Semiclassical Approach to Finite-N Matrix Models

We reformulate the zero-dimensional hermitean one-matrix model as a (nonlocal) collective field theory, for finite~$N$. The Jacobian arising by changing variables from matrix eigenvalues to their density distribution is treated {\it exactly\/}. The semiclassical loop expansion turns out {\it not\/} to coincide with the (topological) ${1\over N}$~expansion, because the classical background has a non-trivial $N$-dependence. We derive a simple integral equation for the classical eigenvalue density, which displays strong non-perturbative behavior around $N\!=\!\infty$. This leads to IR singularities in the large-$N$ expansion, but UV divergencies appear as well, despite remarkable cancellations among the Feynman diagrams. We evaluate the free energy at the two-loop level and discuss its regularization. A simple example serves to illustrate the problems and admits explicit comparison with orthogonal polynomial results.Comment: 27 pages / 3 figures (ps file fixed

Asymptotic flexibility of globally hyperbolic manifolds

In this short note, a question of patching together globally hyperbolic manifolds is adressed which appeared in the context of the construction of Hadamard states.Comment: 2 pages, submitted to 'Mathematische Zeitschrift

Degeneracies and scaling relations in general power-law models for gravitational lenses

The time delay in gravitational lenses can be used to derive the Hubble constant in a relatively simple way. The results of this method are less dependent on astrophysical assumptions than in many other methods. The most important uncertainty is related to the mass model used. We discuss a family of models with a separable radial power-law and an arbitrary angular dependence for the potential psi = r^beta * F(theta). Isothermal potentials are a special case of these models with beta=1. An additional external shear is used to take into account perturbations from other galaxies. Using a simple linear formalism for quadruple lenses, we can derive H0 as a function of the observables and the shear. If the latter is fixed, the result depends on the assumed power-law exponent according to H0 proportional to (2-beta)/beta. The effect of external shear is quantified by introducing a `critical shear' gamma_c as a measure for the amount of shear that changes the result significantly. The analysis shows, that in the general case H0 and gamma_c do not depend on the position of the lens galaxy. We discuss these results and compare with numerical models for a number of real lens systems.Comment: accepted for publication in MNRAS, 10 pages, 4 figures (eps included), uses mn2e.cls, amsmath.sty, times.st

Ln(3) and Black Hole Entropy

We review an idea that uses details of the quasinormal mode spectrum of a black hole to obtain the Bekenstein-Hawking entropy of $A/4$ in Loop Quantum Gravity. We further comment on a recent proposal concerning the quasinormal mode spectrum of rotating black holes. We conclude by remarking on a recent proposal to include supersymmetry.Comment: Contribution to the Proceedings of the 3rd International Symposium on Quantum Theory and Symmetries, Cincinnati, September 2003, added references to the numerical investigations of Kerr quasinormal modes by Berti et. a