The Role of Boundaries in Modern Physics

Abstract

Boundaries play an important role already in classical physics. For example, the main problem of electrostatics is to find the distribution of potential in the presence of conductors, and is equivalent to find solutions of the Laplace equation in a space with boundaries. In the quantization of Maxwell theory, the Casimir effect reflects the possibility to obtain finite values for differences in zero-point energies of the electromagnetic field. Thus, this chapter begins with a review of potential theory in electrostatics and of various boundary effects in quantum electrodynamics. The reader is then introduced to the singularity problem in classical cosmology, viewed as one of the main motivations for studying a quantum theory of the universe. After a review of the path-integral approach to quantum gravity, motivations and problems of the path-integral approach to quantum cosmology are presented. Last, the Hartle-Hawking and tunnelling proposals for the problem of boundary conditions in quantum cosmology are presented in some detail.