20,922 research outputs found
Peaks in the Hartle-Hawking Wave Function from Sums over Topologies
Recent developments in ``Einstein Dehn filling'' allow the construction of
infinitely many Einstein manifolds that have different topologies but are
geometrically close to each other. Using these results, we show that for many
spatial topologies, the Hartle-Hawking wave function for a spacetime with a
negative cosmological constant develops sharp peaks at certain calculable
geometries. The peaks we find are all centered on spatial metrics of constant
negative curvature, suggesting a new mechanism for obtaining local homogeneity
in quantum cosmology.Comment: 16 pages,LaTeX, no figures; v2: some changes coming from revision of
a math reference: wave function peaks sharp but not infinite; v3: added
paragraph in intro on interpretation of wave functio
Entropy vs. Action in the (2+1)-Dimensional Hartle-Hawking Wave Function
In most attempts to compute the Hartle-Hawking ``wave function of the
universe'' in Euclidean quantum gravity, two important approximations are made:
the path integral is evaluated in a saddle point approximation, and only the
leading (least action) extremum is taken into account. In (2+1)-dimensional
gravity with a negative cosmological constant, the second assumption is shown
to lead to incorrect results: although the leading extremum gives the most
important single contribution to the path integral, topologically inequivalent
instantons with larger actions occur in great enough numbers to predominate.
One can thus say that in 2+1 dimensions --- and possibly in 3+1 dimensions as
well --- entropy dominates action in the gravitational path integral.Comment: 17 page
The Sum over Topologies in Three-Dimensional Euclidean Quantum Gravity
In Hawking's Euclidean path integral approach to quantum gravity, the
partition function is computed by summing contributions from all possible
topologies. The behavior such a sum can be estimated in three spacetime
dimensions in the limit of small cosmological constant. The sum over topologies
diverges for either sign of , but for dramatically different reasons:
for , the divergent behavior comes from the contributions of very
low volume, topologically complex manifolds, while for it is a
consequence of the existence of infinite sequences of relatively high volume
manifolds with converging geometries. Possible implications for
four-dimensional quantum gravity are discussed.Comment: 12 pages (LaTeX), UCD-92-1
Smolyak's algorithm: A powerful black box for the acceleration of scientific computations
We provide a general discussion of Smolyak's algorithm for the acceleration
of scientific computations. The algorithm first appeared in Smolyak's work on
multidimensional integration and interpolation. Since then, it has been
generalized in multiple directions and has been associated with the keywords:
sparse grids, hyperbolic cross approximation, combination technique, and
multilevel methods. Variants of Smolyak's algorithm have been employed in the
computation of high-dimensional integrals in finance, chemistry, and physics,
in the numerical solution of partial and stochastic differential equations, and
in uncertainty quantification. Motivated by this broad and ever-increasing
range of applications, we describe a general framework that summarizes
fundamental results and assumptions in a concise application-independent
manner
Planar Ion Trap Geometry for Microfabrication
We describe a novel high aspect ratio radiofrequency linear ion trap geometry
that is amenable to modern microfabrication techniques. The ion trap electrode
structure consists of a pair of stacked conducting cantilevers resulting in
confining fields that take the form of fringe fields from parallel plate
capacitors. The confining potentials are modeled both analytically and
numerically. This ion trap geometry may form the basis for large scale quantum
computers or parallel quadrupole mass spectrometers.
PACS: 39.25.+k, 03.67.Lx, 07.75.+h, 07.10+CmComment: 14 pages, 16 figure
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