1. Quantum Gravity, Quantum Cosmology and Classical Gravity -- 2. Canonical Quantum Gravity -- 3. Perturbative Quantum Gravity -- 4. Global Boundary Conditions and ?(0) Value for the Massless Spin-1/2 Field -- 5. Choice of Boundary Conditions in One-Loop Quantum Cosmology -- 6. Ghost Fields and Gauge Modes in One-Loop Quantum Cosmology -- 7. Local Boundary Conditions for the Weyl Spinor -- 8. One-Loop Results for the Spin-1/2 Field with Local Boundary Conditions -- 9. Local Supersymmetry in Perturbative Quantum Cosmology -- 10. Lorentzian Geometry, U4 Theories and Singularities in Cosmology -- 11. Conclusions -- Problems for the Reader -- Appendices -- Appendix A: Two-Component Spinor Calculus and Its Applications -- Appendix B: The Generalized Zeta-Function -- Appendix C: Euler-Maclaurin Formula and Free Part of the Heat Kernel for the Spin-3/2 Field -- Appendix D: Complex Manifolds -- Appendix E: Lorentzian ADM Formulae for the Curvature -- Appendix F: ?(0) Calculations -- References.

This book is aimed at theoretical and mathematical physicists and mathematicians interested in modern gravitational physics. I have thus tried to use language familiar to readers working on classical and quantum gravity, paying attention both to difficult calculations and to existence theorems, and discussing in detail the current literature. The first aim of the book is to describe recent work on the problern of boundary conditions in one-loop quantum cosmology. The motivation of this research was to under­ stand whether supersymmetric theories are one-loop finite in the presence of boundaries, with application to the boundary-value problems occurring in quantum cosmology. Indeed, higher-loop calculations in the absence of boundaries are already available in the litera­ ture, showing that supergravity is not finite. I believe, however, that one-loop calculations in the presence of boundaries are more fundamental, in that they provide a more direct check of the inconsistency of Supersymmetrie quantum cosmology from the perturbative point of view. It therefore appears that higher-order calculations are not strictly needed, if the one-loop test already yields negative results. Even though the question is not yet settled, this research has led to many interesting, new applications of areas of theoretical and mathematical physics such as twistor theory in flat space, self-adjointness theory, the generalized lliemann zeta-function, and the theory of boundary counterterms in super­ gravity.