Multiple reflection expansion and heat kernel coefficients

Bordag, M.; Vassilevich, D. V.; Falomir, H.; Santangelo, E. M.

doi:10.1103/physrevd.64.045017

Cited by 40 publications

(20 citation statements)

References 35 publications

(33 reference statements)

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“…In a particular case of spherical Σ the heat kernel expansion with transfer boundary conditions was evaluated in [81]. General expressions for a k with k = 0, 1, 2, 3, 4 were obtained in [247].…”

Section: Domain Walls and Brane Worldmentioning

confidence: 99%

“…We should note that not all singular limiting cases of (6.20) are described by the transmittal conditions (6.11), (6.14). The heat kernel expansion for a generalisation of transmittal condition is known in the spherically symmetric case only [81]. Very little is known about the heat kernel if the transfer matrix contains differential operators on Σ (conformal walls of ref.…”

Section: Domain Walls and Brane Worldmentioning

confidence: 99%

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Heat kernel expansion: user's manual

2003

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The heat kernel expansion is a very convenient tool for studying one-loop divergences, anomalies and various asymptotics of the effective action. The aim of this report is to collect useful information on the heat kernel coefficients scattered in mathematical and physical literature. We present explicit expressions for these coefficients on manifolds with and without boundaries, subject to local and non-local boundary conditions, in the presence of various types of singularities (e.g., domain walls). In each case the heat kernel coefficients are given in terms of several geometric invariants. These invariants are derived for scalar and spinor theories with various interactions, Yang-Mills fields, gravity, and open bosonic strings. We discuss the relations between the heat kernel coefficients and quantum anomalies, corresponding anomalous actions, and covariant perturbation expansions of the effective action (both "low-" and "high-energy" ones).Comment: 113 pp, to be submitted to Phys.Repts, v2: added references and corrected typo

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Section: Domain Walls and Brane Worldmentioning

confidence: 99%

Section: Domain Walls and Brane Worldmentioning

confidence: 99%

Heat kernel expansion: user's manual

2003

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“…In [20], the multiple reflection expansion is derived from the from the ansatz of a double boundary layer and this approach has persisted in e.g. [10][11][12][13][14][15][16][17][18][19].…”

Section: Green Functions and Single And Double Boundary Layersmentioning

confidence: 99%

Potential theory, path integrals and the Laplacian of the indicator

Lange

2012

J. High Energ. Phys.

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This paper links the field of potential theory -- i.e. the Dirichlet and Neumann problems for the heat and Laplace equation -- to that of the Feynman path integral, by postulating that the potential is equal to plus/minus the Laplacian of the indicator of the domain D. The Laplacian of the indicator is a generalized function: it is the d-dimensional analogue of the Dirac delta'-function. This function has -- according to the author's best knowledge -- not formally been defined before. We show, first, that the path integral's perturbation series (or Born series) matches the classical single and double boundary layer series of potential theory, thereby connecting two hitherto unrelated fields. Second, we show that the perturbation series is valid for all domains D that allow Green's theorem (i.e. with a finite number of corners, edges and cusps), thereby expanding the classical applicability of boundary layers. Third, we show that the minus (plus) in the potential holds for the Dirichlet (Neumann) boundary condition; showing for the first time a particularly close connection between these two classical problems. Fourth, we demonstrate that the perturbation series of the path integral converges in a monotone/alternating fashion, depending on the convexity/concavity of the domain. We also discuss the third boundary problem (which poses Robin boundary conditions) and discuss an extension to moving domains.Comment: 46 pages, 2 figure

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“…However, S ±± = 0 does not correspond to a free propagation, but rather to two disjoint regions with Neumann boundary conditions on the surface which separates them. On a side note, we remark that this explains the failure of the multiple reflection expansion in the case of these matching conditions [25]. Throughout this subsection we shall suppose that the dimension of M is arbitrary, n = dim M .…”

Section: B Heat Kernel For the Auxiliary Modelmentioning

confidence: 97%

Divergences in the vacuum energy for frequency-dependent interactions

Vassilevich

2009

Phys. Rev. D

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We propose a method for determining ultra-violet divergences in the vacuum energy for systems whose spectrum of perturbations is defined through a non-linear spectrum problem, i.e, when the fluctuation operator itself depends on the frequency. The method is applied to the plasma shell model, which describes some properties of the interaction of electromagnetic field with fullerens. We formulate a scalar model, which simplifies the matrix structure, but keeps the frequency dependence of the plasma shell, and calculate the ultra-violet divergences in the case when the plasma sheet is slightly curved. The divergent terms are expressed in terms of surface integrals of corresponding invariants.Comment: 14 pages, revtex, v2: clarifications adde

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Multiple reflection expansion and heat kernel coefficients

Cited by 40 publications

References 35 publications

Heat kernel expansion: user's manual

Heat kernel expansion: user's manual

Potential theory, path integrals and the Laplacian of the indicator

Divergences in the vacuum energy for frequency-dependent interactions

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