Boundary Condition Perturbation Theory for Use in Spatial Homogenization Methods

1989 ◽  
Vol 102 (2) ◽  
pp. 183-190 ◽  
Author(s):  
F. Rahnema
Keyword(s):  
Boundary Condition ◽  
Perturbation Theory ◽  
Homogenization Methods

scholarly journals A note on boundary conditions in Euclidean gravity

2021 ◽  
pp. 2140004
Author(s):  
Edward Witten
Keyword(s):  
Boundary Condition ◽  
General Relativity ◽  
Perturbation Theory ◽  
Quantum Gravity ◽  
Boundary Conditions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Conformal Geometry ◽  
Extrinsic Curvature ◽  
Euclidean Signature

We review what is known about boundary conditions in General Relativity on a spacetime of Euclidean signature. The obvious Dirichlet boundary condition, in which one specifies the boundary geometry, is actually not elliptic and in general does not lead to a well-defined perturbation theory. It is better-behaved if the extrinsic curvature of the boundary is suitably constrained, for instance if it is positive- or negative-definite. A different boundary condition, in which one specifies the conformal geometry of the boundary and the trace of the extrinsic curvature, is elliptic and always leads formally to a satisfactory perturbation theory. These facts might have interesting implications for semiclassical approaches to quantum gravity. April, 2018

scholarly journals Inconsistencies of the New No-Boundary Proposal

Universe ◽  
2018 ◽  
Vol 4 (10) ◽  
pp. 100 ◽  
Author(s):  
Job Feldbrugge ◽  
Jean-Luc Lehners ◽  
Neil Turok
Keyword(s):  
Boundary Condition ◽  
Perturbation Theory ◽  
Strong Coupling ◽  
Path Integral ◽  
Quantum Cosmology ◽  
Einstein Gravity ◽  
Strong Coupling Regime ◽  
Recent Modification ◽  
Circular Contour ◽  
Made In

In previous works, we have demonstrated that the path integral for real, Lorentzian four-geometries in Einstein gravity yields sensible results in well-understood physical situations, but leads to uncontrolled fluctuations when the “no boundary” condition proposed by Hartle and Hawking is imposed. In order to circumvent our result, new definitions for the gravitational path integral have been sought, involving specific choices for a class of complex four-geometries to be included. In their latest proposal, Diaz Dorronsoro et al. advocate for integrating the lapse over a complex circular contour enclosing the origin. In this note, we show that, like their earlier proposal, this leads to mathematical and physical inconsistencies and thus cannot be regarded as a basis for quantum cosmology. We also comment on Vilenkin and Yamada’s recent modification of the “tunneling" proposal, made in order to avoid the same problems. We show that it leads to the breakdown of perturbation theory in a strong coupling regime.

A Perturbation Theory for the Population of Atomic Energy Levels in Non-Lte Plasmas

1988 ◽  
Vol 102 ◽  
pp. 343-347
Author(s):  
M. Klapisch
Keyword(s):  
Perturbation Theory ◽  
Small Parameter ◽  
Classification Scheme ◽  
Sum Rules ◽  
Energy Levels ◽  
Natural Classification ◽  
Quantum Numbers ◽  
Formal Expansion ◽  
Atomic Energy Levels ◽  
As Effects

AbstractA formal expansion of the CRM in powers of a small parameter is presented. The terms of the expansion are products of matrices. Inverses are interpreted as effects of cascades.It will be shown that this allows for the separation of the different contributions to the populations, thus providing a natural classification scheme for processes involving atoms in plasmas. Sum rules can be formulated, allowing the population of the levels, in some simple cases, to be related in a transparent way to the quantum numbers.

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