Quantization of Field Theories in the Presence of Boundaries

This paper reviews the progress made over the last five years in studying boundary conditions and semiclassical properties of quantum fields about 4-real-dimensional Riemannian backgrounds. For massless spin-$\frac{1}{2}$ fields one has a choice of spectral or supersymmetric boundary conditions, and the corresponding conformal anomalies have been evaluated by using zeta-function regularization. For Euclidean Maxwell theory in vacuum, the mode-by-mode analysis of BRST-covariant Faddeev-Popov amplitudes has been performed for relativistic and non-relativistic gauge conditions. For massless spin-$\frac{3}{2}$ fields, the contribution of physical degrees of freedom to one-loop amplitudes, and the 2-spinor analysis of Dirac and Rarita-Schwinger potentials, have been obtained. In linearized gravity, gauge modes and ghost modes in the de Donder gauge have been studied in detail. This program may lead to a deeper understanding of different quantization techniques for gauge fields and gravitation, to a new vision of gauge invariance, and to new points of view in twistor theory.