But we had hoped ... : The Road We\u27ve Traveled; the Road that Lies Ahead

(Excerpt) It is a privilege to be here with you at this annual gathering to explore matters of consequence affecting our churches at the beginning of this new millennium. I feel honored to be invited into the ongoing conversation of the Institute of Liturgical Studies. I\u27ve had opportunities to cross paths with many in this Lutheran family through associations at the Liturgical Conference, the North American Academy of Liturgy, and Notre Dame, and all have been positive and enriching experiences. My contacts have served to deepen my appreciation of the strong commitment of the Lutheran churches to promoting life-giving worship within congregations. I am inspired by the desire for unity that runs deep in the Lutheran soul

An approach to nonstandard quantum mechanics

We use nonstandard analysis to formulate quantum mechanics in hyperfinite-dimensional spaces. Self-adjoint operators on hyperfinite-dimensional spaces have complete eigensets, and bound states and continuum states of a Hamiltonian can thus be treated on an equal footing. We show that the formalism extends the standard formulation of quantum mechanics. To this end we develop the Loeb-function calculus in nonstandard hulls. The idea is to perform calculations in a hyperfinite-dimensional space, but to interpret expectation values in the corresponding nonstandard hull. We further apply the framework to non-relativistic quantum scattering theory. For time-dependent scattering theory, we identify the starting time and the finishing time of a scattering experiment, and we obtain a natural separation of time scales on which the preparation process, the interaction process, and the detection process take place. For time-independent scattering theory, we derive rigorously explicit formulas for the M{\o}ller wave operators and the S-Matrix

Metric Tests for Curvature from Weak Lensing and Baryon Acoustic Oscillations

We describe a practical measurement of the curvature of the Universe which relies purely on the properties of the Robertson-Walker metric rather than any model for the dynamics or content of the Universe. The cross-correlation between foreground mass and gravitational shear of background galaxies depends upon the angular diameter distances d_A(z_l), d_A(z_s), and d_A(z_s,z_l) on the degenerate triangle formed by observer, source, and lens. In a flat Universe, d_A(z_l,z_s) = d_A(z_s)-d_A(z_l), but in curved Universes an additional term $\propto\Omega_k$ alters the lensing observables even if d_A(z) is fixed. Weak lensing data may be used to solve simultaneously for d_A and the curvature. This method is completely insensitive to: the equation of state; amendments to the General Relativity formulae for the deflection of light or the growth of structure; or biases in the photometric redshift scale. There is however a degeneracy among d_A, $\Omega_k$ and the galaxy bias factors, that can be broken by using measurements of baryon acoustic oscillations with the same imaging data. Ambitious weak-lensing + baryon-oscillation surveys would measure $\Omega_k$ to an accuracy $\approx0.04 f_{\rm sky}^{-1/2} (\sigma_{\ln z}/0.04)^{1/2}$, where $\sigma_{\ln z}$ is the photometric redshift error. We also predict bounds on curvature and other parameters in the context of specific dark-energy models, and compare to other analyses of the weak lensing cross-correlation method. We find both curvature and parametric constraints to be surprisingly insensitive to systematic shear calibration errors.Comment: 26 pages, accepted to ApJ. New notation and minor change