53,848 research outputs found
Quantum Holonomies in (2+1)-Dimensional Gravity
We describe an approach to the quantization of (2+1)--dimensional gravity
with topology R x T^2 and negative cosmological constant, which uses two
quantum holonomy matrices satisfying a q--commutation relation. Solutions of
diagonal and upper--triangular form are constructed, which in the latter case
exhibit additional, non--trivial internal relations for each holonomy matrix.
This leads to the notion of quantum matrix pairs. These are pairs of matrices
with non-commuting entries, which have the same pattern of internal relations,
q-commute with each other under matrix multiplication, and are such that
products of powers of the matrices obey the same pattern of internal relations
as the original pair. This has implications for the classical moduli space,
described by ordered pairs of commuting SL(2,R) matrices modulo simultaneous
conjugation by SL(2,R) matrices.Comment: 5 pages, to appear in the proceedings of 10th Marcel Grossmann
Meeting on Recent Developments in Theoretical and Experimental General
Relativity, Gravitation and Relativistic Field Theories (MG X MMIII), Rio de
Janeiro, Brazil, 20-26 Jul 200
The Quantum Modular Group in (2+1)-Dimensional Gravity
The role of the modular group in the holonomy representation of
(2+1)-dimensional quantum gravity is studied. This representation can be viewed
as a "Heisenberg picture", and for simple topologies, the transformation to the
ADM "Schr{\"o}dinger picture" may be found. For spacetimes with the spatial
topology of a torus, this transformation and an explicit operator
representation of the mapping class group are constructed. It is shown that the
quantum modular group splits the holonomy representation Hilbert space into
physically equivalent orthogonal ``fundamental regions'' that are interchanged
by modular transformations.Comment: 23 pages, LaTeX, no figures; minor changes and clarifications in
response to referee (basic argument and conclusions unaffected
Optical links in the angle-data assembly of the 70-meter antennas
In the precision-pointing mode the 70 meter antennas utilize an optical link provided by an autocollimator. In an effort to improve reliability and performance, commercial instruments were evaluated as replacement candidates, and upgraded versions of the existing instruments were designed and tested. The latter were selected for the Neptune encounter, but commercial instruments with digital output show promise of significant performance improvement for the post-encounter period
Entropy of Folding of the Triangular Lattice
The problem of counting the different ways of folding the planar triangular
lattice is shown to be equivalent to that of counting the possible 3-colorings
of its bonds, a dual version of the 3-coloring problem of the hexagonal lattice
solved by Baxter. The folding entropy Log q per triangle is thus given by
Baxter's formula q=sqrt(3)(Gamma[1/3])^(3/2)/2pi =1.2087...Comment: 9 pages, harvmac, epsf, uuencoded, 5 figures included, Saclay
preprint T/9401
Quantum geometry from 2+1 AdS quantum gravity on the torus
Wilson observables for 2+1 quantum gravity with negative cosmological
constant, when the spatial manifold is a torus, exhibit several novel features:
signed area phases relate the observables assigned to homotopic loops, and
their commutators describe loop intersections, with properties that are not yet
fully understood. We describe progress in our study of this bracket, which can
be interpreted as a q-deformed Goldman bracket, and provide a geometrical
interpretation in terms of a quantum version of Pick's formula for the area of
a polygon with integer vertices.Comment: 19 pages, 11 figures, revised with more explanations, improved
figures and extra figures. To appear GER
Anomalous coupling between topological defects and curvature
We investigate a counterintuitive geometric interaction between defects and
curvature in thin layers of superfluids, superconductors and liquid crystals
deposited on curved surfaces. Each defect feels a geometric potential whose
functional form is determined only by the shape of the surface, but whose sign
and strength depend on the transformation properties of the order parameter.
For superfluids and superconductors, the strength of this interaction is
proportional to the square of the charge and causes all defects to be repelled
(attracted) by regions of positive (negative) Gaussian curvature. For liquid
crystals in the one elastic constant approximation, charges between 0 and
are attracted by regions of positive curvature while all other charges
are repelled.Comment: 5 pages, 4 figures, minor changes, accepted for publication in Phys.
Rev. Let
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