303
  5.2 Coordinate-Momentum Representation: Wigner Distribution Function......Page 304
 References......Page 306
 1 Introduction......Page 307
 2 The Standard Model Algebra......Page 308
 3 Representing Leptons and Quarks......Page 310
 4 The Gauge Symmetries......Page 312
 5 Open Problems......Page 313
 References......Page 314
Foundations of Quantum Theory......Page 315
 1 Introduction......Page 316
 2 Sketch of the Classical Background......Page 317
 3 Setting of the Quantum Theory......Page 320
 4 Lie Algebra of Special Phase Functions......Page 322
 5 Quantum Symmetries......Page 325
 References......Page 332
 1 Introduction and Preliminaries......Page 334
 2 The Frame Optimization Problem......Page 338
  3.1 The Solution to the Quantum State Separability Problem......Page 339
  3.2 Examples of (IC-POVMs)......Page 341
  3.3 The Solution to the Quantum State Estimation Problem......Page 342
  4.1 The Solution to the Quantum State Separability Problem......Page 343
  4.2 Constructing Solutions to the Quantum State Separability Problem......Page 345
  4.3 The Solution to the State Estimation Problem......Page 346
 References......Page 347
 1 Introduction......Page 350
 2 General Structure of Lie Groups: Levi Decomposition......Page 352
 3 A Classification of Poincaré Group Extensions......Page 353
 4 Conservative Extensions of the Poincaré Group......Page 354
 5 Commutation Relations of the Concrete Example......Page 356
 References......Page 359
 1 Introduction......Page 360
 2 The Complement to Noether's Theorem......Page 361
 3 Two-Particle States and Interaction......Page 363
 4 Illustration: A Scattering Experiment......Page 365
 5 Conclusions......Page 367
 References......Page 368
 1 Introduction......Page 369
 2 Associated Laguerre Polynomials......Page 370
 3 SU(2) Representations in the Plane......Page 372
 4 Rigged Hilbert Space Formulation......Page 375
 5 Conclusions......Page 377
 References......Page 378
Nonrelativistic and Classical Theories......Page 380
 1 Introduction......Page 381
 2 Basic Equations......Page 383
 3 Load and Metage......Page 384
 4 Lagrangian variational principle of MHD......Page 388
 5 Non Barotropic Cross Helicity Conservation  via the Noether Theorem......Page 392
 6 Conclusion......Page 394
 References......Page 395
 1 Introduction......Page 397
 2 An Area-Dependent Potential......Page 398
 3 Low-Lying Energy Spectrum......Page 399
 4 Summary......Page 402
 References......Page 404
 1 Introduction......Page 405
 2 Linear Nambu Flows for Classical and Quantum Magnets......Page 406
 3 Computing Transition Probabilities la Nambu......Page 410
 4 Conclusion and Outlook......Page 413
 References......Page 414
 1 Introduction......Page 415
 2 Newton–Cartan Gravity Background......Page 417
 3 Calculation of the Anomaly for Scalars and Fermions......Page 419
 4 Conclusions......Page 420
 References......Page 421
 1 Introduction......Page 424
 2 Time Evolution of a Free Non-relativistic Particle  and a Particle in a Homogeneous Field......Page 425
 3 Generalized Coherent States......Page 427
 4 Unitary Equivalence for Oscillator-Like Solutions  of a Free Non-relativistic Particle with Time-Dependent Mass and a Particle in a Time-Dependent Field......Page 431
 References......Page 432