A Formula for Inserting Point Masses

Let mu be a probability measure on the unit circle and nu be the measure formed by adding a pure point to mu. We give a formula for the Verblunsky coefficients of the perturbed measure, based on a result of Simon

Generalized Bounded Variation and Inserting point masses

Let $d\mu$ be a probability measure on the unit circle and $d\nu$ be the measure formed by adding a pure point to $d\mu$. We give a simple formula for the Verblunsky coefficients of $d\nu$ based on a result of Simon. Then we consider $d\mu_0$, a probability measure on the unit circle with $\ell^2$ Verblunsky coefficients $(\alpha_n (d\mu_0))_{n=0}^{\infty}$ of bounded variation. We insert $m$ pure points to $d\mu$, rescale, and form the probability measure $d\mu_m$. We use the formula above to prove that the Verblunsky coefficients of $d\mu_m$ are in the form \alpha_n(d\mu_0) + \sum_{j=1}^m \frac{\ol{z_j}^{n} c_j}{n} + E_n, where the $c_j$'s are constants of norm 1 independent of the weights of the pure points and independent of $n$; the error term $E_n$ is in the order of $o(1/n)$. Furthermore, we prove that $d\mu_m$ is of $(m+1)$-generalized bounded variation - a notion that we shall introduce in the paper. Then we use this fact to prove that \lim_{n \to \infty} \vp_n^*(z, d\mu_m) is continuous and is equal to $D(z, d\mu_m)^{-1}$ away from the pure points.Comment: To appear in Constructive Approximatio

Gauge Theories in a Bag

We investigate multi-flavour gauge theories confined in d\es 2n-dimensional Euclidean bags. The boundary conditions for the 'quarks' break the axial flavour symmetry and depend on a parameter $\theta$. We determine the $\theta$-dependence of the fermionic correlators and determinants and find that a $CP$-breaking $\theta$-term is generated dynamically. As an application we calculate the chiral condensate in multi-flavour $QED_2$ and the abelian projection of $QCD_2$. In the second model a condensate is generated in the limit where the number of colours, $N_c$, tends to infinity. We prove that the condensate in $QCD_2$ decreases with increasing bag radius $R$ at least as $\sim R^{-1/N_cN_f}$. Finally we determine the correlators of mesonic currents in $QCD_2$.Comment: 40 pages, LATEX-fil

The IR stability of de Sitter QFT: results at all orders

We show that the Hartle-Hawking vacuum for theories of interacting massive scalars in de Sitter space is both perturbatively well-defined and stable in the IR. Correlation functions in this state may be computed on the Euclidean section and Wick-rotated to Lorentz-signature. The results are manifestly de Sitter-invariant and contain only the familiar UV singularities. More importantly, the connected parts of all Lorentz-signature correlators decay at large separations of their arguments. Our results apply to all cases in which the free Euclidean vacuum is well defined, including scalars with masses belonging to both the complementary and principal series of $SO(D,1)$. This suggests that interacting QFTs in de Sitter -- including higher spin fields -- are perturbatively IR-stable at least when i) the Euclidean vacuum of the zero-coupling theory exists and ii) corresponding Lorentz-signature zero-coupling correlators decay at large separations. This work has significant overlap with a paper by Stefan Hollands, which is being released simultaneously.Comment: 30 pp., 4 figures. Small typos fixed, refs adde

Microlensing of Large Sources

We prove a gravitational lensing theorem: the magnification of a source of uniform brightness by a foreground spherical lens is mu =1+pi(2R_E^2-R_L^2)/A, where A is the area of the source and R_E and R_L are the Einstein radius and size of the lens projected into the source plane; this provides an accurate approximation to the exact magnification for R_L^2,R_E^2 << A. Remarkably, this result is independent of the shape of the source or position of the lens (except near the edges). We show that this formula can be generalized to include limb-darkening of a circular source by simply inserting the surface-brightness at the position of the foreground object (divided by the average surface-brightness of the star). We also show that similar formulae apply for a point-mass lens contained in a shear field and mass sheet, and for an ensemble of point masses as long as the Einstein radii are much smaller than the source size. This theorem may be used to compute transit or microlensing lightcurves for which the foreground star or planet has a size and Einstein radius much smaller than the background star.Comment: 9 pages, 8 figures, submitted to Ap

Effective Chiral Lagrangians and Lattice QCD

We propose a general method to obtain accurate estimates for some of the "low-energy constants" in the one-loop effective chiral Lagrangian by means of simulating lattice QCD. In particular, the method is sensitive to those constants whose values are required to test the hypothesis of a massless up-quark. Initial tests performed in the quenched approximation confirm that good statistical precision can be achieved. As a byproduct we obtain an accurate estimate for the ratio of pseudoscalar decay constants, F_K/F_pi, in the quenched approximation, which lies 10% below the experimental result. The quantities that serve to extract the low-energy constants also allow a test of the scaling behaviour of different discretizations of QCD and a search for the effects of dynamical quarks.Comment: 27 pages, 9 postscript files, LaTeX; Corrections in Appendix A and Table 1; numerical results and conclusions unchanged; version to be published in Nucl. Phys.

Discussing the U(1)-Problem of QED_2 without Instantons

We construct QED_2 with mass and flavor and an extra Thirring term. The vacuum expectation values are carefully decomposed into clustering states using the U(1)-axial symmetry of the considered operators and a limiting procedure. The properties of the emerging expectation functional are compared to the proposed theta-vacuum of QCD. The massive theory is bosonized to a generalized Sine-Gordon model (GSG). The structure of the vacuum of QED_2 manifests itself in symmetry properties of the GSG. We study the U(1)-problem and derive a Witten-Veneziano-type formula for the masses of the pseudoscalars determined from a semiclassical approximation.Comment: Accepted for publication in Annals of Physic

On finite-volume gauge theory partition functions

We prove a Mahoux–Mehta-type theorem for finite-volume partition functions of SU(Nc≥3) gauge theories coupled to fermions in the fundamental representation. The large-volume limit is taken with the constraint V1/mπ4. The theorem allows one to express any k-point correlation function of the microscopic Dirac operator spectrum entirely in terms of the 2-point function. The sum over topological charges of the gauge fields can be explicitly performed for these k-point correlation functions. A connection to an integrable KP hierarchy, for which the finite-volume partition function is a τ-function, is pointed out. Relations between the effective partition functions for these theories in 3 and 4 dimensions are derived. We also compute analytically, and entirely from finite-volume partition functions, the microscopic spectral density of the Dirac operator in SU(Nc) gauge theories coupled to quenched fermions in the adjoint representation. The result coincides exactly with earlier results based on Random Matrix Theory

Particle Physics from Almost Commutative Spacetimes

Our aim in this review article is to present the applications of Connes' noncommutative geometry to elementary particle physics. Whereas the existing literature is mostly focused on a mathematical audience, in this article we introduce the ideas and concepts from noncommutative geometry using physicists' terminology, gearing towards the predictions that can be derived from the noncommutative description. Focusing on a light package of noncommutative geometry (so-called 'almost commutative manifolds'), we shall introduce in steps: electrodynamics, the electroweak model, culminating in the full Standard Model. We hope that our approach helps in understanding the role noncommutative geometry could play in describing particle physics models, eventually unifying them with Einstein's (geometrical) theory of gravity.Comment: 104 pages, 5 figures, version 2 (minor changes and some additional references