## Euclidean Quantum GravityThe Euclidean approach to Quantum Gravity was initiated almost 15 years ago in an attempt to understand the difficulties raised by the spacetime singularities of classical general relativity which arise in the gravitational collapse of stars to form black holes and the entire universe in the Big Bang. An important motivation was to develop an approach capable of dealing with the nonlinear, non-perturbative aspects of quantum gravity due to topologically non-trivial spacetimes. There are important links with a Riemannian geometry. Since its inception the theory has been applied to a number of important physical problems including the thermodynamic properties of black holes, quantum cosmology and the problem of the cosmological constant. It is currently at the centre of a great deal of interest.This is a collection of survey lectures and reprints of some important lectures on the Euclidean approach to quantum gravity in which one expresses the Feynman path integral as a sum over Riemannian metrics. As well as papers on the basic formalism there are sections on Black Holes, Quantum Cosmology, Wormholes and Gravitational Instantons. |

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### Contents

GENERAL FORMALISM | x |

OneLoop Divergencies in the Theory of Gravitation | 69 |

The PathIntegral Approach to Quantum Gravity | 69 |

Path Integrals and the Indefiniteness of the Gravitational Action | 102 |

Proof of the Positive Action Conjecture in Quantum Relativity | 112 |

The Conformal Rotation in Perturbative Gravity | 143 |

Quantum Tunneling and Negative Eigenvalues | 152 |

A Relation Between Volume Mean Curvature and Diameter | 161 |

Gravitational Effects on and of Vacuum Decay | 295 |

Supercooled Phase Transitions in the Very Early Universe | 306 |

The Quantum State of the Universe | 329 |

Origin of Structure in the Universe | 347 |

Wormholes in Spacetime | 366 |

A Theory | 388 |

Wormholes and the Cosmological Constant | 414 |

Wormholes in Spacetime and the Constants of Nature | 442 |

Particle Creation by Black Holes | 167 |

Black Holes and Thermal Green Functions | 205 |

Action Integrals and Partition Functions in Quantum Gravity | 233 |

Instability of Flat Space at Finite Temperature | 257 |

Thermal Stress Tensor in Static Einstein Spaces | 274 |

Cosmological Event Horizons Thermodynamics and Particle Creation | 281 |

Asymptotically Flat SelfDual Solutions to Euclidean Gravity | 497 |

The Positive Action Conjecture and Asymptotically Euclidean Metrics | 503 |

Polygons and Gravitons | 531 |

The Construction of ALE Spaces as HyperKahler Quotients | 539 |

A Compact Rotating Gravitational Instanton | 558 |

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amplitude analytic continuation asymptotically Euclidean asymptotically flat baby universe background metric black hole boundary conditions calculate classical closed universes Coleman's compact configurations contribution coordinates corresponding cosmological constant curvature defined density derivatives effects eigenvalues Einstein equations energy Euclidean action Euclidean functional integrals euclidean path integral Euclidean space evaluate event horizon field equations finite flat space fluctuations functional integral gauge geometry given gravitational field gravitational instantons gravitons ground-state wave function Hamiltonian infinity instantons interactions invariant Killing vector Lagrangian Lett Lorentzian manifold mass matter fields minisuperspace modes negative observer obtained operator parameter particle partition function path integral perturbation Phys physical Planck propagator quantum gravity radius region renormalization result rotation S.W. Hawking scalar field scale Schwarzschild self-dual singularity Sitter space solution spacelike spacetime stationary surface symmetry temperature tensor thermal tion topology transformation vacuum vanishes variables wave function Wheeler-DeWitt equation wormhole zero