The C_2 heat-kernel coefficient in the presence of boundary discontinuities

We consider the heat-kernel on a manifold whose boundary is piecewise smooth. The set of independent geometrical quantities required to construct an expression for the contribution of the boundary discontinuities to the C_{2} heat-kernel coefficient is derived in the case of a scalar field with Dirichlet and Robin boundary conditions. The coefficient is then determined using conformal symmetry and evaluation on some specific manifolds. For the Robin case a perturbation technique is also developed and employed. The contributions to the smeared heat-kernel coefficient and cocycle function are calculated. Some incomplete results for spinor fields with mixed conditions are also presented.Comment: 25 pages, LaTe

Imaging Invasion: Micro-CT imaging of adamantinomatous craniopharyngioma highlights cell type specific spatial relationships of tissue invasion.

Tissue invasion and infiltration by brain tumours poses a clinical challenge, with destruction of structures leading to morbidity. We assessed whether micro-CT could be used to map tumour invasion in adamantinomatous craniopharyngioma (ACP), and whether it could delineate ACPs and their intrinsic components from surrounding tissue.Three anonymised archival frozen ACP samples were fixed, iodinated and imaged using a micro-CT scanner prior to the use of standard histological processing and immunohistochemical techniques.We demonstrate that micro-CT imaging can non-destructively give detailed 3D structural information of tumours in volumes with isotropic voxel sizes of 4-6 microns, which can be correlated with traditional histology and immunohistochemistry.Such information complements classical histology by facilitating virtual slicing of the tissue in any plane and providing unique detail of the three dimensional relationships of tissue compartments

Hyperspherical entanglement entropy

The coefficient of the log term in the entanglement entropy associated with hyperspherical surfaces in flat space-time is shown to equal the conformal anomaly by conformally transforming Euclideanised space--time to a sphere and using already existing formulae for the relevant heat--kernel coefficients after cyclic factoring. The analytical reason for the result is that the conformal anomaly on the lune has an extremum at the ordinary sphere limit. A proof is given. Agreement with a recent evaluation of the coefficient is found.Comment: 7 pages. Final revision. Historical comments amended. Minor remarks adde

Heat Kernel Expansion for Semitransparent Boundaries

We study the heat kernel for an operator of Laplace type with a $\delta$-function potential concentrated on a closed surface. We derive the general form of the small $t$ asymptotics and calculate explicitly several first heat kernel coefficients.Comment: 16 page

Whightman function and scalar Casimir densities for a wedge with a cylindrical boundary

Whightman function, vacuum expectation values of the field square, and the energy-momentum tensor are investigated for a scalar field inside a wedge with and without a coaxial cylindrical boundary. Dirichlet boundary conditions are assumed on the bounding surfaces. The vacuum energy-momentum tensor is evaluated in the general case of the curvature coupling parameter. Making use of a variant of the generalized Abel-Plana formula, expectation values are presented as the sum of two terms. The first one corresponds to the geometry without a cylindrical boundary and the second one is induced by the presence of this boundary. The asymptotic behaviour of the field square, vacuum energy density and stresses near the boundaries are investigated. The additional vacuum forces acting on the wedge sides due the presence of the cylindrical boundary are evaluated and it is shown that these forces are attractive. As a limiting case, the geometry of two parallel plates perpendicularly intersected by a third one is analyzed.Comment: 19 pages, 6 figures, new section is added on the VEVs for the region outside the cylidrical shell, discussion and references added, accepted for publication in J. Phys.

Casimir energy in the MIT bag model

The vacuum energies corresponding to massive Dirac fields with the boundary conditions of the MIT bag model are obtained. The calculations are done with the fields occupying the regions inside and outside the bag, separately. The renormalization procedure for each of the situations is studied in detail, in particular the differences occurring with respect to the case when the field extends over the whole space. The final result contains several constants undergoing renormalization, which can be determined only experimentally. The non-trivial finite parts which appear in the massive case are found exactly, providing a precise determination of the complete, renormalized zero-point energy for the first time, in the fermionic case. The vacuum energy behaves like inverse powers of the mass for large masses.Comment: 19 pages, Latex, 1 Postscript figure, submitted to J. Phys.

The hybrid spectral problem and Robin boundary conditions

The hybrid spectral problem where the field satisfies Dirichlet conditions (D) on part of the boundary of the relevant domain and Neumann (N) on the remainder is discussed in simple terms. A conjecture for the C_1 coefficient is presented and the conformal determinant on a 2-disc, where the D and N regions are semi-circles, is derived. Comments on higher coefficients are made. A hemisphere hybrid problem is introduced that involves Robin boundary conditions and leads to logarithmic terms in the heat--kernel expansion which are evaluated explicitly.Comment: 24 pages. Typos and a few factors corrected. Minor comments added. Substantial Robin additions. Substantial revisio

Further functional determinants

Functional determinants for the scalar Laplacian on spherical caps and slices, flat balls, shells and generalised cylinders are evaluated in two, three and four dimensions using conformal techniques. Both Dirichlet and Robin boundary conditions are allowed for. Some effects of non-smooth boundaries are discussed; in particular the 3-hemiball and the 3-hemishell are considered. The edge and vertex contributions to the $C_{3/2}$ coefficient are examined.Comment: 25 p,JyTex,5 figs. on request