A general method is known to exist for studying Abelian and non-Abelian gauge theories, as well as Euclidean quantum gravity, at one-loop level on manifolds with boundary. In the latter case, boundary conditions on metric perturbations h can be chosen to be completely invariant under infinitesimal diffeomorphisms, to preserve the invariance group of the theory and BRST symmetry.In the de Donder gauge, however, the resulting boundary-value problem for the Laplace type operator acting on h is known to be self-adjoint but not strongly elliptic. The latter is a technical condition ensuring that a unique smooth solution of the boundary-value problem exists, which implies, in turn, that the global heat-kernel asymptotics yielding one-loop divergences and oneloop effective action actually exists. The present paper shows that, on the Euclidean four-ball, only the scalar part of perturbative modes for quantum gravity are affected by the lack of strong ellipticity. Further evidence for lack of strong ellipticity, from an analytic point of view, is therefore obtained. Interestingly, three sectors of the scalar-perturbation problem remain elliptic, while lack of strong ellipticity is "confined" to the remaining fourth sector. The integral representation of the resulting ζ-function asymptotics is also obtained; this remains regular at the origin by virtue of a spectral identity here obtained for the first time.

In this paper we utilize ζ-function regularization techniques in order to compute the Casimir force for massless scalar fields subject to Dirichlet and Neumann boundary conditions in the setting of the conical piston. The piston geometry is obtained by dividing the bounded generalized cone into two regions separated by its cross section positioned at a with a ∈ (0, b) with b > 0. We obtain expressions for the Casimir force that are valid in any dimension for both Dirichlet and Neumann boundary conditions in terms of the spectral ζ-function of the piston. As a particular case, we specify the piston to be a d-dimensional sphere and present explicit results for d = 2, 3, 4, 5. I. INTRODUCTIONThe Casimir effect is one of the most important macroscopic manifestations of the zero point energy of quantized fields under the influence of external conditions [11,38] or in spaces with non-trivial topology. In recent years, a vast amount of literature has been produced on the Casimir effect, which was first predicted in the seminal paper [13], especially for its relevance in nanoscale physics [10,11,38]. Due to its nature, calculations of the vacuum energy lead to divergencies which need to be regularized and subsequently renormalized. Several regularization methods exist, amongst the most important ones are frequency cutoff, point splitting and zeta function regularization [6,10,12,22,23,38]. For many configurations, these techniques yield the same finite renormalized result, however the way divergencies are removed is different in each scheme. The non-uniqueness of the removal procedure raises the question, which of them is the physically best motivated one. Technical and interpretational problems of this nature can actually be avoided if one considers the Casimir effect between separate objects. In this case, the divergent part of the energy (for massless fields) depends on the heat kernel coefficient a D/2 related to the geometry of the objects.These coefficients, in turn, do not depend on the distance between the bodies and, hence, the Casimir force between them is free of divergencies [10]. Belonging to the class of configurations for which the Casimir force has been unambiguously evaluated are pistons of certain types.These piston configurations, introduced in [14], have become increasingly important because of this fact. A large variety of piston configurations and boundary conditions have been studied throughout the lit- *

Recent work on Euclidean quantum gravity on the four-ball has proved regularity at the origin of the generalized ζ-function built from eigenvalues for metric and ghost modes, when diffeomorphism-invariant boundary conditions are imposed in the de Donder gauge. The hardest part of the analysis involves one of the four sectors for scalar-type perturbations, the eigenvalues of which are obtained by squaring up roots of a linear combination of Bessel functions of integer adjacent orders, with a coefficient of linear combination depending on the unknown roots. This paper obtains, first, approximate analytic formulae for such roots for all values of the order of Bessel functions. For this purpose, both the descending series for Bessel functions and their uniform asymptotic expansion at large order are used. The resulting generalized ζfunction is also built, and another check of regularity at the origin is obtained. For the first time in the literature on quantum gravity on manifolds with boundary, a vanishing one-loop wave function of the Universe is found in the limit of small threegeometry, which suggests a quantum avoidance of the cosmological singularity driven by full diffeomorphism invariance of the boundary-value problem for one-loop quantum theory.

In this work we examine a system consisting of a confined one-dimensional arrangement of atoms that we describe by using the 2-dimensional CP N −1 model, restricted to an interval and at finite temperature. We develop a method to obtain the bulk and boundary parts of the one-loop effective action as a function of the effective mass of the fluctuations. The formalism has the advantage of allowing for a systematic analysis of a large class of boundary conditions and to model the (adiabatic) response of the ground state to changes in the boundary conditions. In the case of periodic boundary conditions, we find that inhomogeneous phases are disfavored for intervals of large size. Away from periodic boundary conditions, our numerical results show that the ground state has a generic crystal-like structure that can be modulated by variations of the boundary conditions. The results presented here could be relevant for experimental implementations of nonlinear sigma models and could be tested by lattice numerical simulations.

In this work we analyze the Casimir energy and force for a scalar field endowed with general self-adjoint boundary conditions propagating in a higher dimensional piston configuration. The piston is constructed as a direct product I × N , with I = [0, L] ⊂ R and N a smooth, compact Riemannian manifold with or without boundary. The study of the Casimir energy and force for this configuration is performed by employing the spectral zeta function regularization technique. The obtained analytic results depend explicitly on the spectral zeta function associated with the manifold N and the parameters describing the general boundary conditions imposed. These results are then specialized to the case in which the manifold N is a d-dimensional sphere.