Bicyclic Semigroup (Monoid)

Abstract

Although Einstein's general relativity leads to a theory of quantum gravity which is not perturbatively renormalizable, the analysis of the semiclassical approximation remains of crucial importance to test the internal consistency of any theory of quantum gravity. For this purpose, it is necessary to achieve a thorough understanding of the problem of boundary conditions in the theory of quantized fields. Indeed, the path-integral representation of the propagator, the general theory of the effective action, and the recent attempts to define a quantum state of the universe [1], provide three relevant examples where the appropriate formulation of boundary conditions plays a crucial role to obtain a well defined model of some properties of quantum gravity.For gauge fields and gravitation, one may reduce the theory to its physical degrees of freedom by imposing a gauge condition before quantization, or one may use the Faddeev-Popov formalism for quantum amplitudes, or the extended-phase-space Hamiltonian formalism of Batalin, Fradkin and Vilkovisky. Moreover, a powerful non-diagrammatic method to perform the one-loop analysis is the one which relies on ζ-function regularization. This is a naturally occurring technique, since semiclassical amplitudes involve definition and calculation of determinants of elliptic, self-adjoint differential operators. Once these choices are made, there are still many problems which deserve a careful consideration. They are as follows.(i) Choice of background four-geometry. This may be flat Euclidean fourspace, which is relevant for massless theories, or the de Sitter four-sphere, which is relevant for inflationary cosmology, or more general curved backgrounds. The latter appear interesting for a better understanding of quantum field theory in curved space-time.(ii) Choice of boundary three-geometry. This may consist of two threesurfaces (e.g. two concentric three-spheres), motivated by quantum field theory, or just one three-surface (e.g. one three-sphere), motivated by quantum cosmology [1], or more complicated examples of boundary three-geometries.(iii) Choice of gauge-averaging functional. For example, one may study Lorenz or Coulomb gauge for Euclidean Maxwell theory, or de Donder or axial gauge for gravitation, or non-covariant gauges which take explicitly into account 1