The Asymptotics of The Laplacian on a Manifold with Boundary

Branson, Thomas; Gilkey, Peter Β.

doi:10.1080/03605309908820686

Cited by 266 publications

(418 citation statements)

References 13 publications

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“…The coefficients for Dirichlet and Neumann boundary conditions can be found in [75]. For a general surface the coefficients are given in [76][77][78][79][80]. In [81] the coefficients for a d-dimensional sphere with Dirichlet and Robin boundary conditions are given up to n = 10.…”

Section: The Heat Kernel Expansionmentioning

confidence: 99%

“…It is possible, however, to obtain a much more simple system in the case when the boundary undergoes small harmonic oscillations under the condition of a parametric resonance. Let us consider, following [61], the motion of the boundary according to the law 76) where ω 1 = cπ/a 0 , and the non-dimensional amplitude of the oscillations is ǫ << 1 (in realistic situations ǫ ∼ 10 −7 ). In the framework of the theory of the parametrically excited systems [62] the coefficients α nm , β nm can be considered as slowly varying functions of time.…”

Section: Moving Boundaries In a Two-dimensional Space-timementioning

confidence: 99%

“…Using a Mellin transform we obtain from Eqs. (3.74) and (3.75) 76) which is an expression of the ground state energy in cutoff regularization and given as an integral of the Mellin-Barnes type of the zeta function. Here the integration path goes parallel to the imaginary axis to the right of the poles of the integrand.…”

Section: The Divergent Part Of the Ground State Energymentioning

confidence: 99%

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New developments in the Casimir effect

2001

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We provide a review of both new experimental and theoretical developments in the Casimir effect. The Casimir effect results from the alteration by the boundaries of the zero-point electromagnetic energy. Unique to the Casimir force is its strong dependence on shape, switching from attractive to repulsive as function of the size, geometry and topology of the boundary. Thus the Casimir force is a direct manifestation of the boundary dependence of quantum vacuum. We discuss in depth the general structure of the infinities in the field theory which are removed by a combination of zeta-functional regularization and heat kernel expansion. Different representations for the regularized vacuum energy are given. The Casimir energies and forces in a number of configurations of interest to applications are calculated. We stress the development of the Casimir force for real media including effects of nonzero temperature, finite conductivity of the boundary metal and surface roughness. Also the combined effect of these important factors is investigated in detail on the basis of condensed matter physics and quantum field theory at nonzero temperature. The experiments on measuring the Casimir force are also reviewed, starting first with the older measurements and finishing with a detailed presentation of modern precision experiments. The latter are accurately compared with the theoretical results for real media. At the end of the review we provide the most recent constraints on the corrections to Newtonian gravitational law and other hypothetical long-range interactions at submillimeter range obtained from the Casimir force measurements.

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Section: The Heat Kernel Expansionmentioning

confidence: 99%

Section: Moving Boundaries In a Two-dimensional Space-timementioning

confidence: 99%

Section: The Divergent Part Of the Ground State Energymentioning

confidence: 99%

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New developments in the Casimir effect

2001

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“…This approach appears somewhat simpler, although is less "algorithmic" since one has to invent new functional relations appropriate for a particular problem. The full power of this method has been demostrated on manifolds with boundaries [52] (cf. also [419] for minor corrections).…”

Section: B Heat Kernel Expansionmentioning

confidence: 99%

“…"Special" geodesicsÃ = 0 with 52) and "degenerate" ones obeying A = ξ(r) = const. must be considered separately.…”

Section: Global Structurementioning

confidence: 99%

Dilaton gravity in two dimensions

2002

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The study of general two dimensional models of gravity allows to tackle basic questions of quantum gravity, bypassing important technical complications which make the treatment in higher dimensions difficult. As the physically important examples of spherically symmetric Black Holes, together with string inspired models, belong to this class, valuable knowledge can also be gained for these systems in the quantum case. In the last decade new insights regarding the exact quantization of the geometric part of such theories have been obtained. They allow a systematic quantum field theoretical treatment, also in interactions with matter, without explicit introduction of a specific classical background geometry. The present review tries to assemble these results in a coherent manner, putting them at the same time into the perspective of the quite large literature on this subject.

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Scalar Quantum Fields Confined by Rectangular Boundaries

Actor

1995

Fortschr. Phys.

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A collection of local Casimir problems is studied in detail. These problems are generated by confining a scalar quantum field within boundaries consisting of planes, edges and corners. The boundaries include isolated infinite planes, rectangular edges and corners, parallel planes, the rectangular waveguide and the rectangular cavity. Calculations are done for massless and massive confined fields and for fields at zero and nonzero temperature. Each example has its finite global Casimir effect, which must be computable from the exact local description. The latter however contains known boundary divergences which seem to imply an infinite global Casimir effect. A physical picture and mathematically explicit method is presented for computing finite global Casimir effects from the divergent local description. Divergent boundary functions extending outward in space away from a given boundary ϱ𝔐 represent the distortion of the virtual particle sea of the confined field caused by ϱ𝔐. This distortion of the virtual sea is inseparable from the existence of ϱ𝔐, and may be regarded as the cost of construction of ϱ𝔐 out of nothing. Thus a given divergent boundary function is intrinsic to its ϱ𝔐 — where here is meant specifically the complete boundary function extending (if it does) all the way to infinity. Different boundaries cut off parts of each other's boundary functions, and this plays a vital role in the global Casimir effect as will be explained. It is shown in the examples considered how to compute the global Casimir effect by integration of the exact local description over the region within which the quantum field is confined. Rectangular boundary geometry makes calculations relatively simple; however the method can be applied to boundaries of arbitrary shape. At finite temperature the confined quantum field represents a thermal gas as well as a virtual particle sea. Boundaries also distort this gas of real particles in a way which can be computed. Exact local calculations are done for the rectangular boundaries considered. Beyond these considerations for free fields, systems with interacting fields present additional very interesting local Casimir phenomena. A model two‐field interacting theory is discussed briefly for illustration.

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The Asymptotics of The Laplacian on a Manifold with Boundary

Cited by 266 publications

References 13 publications

New developments in the Casimir effect

New developments in the Casimir effect

Dilaton gravity in two dimensions

Scalar Quantum Fields Confined by Rectangular Boundaries

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