## Geometric QuantizationThe geometric approach to quantization was introduced by Konstant and Souriau more than 20 years ago. It has given valuable and lasting insights into the relationship between classical and quantum systems, and continues to be a popular research topic. The ideas have proved useful in pure mathematics, notably in representation theory, as well as in theoretical physics. The most recent applications have been in conformal field theory and in the Jones-Witten theory of knots. The successful original edition of this book was published in 1980. Now it has been completely revised and extensively rewritten. The presentation has been simplified and many new examples have been added. The material on field theory has been expanded. |

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### Contents

SYMPLECTIC GEOMETRY | 1 |

LAGRANGIAN AND HAMILTONIAN MECHANICS | 16 |

SYMMETRY | 38 |

HAMILTONJACOBI THEORY | 60 |

COMPLEX POLARIZATIONS | 89 |

ELEMENTARY RELATIVISTIC SYSTEMS | 107 |

CLASSICAL FIELDS | 130 |

PREQUANTIZATION | 155 |

THE METAPLECTIC CORRECTION | 218 |

APPENDIX | 257 |

Vector bundles | 263 |

Distributions and foliations | 269 |

Stationary phase | 276 |

291 | |

301 | |

QUANTIZATION | 171 |

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### Common terms and phrases

action canonical coordinates canonical transformations Cauchy surface classical coadjoint cocycle coisotropic compact complex structure components condition conjugate constraint construction coordinate system corresponding cotangent bundle curvature defined denote derivative determined dimension equivalent example fibres field equations Fock space foliation follows given Guillemin and Sternberg half,form Hamilton,Jacobi equation Hamiltonian vector fields hence Hilbert space holomorphic inner product integral curves invariant isotropic Kahler Lagrangian subspace Lie algebra line bundle linear metaplectic metric momentum neighbourhood nonnegative orbits pairing particle phase space polarized sections positive prequantum bundle projection Proposition pull,back quantization quantum real polarization reduction representation right,hand side satisfies scalar simply connected solutions space,time spanned spinor square root symmetric symplectic form symplectic frame symplectic manifold symplectic potential symplectic structure symplectic vector space tensor theory time,dependent transition functions transverse trivial unitary values vanishes vector bundle wave functions