Comprehensive Solution to the Cosmological Constant, Zero-Point Energy, and Quantum Gravity Problems

We present a solution to the cosmological constant, the zero-point energy, and the quantum gravity problems within a single comprehensive framework. We show that in quantum theories of gravity in which the zero-point energy density of the gravitational field is well-defined, the cosmological constant and zero-point energy problems solve each other by mutual cancellation between the cosmological constant and the matter and gravitational field zero-point energy densities. Because of this cancellation, regulation of the matter field zero-point energy density is not needed, and thus does not cause any trace anomaly to arise. We exhibit our results in two theories of gravity that are well-defined quantum-mechanically. Both of these theories are locally conformal invariant, quantum Einstein gravity in two dimensions and Weyl-tensor-based quantum conformal gravity in four dimensions (a fourth-order derivative quantum theory of the type that Bender and Mannheim have recently shown to be ghost-free and unitary). Central to our approach is the requirement that any and all departures of the geometry from Minkowski are to be brought about by quantum mechanics alone. Consequently, there have to be no fundamental classical fields, and all mass scales have to be generated by dynamical condensates. In such a situation the trace of the matter field energy-momentum tensor is zero, a constraint that obliges its cosmological constant and zero-point contributions to cancel each other identically, no matter how large they might be. Quantization of the gravitational field is caused by its coupling to quantized matter fields, with the gravitational field not needing any independent quantization of its own. With there being no a priori classical curvature, one does not have to make it compatible with quantization.Comment: Final version, to appear in General Relativity and Gravitation (the final publication is available at http://www.springerlink.com). 58 pages, revtex4, some additions to text and some added reference

Wegner estimate for discrete alloy-type models

We study discrete alloy-type random Schr\"odinger operators on $\ell^2(\mathbb{Z}^d)$. Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. If the single site potential is compactly supported and the distribution of the coupling constant is of bounded variation a Wegner estimate holds. The bound is polynomial in the volume of the box and thus applicable as an ingredient for a localisation proof via multiscale analysis.Comment: Accepted for publication in AHP. For an earlier version see http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=09-10

Black Hole Interaction Energy

The interaction energy between two black holes at large separation distance is calculated. The first term in the expansion corresponds to the Newtonian interaction between the masses. The second term corresponds to the spin-spin interaction. The calculation is based on the interaction energy defined on the two black holes initial data. No test particle approximation is used. The relation between this formula and cosmic censorship is discussed.Comment: 18 pages, 2 figures, LaTeX2

Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes

We present an abstract framework for analyzing the weak error of fully discrete approximation schemes for linear evolution equations driven by additive Gaussian noise. First, an abstract representation formula is derived for sufficiently smooth test functions. The formula is then applied to the wave equation, where the spatial approximation is done via the standard continuous finite element method and the time discretization via an I-stable rational approximation to the exponential function. It is found that the rate of weak convergence is twice that of strong convergence. Furthermore, in contrast to the parabolic case, higher order schemes in time, such as the Crank-Nicolson scheme, are worthwhile to use if the solution is not very regular. Finally we apply the theory to parabolic equations and detail a weak error estimate for the linearized Cahn-Hilliard-Cook equation as well as comment on the stochastic heat equation

Expanded Vandermonde powers and sum rules for the two-dimensional one-component plasma

The two-dimensional one-component plasma (2dOCP) is a system of $N$ mobile particles of the same charge $q$ on a surface with a neutralising background. The Boltzmann factor of the 2dOCP at temperature $T$ can be expressed as a Vandermonde determinant to the power $\Gamma=q^{2}/(k_B T)$. Recent advances in the theory of symmetric and anti-symmetric Jack polymonials provide an efficient way to expand this power of the Vandermonde in their monomial basis, allowing the computation of several thermodynamic and structural properties of the 2dOCP for $N$ values up to 14 and $\Gamma$ equal to 4, 6 and 8. In this work, we explore two applications of this formalism to study the moments of the pair correlation function of the 2dOCP on a sphere, and the distribution of radial linear statistics of the 2dOCP in the plane

An integral method for solving nonlinear eigenvalue problems

We propose a numerical method for computing all eigenvalues (and the corresponding eigenvectors) of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane. The method uses complex integrals of the resolvent operator, applied to at least $k$ column vectors, where $k$ is the number of eigenvalues inside the contour. The theorem of Keldysh is employed to show that the original nonlinear eigenvalue problem reduces to a linear eigenvalue problem of dimension $k$. No initial approximations of eigenvalues and eigenvectors are needed. The method is particularly suitable for moderately large eigenvalue problems where $k$ is much smaller than the matrix dimension. We also give an extension of the method to the case where $k$ is larger than the matrix dimension. The quadrature errors caused by the trapezoid sum are discussed for the case of analytic closed contours. Using well known techniques it is shown that the error decays exponentially with an exponent given by the product of the number of quadrature points and the minimal distance of the eigenvalues to the contour

The role of laboratory testing in hospitalised and critically ill COVID-19-positive patients

The COVID-19 pandemic has placed healthcare resources around the world under immense pressure. South Africa, given the condition of its healthcare system, is particularly vulnerable. There has been much discussion around rational healthcare utilisation, ranging from diagnostic testing and personal protective equipment to triage and appropriate use of ventilation strategies. There has, however, been little guidance around use of laboratory tests once COVID-19 positive patients have been admitted to hospital. We present a working guide to rational laboratory test use, specifically for COVID-19, among hospitalised patients, including the critically ill. The specific tests, the reasons for testing, their clinical usefulness, timing and frequency are addressed. We also provide a discussion around evidence for the use of these tests from a clinical perspective.The Critical Care Society of Southern Africahttp://www.sajcc.org.za/index.php/SAJCCam2021Critical Car

Low lying spectrum of weak-disorder quantum waveguides

We study the low-lying spectrum of the Dirichlet Laplace operator on a randomly wiggled strip. More precisely, our results are formulated in terms of the eigenvalues of finite segment approximations of the infinite waveguide. Under appropriate weak-disorder assumptions we obtain deterministic and probabilistic bounds on the position of the lowest eigenvalue. A Combes-Thomas argument allows us to obtain so-called 'initial length scale decay estimates' at they are used in the proof of spectral localization using the multiscale analysis.Comment: Accepted for publication in Journal of Statistical Physics http://www.springerlink.com/content/0022-471

A gauge model for quantum mechanics on a stratified space

In the Hamiltonian approach on a single spatial plaquette, we construct a quantum (lattice) gauge theory which incorporates the classical singularities. The reduced phase space is a stratified K\"ahler space, and we make explicit the requisite singular holomorphic quantization procedure on this space. On the quantum level, this procedure furnishes a costratified Hilbert space, that is, a Hilbert space together with a system which consists of the subspaces associated with the strata of the reduced phase space and of the corresponding orthoprojectors. The costratified Hilbert space structure reflects the stratification of the reduced phase space. For the special case where the structure group is $\mathrm{SU}(2)$, we discuss the tunneling probabilities between the strata, determine the energy eigenstates and study the corresponding expectation values of the orthoprojectors onto the subspaces associated with the strata in the strong and weak coupling approximations.Comment: 38 pages, 9 figures. Changes: comments on the heat kernel and coherent states have been adde

Path Integral Monte Carlo Approach to the U(1) Lattice Gauge Theory in (2+1) Dimensions

Path Integral Monte Carlo simulations have been performed for U(1) lattice gauge theory in (2+1) dimensions on anisotropic lattices. We extractthe static quark potential, the string tension and the low-lying "glueball" spectrum.The Euclidean string tension and mass gap decrease exponentially at weakcoupling in excellent agreement with the predictions of Polyakov and G{\" o}pfert and Mack, but their magnitudes are five times bigger than predicted. Extrapolations are made to the extreme anisotropic or Hamiltonian limit, and comparisons are made with previous estimates obtained in the Hamiltonian formulation.Comment: 12 pages, 16 figure