![]() The matter density and its first two derivatives are continuous across the interface. stars refer to the vacuum polarization around spherically symmetric objects, and divergences show up for sharp interfaces, where the matter density is discontinuous. ⟨ ϕ 2 ( z ) ⟩ r e n and ⟨ T μ ν ( z ) ⟩ r e nįor a free quantum field in a curved background, both quantities diverge at the points where the background is not sufficiently smooth. We stress at this point that the results obtained in this section have their gravitational counterparts:Īs shown in Ref. We will provide some examples to illustrate the appearance of divergences for sharp interfaces. We will show that, for smooth σ, the model is renormalizable using the standard renormalization procedure for quantum fields in curved spacetimes. We will consider a scalar vacuum field in curved spacetimes, coupled to a classical field σ that models the “mirror”. The aim of the present paper is to provide suchĪnalysis. The divergences associated to the unphysical limits will reappear when consideringĭiscontinuous classical fields. Replace the boundary conditions by interactions with a classical field, and show not only that the matter sector of the theory is renormalizable, but also that the usual divergences in the energy-momentum tensor can be absorbed in the coupling constants of the gravitational sector, resulting inįinite and well defined semiclassical Einstein equations when the classical field is sufficiently Performed in the context of quantum field theory in curved spacetimes. Ī complete analysis of the Casimir self-energies and their eventual gravitational implications must be For a discussion of some aspects of the coupling of Near a potential barrier is finite for sufficiently smooth potentials. Recently, Milton Milton and Bouas et al Bouas considered a similar problem, computing the energyĭensity for a scalar vacuum field in some particular potentials (“soft walls”), showing that the energy density Specific interaction between the high energy modes of the quantum field and the microscopic degrees ofįreedom of the body. When these limits are taken, the self-energy depends on the ultraviolet cutoff, i.e. Σ → ∞ at a particular point, imposing Dirichlet boundary conditions on the vacuum field at that Limit corresponds to consider a discontinuous σ, and the “perfect conductor” limit to take In those works, the authors considered a toy model consisting of aĪ background (classical field) σ, with an interaction of the form ϕ 2 σ. It was pointed out that, if the boundary conditions are replaced by an interaction with a second field, the model is renormalizable and the The problem of the divergences in the self-energies has been considered in flat spacetime in a series of works by Graham et al Since the transition region between media becomes larger than the wavelength of the high Modes with extremely small wavelengths, it is unphysical to assume a sharp boundary, In this case the origin of the divergences is that, for More generally, divergences in the renormalized energy-momentum-tensor are presentĮven for non-perfect conductors, as long as there is a sharp boundary between two media with differentĮlectromagnetic properties. One expects any material to become transparent at high energies. The origin of the divergences is the unphysicalĪssumption of perfect conductivity for all modes of the electromagnetic field since, on physical grounds, These divergences are not the usual ones in quantum field theory,īecause they are present in the already renormalized energy-momentum tensor. In the case of perfect conductors, the vacuum energy density, or more generally the energy-momentum tensor, diverges near the boundaries, as notedįor the first time by Deutsch and Candelas a long time ago DC.
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