Quantized Maxwell Theory in a Conformally Invariant Gauge

Maxwell theory can be studied in a gauge which is invariant under conformal rescalings of the metric, and first proposed by Eastwood and Singer. This paper studies the corresponding quantization in flat Euclidean 4-space. The resulting ghost operator is a fourth-order elliptic operator, while the operator P on perturbations of the potential is a sixth-order elliptic operator. The operator P may be reduced to a second-order non-minimal operator if a dimensionless gauge parameter tends to infinity. Gauge-invariant boundary conditions are obtained by setting to zero at the boundary the whole set of perturbations of the potential, jointly with ghost perturbations and their normal derivative. This is made possible by the fourth-order nature of the ghost operator. An analytic representation of the ghost basis functions is also obtained.Comment: 8 pages, plain Tex. In this revised version, the calculation of ghost basis functions has been amended, and the presentation has been improve

Essential self-adjointness in one-loop quantum cosmology

The quantization of closed cosmologies makes it necessary to study squared Dirac operators on closed intervals and the corresponding quantum amplitudes. This paper proves self-adjointness of these second-order elliptic operators.Comment: 14 pages, plain Tex. An Erratum has been added to the end, which corrects section

Boundary Operators in Quantum Field Theory

The fundamental laws of physics can be derived from the requirement of invariance under suitable classes of transformations on the one hand, and from the need for a well-posed mathematical theory on the other hand. As a part of this programme, the present paper shows under which conditions the introduction of pseudo-differential boundary operators in one-loop Euclidean quantum gravity is compatible both with their invariance under infinitesimal diffeomorphisms and with the requirement of a strongly elliptic theory. Suitable assumptions on the kernel of the boundary operator make it therefore possible to overcome problems resulting from the choice of purely local boundary conditions.Comment: 23 pages, plain Tex. The revised version contains a new section, and the presentation has been improve

Quantum Effects in Friedmann-Robertson-Walker Cosmologies

Electrodynamics for self-interacting scalar fields in spatially flat Friedmann-Robertson-Walker space-times is studied. The corresponding one-loop field equation for the expectation value of the complex scalar field in the conformal vacuum is derived. For exponentially expanding universes, the equations for the Bogoliubov coefficients describing the coupling of the scalar field to gravity are solved numerically. They yield a non-local correction to the Coleman-Weinberg effective potential which does not modify the pattern of minima found in static de Sitter space. Such a correction contains a dissipative term which, accounting for the decay of the classical configuration in scalar field quanta, may be relevant for the reheating stage. The physical meaning of the non-local term in the semiclassical field equation is investigated by evaluating this contribution for various background field configurations.Comment: 17 pages, plain TeX + 5 uuencoded figure

Non-Local Boundary Conditions in Euclidean Quantum Gravity

Non-local boundary conditions for Euclidean quantum gravity are proposed, consisting of an integro-differential boundary operator acting on metric perturbations. In this case, the operator P on metric perturbations is of Laplace type, subject to non-local boundary conditions; by contrast, its adjoint is the sum of a Laplacian and of a singular Green operator, subject to local boundary conditions. Self-adjointness of the boundary-value problem is correctly formulated by looking at Dirichlet-type and Neumann-type realizations of the operator P, following recent results in the literature. The set of non-local boundary conditions for perturbative modes of the gravitational field is written in general form on the Euclidean four-ball. For a particular choice of the non-local boundary operator, explicit formulae for the boundary-value problem are obtained in terms of a finite number of unknown functions, but subject to some consistency conditions. Among the related issues, the problem arises of whether non-local symmetries exist in Euclidean quantum gravity.Comment: 23 pages, plain Tex. The revised version is much longer, and new original calculations are presented in section

Euclidean Maxwell Theory in the Presence of Boundaries. II

Zeta-function regularization is applied to complete a recent analysis of the quantized electromagnetic field in the presence of boundaries. The quantum theory is studied by setting to zero on the boundary the magnetic field, the gauge-averaging functional and hence the Faddeev-Popov ghost field. Electric boundary conditions are also studied. On considering two gauge functionals which involve covariant derivatives of the 4-vector potential, a series of detailed calculations shows that, in the case of flat Euclidean 4-space bounded by two concentric 3-spheres, one-loop quantum amplitudes are gauge independent and their mode-by-mode evaluation agrees with the covariant formulae for such amplitudes and coincides for magnetic or electric boundary conditions. By contrast, if a single 3-sphere boundary is studied, one finds some inconsistencies, i.e. gauge dependence of the amplitudes.Comment: 24 pages, plain-tex, recently appearing in Classical and Quantum Gravity, volume 11, pages 2939-2950, December 1994. The authors apologize for the delay in circulating the file, due to technical problems now fixe

Non-Locality and Ellipticity in a Gauge-Invariant Quantization

The quantum theory of a free particle in two dimensions with non-local boundary conditions on a circle is known to lead to surface and bulk states. Such a scheme is here generalized to the quantized Maxwell field, subject to mixed boundary conditions. If the Robin sector is modified by the addition of a pseudo-differential boundary operator, gauge-invariant boundary conditions are obtained at the price of dealing with gauge-field and ghost operators which become pseudo-differential. A good elliptic theory is then obtained if the kernel occurring in the boundary operator obeys certain summability conditions, and it leads to a peculiar form of the asymptotic expansion of the symbol. The cases of ghost operator of negative and positive order are studied within this framework.Comment: 17 pages, plain Te

One-Loop Divergences in Simple Supergravity: Boundary Effects

This paper studies the semiclassical approximation of simple supergravity in Riemannian four-manifolds with boundary, within the framework of $\zeta$-function regularization. The massless nature of gravitinos, jointly with the presence of a boundary and a local description in terms of potentials for spin ${3\over 2}$, force the background to be totally flat. First, nonlocal boundary conditions of the spectral type are imposed on spin-${3\over 2}$ potentials, jointly with boundary conditions on metric perturbations which are completely invariant under infinitesimal diffeomorphisms. The axial gauge-averaging functional is used, which is then sufficient to ensure self-adjointness. One thus finds that the contributions of ghost and gauge modes vanish separately. Hence the contributions to the one-loop wave function of the universe reduce to those $\zeta(0)$ values resulting from physical modes only. Another set of mixed boundary conditions, motivated instead by local supersymmetry and first proposed by Luckock, Moss and Poletti, is also analyzed. In this case the contributions of gauge and ghost modes do not cancel each other. Both sets of boundary conditions lead to a nonvanishing $\zeta(0)$ value, and spectral boundary conditions are also studied when two concentric three-sphere boundaries occur. These results seem to point out that simple supergravity is not even one-loop finite in the presence of boundaries.Comment: 37 pages, Revtex. Equations (5.2), (5.3), (5.5), (5.7), (5.8) and (5.13) have been amended, jointly with a few misprint