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Coherent.States.and.Applications.in.Mathematical.Physics
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1 Introduction to Coherent States . . . . . . . . . . . . . . . . . . . . . 1 1.1 The Weyl–Heisenberg Group and the Canonical Coherent States . . 2 1.1.1 The Weyl–Heisenberg Translation Operator . . . . . . . . 2 1.1.2 The Coherent States of Arbitrary Profile . . . . . . . . . . 6 1.2 The Coherent States of the Harmonic Oscillator . . . . . . . . . . 7 1.2.1 Definition and Properties . . . . . . . . . . . . . . . . . . 7 1.2.2 The Time Evolution of the Coherent State for the Harmonic Oscillator Hamiltonian . . . . . . . . . . 11 1.2.3 AnOver-completeSystem. . . . . . . . . . . . . . . . . . 12 1.3 From Schrödinger to Bargmann–Fock Representation . . . . . . . 16 2 Weyl Quantization and Coherent States . . . . . . . . . . . . . . . . 23 2.1 Classical and Quantum Observables . . . . . . . . . . . . . . . . . 23 2.1.1 Group Invariance of Weyl Quantization . . . . . . . . . . . 27 2.2 Wigner Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3 Coherent States and Operator Norms Estimates . . . . . . . . . . . 35 2.4 Product Rule and Applications . . . . . . . . . . . . . . . . . . . 40 2.4.1 The Moyal Product . . . . . . . . . . . . . . . . . . . . . 40 2.4.2 Functional Calculus . . . . . . . . . . . . . . . . . . . . . 43 2.4.3 Propagation of Observables . . . . . . . . . . . . . . . . . 45 2.4.4 Return to Symplectic Invariance of Weyl Quantization . . . 47 2.5 Husimi Functions, Frequency Sets and Propagation . . . . . . . . 49 2.5.1 Frequency Sets . . . . . . . . . . . . . . . . . . . . . . . . 49 2.5.2 About Frequency Set of Eigenstates . . . . . . . . . . . . . 52 2.6 WickQuantization . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.6.1 General Properties . . . . . . . . . . . . . . . . . . . . . . 52 2.6.2 ApplicationtoSemi-classicalMeasures . . . . . . . . . . . 55 3 The Quadratic Hamiltonians . . . . . . . . . . . . . . . . . . . . . . 59 3.1 The Propagator of Quadratic Quantum Hamiltonians . . . . . . . . 59 3.2 The Propagation of Coherent States . . . . . . . . . . . . . . . . . 61 3.3 TheMetaplecticTransformations . . . . . . . . . . . . . . . . . . 69 ix x Contents 3.4 Representation of the Quantum Propagator in Terms of the Generator of Squeezed States . . . . . . . . . . . . . . . . . 71 3.5 Representation of the Weyl Symbol of the Metaplectic Operators . 78 3.6 Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.6.1 TheClassicalMotion . . . . . . . . . . . . . . . . . . . . 81 3.7 The Quantum Evolution . . . . . . . . . . . . . . . . . . . . . . . 82 4 The Semiclassical Evolution of Gaussian Coherent States . . . . . . 87 4.1 General Results and Assumptions . . . . . . . . . . . . . . . . . . 87 4.1.1 Assumptions andNotations . . . . . . . . . . . . . . . . . 88 4.1.2 The Semiclassical Evolution of Generalized Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.1.3 RelatedWorks andOtherResults . . . . . . . . . . . . . . 100 4.2 Application to the Spreading of Quantum Wave Packets . . . . . . 100 4.3 Evolution of Coherent States and Bargmann Transform . . . . . . 103 4.3.1 FormalComputations . . . . . . . . . . . . . . . . . . . . 103 4.3.2 Weighted Estimates and Fourier–Bargmann Transform . . . 105 4.3.3 Large Time Estimates and Fourier–Bargmann Analysis . . 107 4.3.4 Exponentially Small Estimates . . . . . . . . . . . . . . . 110 4.4 Application to the Scattering Theory . . . . . . . . . . . . . . . . 114 5 Trace Formulas and Coherent States . . . . . . . . . . . . . . . . . . 123 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.2 The Semi-classical Gutzwiller Trace Formula . . . . . . . . . . . . 127 5.3 Preparations for the Proof . . . . . . . . . . . . . . . . . . . . . . 131 5.4 The Stationary Phase Computation . . . . . . . . . . . . . . . . . 135 5.5 A Pointwise Trace Formula and Quasi-modes . . . . . . . . . . . 143 5.5.1 APointwiseTraceFormula . . . . . . . . . . . . . . . . . 144 5.5.2 Quasi-modes and Bohr–Sommerfeld Quantization Rules . . 145 6 Quantization and Coherent States on the 2-Torus . . . . . . . . . . . 151 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.2 TheAutomorphismsof the2-Torus . . . . . . . . . . . . . . . . . 151 6.3 TheKinematicsFrameworkandQuantization . . . . . . . . . . . 155 6.4 The Coherent States of the Torus . . . . . . . . . . . . . . . . . . 162 6.5 TheWeyl andAnti-WickQuantizationsonthe2-Torus . . . . . . . 166 6.5.1 TheWeylQuantizationon the 2-Torus . . . . . . . . . . . 166 6.5.2 TheAnti-WickQuantizationonthe2-Torus . . . . . . . . 168 6.6 Quantum Dynamics and Exact Egorov’s Theorem . . . . . . . . . 170 6.6.1 Quantization of SL(2,Z) . . . . . . . . . . . . . . . . . . 170 6.6.2 The Egorov Theorem Is Exact . . . . . . . . . . . . . . . . 173 6.6.3 Propagation of Coherent States . . . . . . . . . . . . . . . 174 6.7 Equipartition of the Eigenfunctions of Quantized Ergodic Maps onthe2-Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.8 Spectral Analysis of Hamiltonian Perturbations . . . . . . . . . . . 177 Contents xi 7 Spin-Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.2 The Groups SO(3) and SU(2) . . . . . . . . . . . . . . . . . . . . 183 7.3 The Irreducible Representations of SU(2) . . . . . . . . . . . . . . 187 7.3.1 The Irreducible Representations of su(2) . . . . . . . . . . 187 7.3.2 The Irreducible Representations of SU(2) . . . . . . . . . . 191 7.3.3 Irreducible Representations of SO(3) and Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . 196 7.4 The Coherent States of SU(2) . . . . . . . . . . . . . . . . . . . . 199 7.4.1 Definition and First Properties . . . . . . . . . . . . . . . . 199 7.4.2 SomeExplicitFormulas . . . . . . . . . . . . . . . . . . . 203 7.5 Coherent States on the Riemann Sphere . . . . . . . . . . . . . . . 213 7.6 Application to High Spin Inequalities . . . . . . . . . . . . . . . . 216 7.6.1 Berezin–Lieb Inequalities . . . . . . . . . . . . . . . . . . 216 7.6.2 HighSpinEstimates . . . . . . . . . . . . . . . . . . . . . 217 7.7 More on High Spin Limit: From Spin-Coherent States to Harmonic-Oscillator Coherent States . . . . . . . . . . . . . . . 220 8 Pseudo-Spin-Coherent States . . . . . . . . . . . . . . . . . . . . . . 225 8.1 Introduction to the Geometry of the Pseudo-Sphere, SO(2, 1) and SU(1, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 8.1.1 Minkowski Model . . . . . . . . . . . . . . . . . . . . . . 225 8.1.2 Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 227 8.1.3 The Disc and the Half-Plane Poincaré Representations of the Pseudo-Sphere . . . . . . . . . . . . . . . . . . . . 229 8.2 Unitary Representations of SU(1, 1) . . . . . . . . . . . . . . . . 231 8.2.1 Classification of the Possible Representations of SU(1, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . 233 8.2.2 Discrete Series Representations of SU(1, 1) . . . . . . . . 234 8.2.3 Irreducibility of Discrete Series . . . . . . . . . . . . . . . 238 8.2.4 PrincipalSeries . . . . . . . . . . . . . . . . . . . . . . . 239 8.2.5 ComplementarySeries . . . . . . . . . . . . . . . . . . . . 241 8.2.6 BosonsSystemsRealizations . . . . . . . . . . . . . . . . 243 8.3 Pseudo-Coherent States for Discrete Series . . . . . . . . . . . . . 245 8.3.1 Definition of Coherent States for Discrete Series . . . . . . 245 8.3.2 SomeExplicitFormula . . . . . . . . . . . . . . . . . . . 246 8.3.3 Bargmann Transform and Large k Limit . . . . . . . . . . 250 8.4 Coherent States for the Principal Series . . . . . . . . . . . . . . . 252 8.5 Generator of Squeezed States. Application . . . . . . . . . . . . . 252 8.5.1 The Generator of Squeezed States . . . . . . . . . . . . . . 253 8.5.2 Application to Quantum Dynamics . . . . . . . . . . . . . 254 8.6 Wavelets and Pseudo-Spin-Coherent States . . . . . . . . . . . . . 258 9 The Coherent States of the Hydrogen Atom . . . . . . . . . . . . . . 263 9.1 The S3 Sphere and the Group SO(4) . . . . . . . . . . . . . . . . 263 9.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 263 xii Contents 9.1.2 Irreducible Representations of SO(4) . . . . . . . . . . . . 264 9.1.3 Hyperspherical Harmonics and Spectral Decomposition of ΔS3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 9.1.4 The Coherent States for S3 . . . . . . . . . . . . . . . . . 268 9.2 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . 271 9.2.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . 271 9.2.2 The Fock Transformation: A Map from L2(S3) to the Pure-Point Subspace of Hˆ . . . . . . . . . . . . . . 273 9.3 The Coherent States of the Hydrogen Atom . . . . . . . . . . . . . 277 10 Bosonic Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . 285 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 10.2 Fock Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 10.2.1 Bosons andFermions . . . . . . . . . . . . . . . . . . . . 286 10.2.2 Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 10.3 The Bosons Coherent States . . . . . . . . . . . . . . . . . . . . . 292 10.4 TheClassicalLimit forLargeSystemsofBosons . . . . . . . . . . 296 10.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 296 10.4.2 Hepp’s Method . . . . . . . . . . . . . . . . . . . . . . . . 297 10.4.3 Remainder Estimates in the Hepp Method . . . . . . . . . 303 10.4.4 Time Evolution of Coherent States . . . . . . . . . . . . . 306 11 Fermionic Coherent States . . . . . . . . . . . . . . . . . . . . . . . . 311 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 11.2 From Fermionic Fock Spaces to Grassmann Algebras . . . . . . . 312 11.3 Integration on Grassmann Algebra . . . . . . . . . . . . . . . . . 315 11.3.1 More Properties on Grassmann Algebras . . . . . . . . . . 315 11.3.2 CalculuswithGrassmannNumbers . . . . . . . . . . . . . 317 11.3.3 Gaussian Integrals . . . . . . . . . . . . . . . . . . . . . . 318 11.4 Super-Hilbert Spaces and Operators . . . . . . . . . . . . . . . . . 320 11.4.1 A Space for Fermionic States . . . . . . . . . . . . . . . . 320 11.4.2 IntegralKernels . . . . . . . . . . . . . . . . . . . . . . . 322 11.4.3 AFourierTransform . . . . . . . . . . . . . . . . . . . . . 323 11.5 Coherent States for Fermions . . . . . . . . . . . . . . . . . . . . 324 11.5.1 WeylTranslations . . . . . . . . . . . . . . . . . . . . . . 324 11.5.2 Fermionic Coherent States . . . . . . . . . . . . . . . . . . 325 11.6 RepresentationsofOperators . . . . . . . . . . . . . . . . . . . . 327 11.6.1 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 11.6.2 Representation by Translations and Weyl Quantization . . . 331 11.6.3 Wigner–Weyl Functions . . . . . . . . . . . . . . . . . . . 334 11.6.4 The Moyal Product for Fermions . . . . . . . . . . . . . . 339 11.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 11.7.1 The Fermi Oscillator . . . . . . . . . . . . . . . . . . . . . 341 11.7.2 TheFermi–DiracStatistics . . . . . . . . . . . . . . . . . 342 11.7.3 Quadratic Hamiltonians and Coherent States . . . . . . . . 343 11.7.4 More on Quadratic Propagators . . . . . . . . . . . . . . . 348 Contents xiii 12 Supercoherent States—An Introduction . . . . . . . . . . . . . . . . 353 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 12.2 Quantum Supersymmetry . . . . . . . . . . . . . . . . . . . . . . 354 12.3 Classical Superspaces . . . . . . . . . . . . . . . . . . . . . . . . 356 12.3.1 Morphisms and Spaces . . . . . . . . . . . . . . . . . . . 356 12.3.2 Superalgebra Notions . . . . . . . . . . . . . . . . . . . . 357 12.3.3 ExamplesofMorphisms . . . . . . . . . . . . . . . . . . . 358 12.4 Super-Lie Algebras and Groups . . . . . . . . . . . . . . . . . . . 359 12.4.1 Super-Lie Algebras . . . . . . . . . . . . . . . . . . . . . 359 12.4.2 Supermanifolds, a Very Brief Presentation . . . . . . . . . 361 12.4.3 Super-Lie Groups . . . . . . . . . . . . . . . . . . . . . . 362 12.5 Classical Supersymmetry . . . . . . . . . . . . . . . . . . . . . . 366 12.5.1 A Short Overview of Classical Mechanics . . . . . . . . . 366 12.5.2 Supersymmetric Mechanics . . . . . . . . . . . . . . . . . 369 12.5.3 Supersymmetric Quantization . . . . . . . . . . . . . . . . 374 12.6 Supercoherent States . . . . . . . . . . . . . . . . . . . . . . . . . 376 12.7 Phase Space Representations of Super Operators . . . . . . . . . . 378 12.8 Application to the Dicke Model . . . . . . . . . . . . . . . . . . . 379 Appendix A Tools for Integral Computations . . . . . . . . . . . . . . . 383 A.1 Fourier Transform of Gaussian Functions . . . . . . . . . . . . . . 383 A.2 Sketch of Proof for Theorem 29 . . . . . . . . . . . . . . . . . . . 383 A.3 A Determinant Computation . . . . . . . . . . . . . . . . . . . . . 384 A.4 The Saddle Point Method . . . . . . . . . . . . . . . . . . . . . . 387 A.4.1 TheOneRealVariableCase . . . . . . . . . . . . . . . . . 387 A.4.2 TheComplexVariablesCase . . . . . . . . . . . . . . . . 387 A.5 KählerGeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 Appendix B Lie Groups and Coherent States . . . . . . . . . . . . . . . 391 B.1 Lie Groups and Coherent States . . . . . . . . . . . . . . . . . . . 391 B.2 On Lie Groups and Lie Algebras . . . . . . . . . . . . . . . . . . 391 B.2.1 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 391 B.2.2 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 392 B.3 Representations of Lie Groups . . . . . . . . . . . . . . . . . . . . 395 B.3.1 General Properties of Representations . . . . . . . . . . . . 395 B.3.2 The Compact Case . . . . . . . . . . . . . . . . . . . . . . 397 B.3.3 The Non-compact Case . . . . . . . . . . . . . . . . . . . 398 B.4 Coherent States According Gilmore–Perelomov . . . . . . . . . . 398 Appendix C Berezin Quantization and Coherent States . . . . . . . . . 401 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 |
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