7,505 research outputs found
Diffeomorphism invariant eigenvalue problem for metric perturbations in a bounded region
We suggest a method of construction of general diffeomorphism invariant
boundary conditions for metric fluctuations. The case of dimensional
Euclidean disk is studied in detail. The eigenvalue problem for the Laplace
operator on metric perturbations is reduced to that on -dimensional vector,
tensor and scalar fields. Explicit form of the eigenfunctions of the Laplace
operator is derived. We also study restrictions on boundary conditions which
are imposed by hermiticity of the Laplace operator.Comment: LATeX file, no figures, no special macro
Spectral Action for Robertson-Walker metrics
We use the Euler-Maclaurin formula and the Feynman-Kac formula to extend our
previous method of computation of the spectral action based on the Poisson
summation formula. We show how to compute directly the spectral action for the
general case of Robertson-Walker metrics. We check the terms of the expansion
up to a_6 against the known universal formulas of Gilkey and compute the
expansion up to a_{10} using our direct method
A note on a gauge-gravity relation and functional determinants
We present a refinement of a recently found gauge-gravity relation between
one-loop effective actions: on the gauge side, for a massive charged scalar in
2d dimensions in a constant maximally symmetric electromagnetic field; on the
gravity side, for a massive spinor in d-dimensional (Euclidean) anti-de Sitter
space. The inclusion of the dimensionally regularized volume of AdS leads to
complete mapping within dimensional regularization. In even-dimensional AdS, we
get a small correction to the original proposal; whereas in odd-dimensional
AdS, the mapping is totally new and subtle, with the `holographic trace
anomaly' playing a crucial role.Comment: 6 pages, io
Effective action and heat kernel in a toy model of brane-induced gravity
We apply a recently suggested technique of the Neumann-Dirichlet reduction to
a toy model of brane-induced gravity for the calculation of its quantum
one-loop effective action. This model is represented by a massive scalar field
in the -dimensional flat bulk supplied with the -dimensional kinetic
term localized on a flat brane and mimicking the brane Einstein term of the
Dvali-Gabadadze-Porrati (DGP) model. We obtain the inverse mass expansion of
the effective action and its ultraviolet divergences which turn out to be
non-vanishing for both even and odd spacetime dimensionality . For the
massless case, which corresponds to a limit of the toy DGP model, we obtain the
Coleman-Weinberg type effective potential of the system. We also obtain the
proper time expansion of the heat kernel in this model associated with the
generalized Neumann boundary conditions containing second order tangential
derivatives. We show that in addition to the usual integer and half-integer
powers of the proper time this expansion exhibits, depending on the dimension
, either logarithmic terms or powers multiple of one quarter. This property
is considered in the context of strong ellipticity of the boundary value
problem, which can be violated when the Euclidean action of the theory is not
positive definite.Comment: LaTeX, 20 pages, new references added, typos correcte
Heat Kernel Expansion for Semitransparent Boundaries
We study the heat kernel for an operator of Laplace type with a
-function potential concentrated on a closed surface. We derive the
general form of the small asymptotics and calculate explicitly several
first heat kernel coefficients.Comment: 16 page
Phase transition in a static granular system
We find that a column of glass beads exhibits a well-defined transition
between two phases that differ in their resistance to shear. Pulses of
fluidization are used to prepare static states with well-defined particle
volume fractions in the range 0.57-0.63. The resistance to shear is
determined by slowly inserting a rod into the column of beads. The transition
occurs at for a range of speeds of the rod.Comment: 4 pages, 4 figures. The paper is significantly extended, including
new dat
Prolongations of Geometric Overdetermined Systems
We show that a wide class of geometrically defined overdetermined semilinear
partial differential equations may be explicitly prolonged to obtain closed
systems. As a consequence, in the case of linear equations we extract sharp
bounds on the dimension of the solution space.Comment: 22 pages. In the second version, a comparison with the classical
theory of prolongations was added. In this third version more details were
added concerning our construction and especially the use of Kostant's
computation of Lie algebra cohomolog
Boundary dynamics and multiple reflection expansion for Robin boundary conditions
In the presence of a boundary interaction, Neumann boundary conditions should
be modified to contain a function S of the boundary fields: (\nabla_N +S)\phi
=0. Information on quantum boundary dynamics is then encoded in the
-dependent part of the effective action. In the present paper we extend the
multiple reflection expansion method to the Robin boundary conditions mentioned
above, and calculate the heat kernel and the effective action (i) for constant
S, (ii) to the order S^2 with an arbitrary number of tangential derivatives.
Some applications to symmetry breaking effects, tachyon condensation and brane
world are briefly discussed.Comment: latex, 22 pages, no figure
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 coefficient are examined.Comment: 25 p,JyTex,5 figs. on request
Heat-kernel coefficients for oblique boundary conditions
We calculate the heat-kernel coefficients, up to , for a U(1) bundle on
the 4-Ball for boundary conditions which are such that the normal derivative of
the field at the boundary is related to a first-order operator in boundary
derivatives acting on the field. The results are used to place restrictions on
the general forms of the coefficients. In the specific case considered, there
can be a breakdown of ellipticity.Comment: 9 pages, JyTeX. One reference added and minor corrections mad
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