On the issue of imposing boundary conditions on quantum fields

An interesting example of the deep interrelation between Physics and Mathematics is obtained when trying to impose mathematical boundary conditions on physical quantum fields. This procedure has recently been re-examined with care. Comments on that and previous analysis are here provided, together with considerations on the results of the purely mathematical zeta-function method, in an attempt at clarifying the issue. Hadamard regularization is invoked in order to fill the gap between the infinities appearing in the QFT renormalized results and the finite values obtained in the literature with other procedures.Comment: 13 pages, no figure

The ground state energy of a massive scalar field in the background of a semi-transparent spherical shell

We calculate the zero point energy of a massive scalar field in the background of an infinitely thin spherical shell given by a potential of the delta function type. We use zeta functional regularization and express the regularized ground state energy in terms of the Jost function of the related scattering problem. Then we find the corresponding heat kernel coefficients and perform the renormalization, imposing the normalization condition that the ground state energy vanishes when the mass of the quantum field becomes large. Finally the ground state energy is calculated numerically. Corresponding plots are given for different values of the strength of the background potential, for both attractive and repulsive potentials.Comment: 15 pages, 5 figure

Casimir Effect for Spherical Shell in de Sitter Space

The Casimir stress on a spherical shell in de Sitter background for massless scalar field satisfying Dirichlet boundary conditions on the shell is calculated. The metric is written in conformally flat form. Although the metric is time dependent no particles are created. The Casimir stress is calculated for inside and outside of the shell with different backgrounds corresponding to different cosmological constants. The detail dynamics of the bubble depends on different parameter of the model. Specifically, bubbles with true vacuum inside expand if the difference in the vacuum energies is small, otherwise they collapse.Comment: 9 pages, submitted to Class. Quantum Gra

Explicit Zeta Functions for Bosonic and Fermionic Fields on a Noncommutative Toroidal Spacetime

Explicit formulas for the zeta functions $\zeta_\alpha (s)$ corresponding to bosonic ($\alpha =2$) and to fermionic ($\alpha =3$) quantum fields living on a noncommutative, partially toroidal spacetime are derived. Formulas for the most general case of the zeta function associated to a quadratic+linear+constant form (in {\bf Z}) are obtained. They provide the analytical continuation of the zeta functions in question to the whole complex $s-$plane, in terms of series of Bessel functions (of fast, exponential convergence), thus being extended Chowla-Selberg formulas. As well known, this is the most convenient expression that can be found for the analytical continuation of a zeta function, in particular, the residua of the poles and their finite parts are explicitly given there. An important novelty is the fact that simple poles show up at $s=0$, as well as in other places (simple or double, depending on the number of compactified, noncompactified, and noncommutative dimensions of the spacetime), where they had never appeared before. This poses a challenge to the zeta-function regularization procedure.Comment: 15 pages, no figures, LaTeX fil

Dynamical Casimir Effect with Semi-Transparent Mirrors, and Cosmology

After reviewing some essential features of the Casimir effect and, specifically, of its regularization by zeta function and Hadamard methods, we consider the dynamical Casimir effect (or Fulling-Davis theory), where related regularization problems appear, with a view to an experimental verification of this theory. We finish with a discussion of the possible contribution of vacuum fluctuations to dark energy, in a Casimir like fashion, that might involve the dynamical version.Comment: 11 pages, Talk given in the Workshop ``Quantum Field Theory under the Influence of External Conditions (QFEXT07)'', Leipzig (Germany), September 17 - 21, 200

Vacuum energy in the presence of a magnetic string with delta function profile

We present a calculation of the ground state energy of massive spinor fields and massive scalar fields in the background of an inhomogeneous magnetic string with potential given by a delta function. The zeta functional regularization is used and the lowest heat kernel coefficients are calculated. The rest of the analytical calculation adopts the Jost function formalism. In the numerical part of the work the renormalized vacuum energy as a function of the radius $R$ of the string is calculated and plotted for various values of the strength of the potential. The sign of the energy is found to change with the radius. For both scalar and spinor fields the renormalized energy shows no logarithmic behaviour in the limit $R\to 0$, as was expected from the vanishing of the heat kernel coefficient $A_2$, which is not zero for other types of profiles.Comment: 30 pages, 10 figure

Casimir effect for scalar fields with Robin boundary conditions in Schwarzschild background

The stress tensor of a massless scalar field satisfying Robin boundary conditions on two one-dimensional wall in two-dimensional Schwarzschild background is calculated. We show that vacuum expectation value of stress tensor can be obtained explicitly by Casimir effect, trace anomaly and Hawking radiation.Comment: 10 pages, no figure

Casimir stress on parallel plates in de Sitter space

The Casimir stress on two parallel plates in de Sitter background for massless scalar field satisfying Robin boundary conditions on the plates is calculated. The metric is written in conformally flat form to make maximum use of the Minkowski space calculations. Different cosmological constants are assumed for the space between and outside of the plates to have general results applicable to the case of domain wall formations in the early universe.Comment: 6 page

The renormalization group and spontaneous compactification of a higher-dimensional scalar field theory in curved spacetime

The renormalization group (RG) is used to study the asymptotically free $\phi_6^3$-theory in curved spacetime. Several forms of the RG equations for the effective potential are formulated. By solving these equations we obtain the one-loop effective potential as well as its explicit forms in the case of strong gravitational fields and strong scalar fields. Using zeta function techniques, the one-loop and corresponding RG improved vacuum energies are found for the Kaluza-Klein backgrounds $R^4\times S^1\times S^1$ and $R^4\times S^2$. They are given in terms of exponentially convergent series, appropriate for numerical calculations. A study of these vacuum energies as a function of compactification lengths and other couplings shows that spontaneous compactification can be qualitatively different when the RG improved energy is used.Comment: LaTeX, 15 pages, 4 figure

Heat-kernel expansion on non compact domains and a generalised zeta-function regularisation procedure

Heat-kernel expansion and zeta function regularisation are discussed for Laplace type operators with discrete spectrum in non compact domains. Since a general theory is lacking, the heat-kernel expansion is investigated by means of several examples. It is pointed out that for a class of exponential (analytic) interactions, generically the non-compactness of the domain gives rise to logarithmic terms in the heat-kernel expansion. Then, a meromorphic continuation of the associated zeta function is investigated. A simple model is considered, for which the analytic continuation of the zeta function is not regular at the origin, displaying a pole of higher order. For a physically meaningful evaluation of the related functional determinant, a generalised zeta function regularisation procedure is proposed.Comment: Latex, 14 pages, no figures. The version to be published in JM