Operator product expansion algebra

We establish conceptually important properties of the operator product expansion (OPE) in the context of perturbative, Euclidean $\varphi^{4}$-quantum field theory. First, we demonstrate, generalizing earlier results and techniques of arXiv:1105.3375, that the 3-point OPE, $< O_{A_1} O_{A_2} O_{A_3} > = \sum_{C} \cal{C}_{A_1 A_2 A_3}^{C}$, usually interpreted only as an asymptotic short distance expansion, actually converges at finite, and even large, distances. We further show that the factorization identity $\cal{C}_{A_1 A_2 A_3}^{B}=\sum_{C}\cal{C}_{A_1 A_2}^{C} \cal{C}_{C A_3}^{B}$ is satisfied for suitable configurations of the spacetime arguments. Again, the infinite sum is shown to be convergent. Our proofs rely on explicit bounds on the remainders of these expansions, obtained using refined versions, mostly due to Kopper et al., of the renormalization group flow equation method. These bounds also establish that each OPE coefficient is a real analytic function in the spacetime arguments for non-coinciding points. Our results hold for arbitrary but finite loop orders. They lend support to proposals for a general axiomatic framework of quantum field theory, based on such `consistency conditions' and akin to vertex operator algebras, wherein the OPE is promoted to the defining structure of the theory.Comment: 53 pages, v2: typos removed, minor corrections, v3: typos removed, minor changes in introduction, v4: Note added in proo

Multigrid Waveform Relaxation on Spatial Finite Element Meshes: The Discrete-Time Case

The efficiency of numerically solving time-dependent partial differential equations on parallel computers can be greatly improved by computing the solution on many time levels simultaneously. The theoretical properties of one such method, namely the discrete-time multigrid waveform relaxation method, are investigated for systems of ordinary differential equations obtained by spatial finite-element discretisation of linear parabolic initial-boundary value problems. The results are compared to the corresponding continuous-time results. The theory is illustrated for a one-dimensional and a two-dimensional model problem and checked against results obtained by numerical experiments

Passive Newtonian noise suppression for gravitational-wave observatories based on shaping of the local topography

In this article we propose a new method for reducing Newtonian noise in laser-interferometric gravitational-wave detectors located on the Earth's surface. We show that by excavating meter-scale recesses in the ground around the main test masses of a gravitational wave detector it is possible to reduce the coupling of Rayleigh wave driven seismic disturbances to test mass displacement. A discussion of the optimal recess shape is given and we use finite element simulations to derive the scaling of the Newtonian noise suppression with the parameters of the recess as well as the frequency of the seismic excitation. Considering an interferometer similar to an Advance LIGO configuration, our simulations indicate a frequency dependent Newtonian noise suppression factor of 2 to 4 in the relevant frequency range for a recesses of 4m depth and a width and length of 11m and 5m, respectively. Though a retrofit to existing interferometers seems not impossible, the application of our concept to future infrastructures seems to provide a better benefit/cost ratio and therefore a higher feasibility.Comment: 12 pages, 5 figure

At Ease with Your Warnings: The Principles of the Salutogenesis Model Applied to Automatic Static Analysis

The results of an automatic static analysis run can be overwhelming, especially for beginners. The overflow of information and the resulting need for many decisions is mentally tiring and can cause stress symptoms. There are several models in health care which are designed to fight stress. One of these is the salutogenesis model created by Aaron Antonovsky. In this paper, we will present an idea on how to transfer this model into a triage and recommendation model for static analysis tools and give an example of how this can be implemented in FindBugs, a static analysis tool for Java.Comment: 5 pages, 4 figure