Perturbation theory of transformed quantum fields

We consider a scalar quantum field $\phi$ with arbitrary polynomial self-interaction in perturbation theory. If the field variable $\phi$ is repaced by a local diffeomorphism $\phi(x) = \rho(x) + a_1 \rho^2(x) +\ldots$, this field $\rho$ obtains infinitely many additional interaction vertices. We show that the $S$-matrix of $\rho$ coincides with the one of $\phi$ without using path-integral arguments. This result holds even if the underlying field has a propagator of higher than quadratic order in the momentum. If tadpole diagrams vanish, the diffeomorphism can be tuned to cancel all contributions of an underlying $\phi^s$-type self interaction at one fixed external offshell momentum, rendering $\rho$ a free theory at this momentum. Finally, we propose one way to extend the diffeomorphism to a non-local transformation involving derivatives without spoiling the combinatoric structure of the local diffeomorphism.Comment: 28 pages, 9 figure

On Dihedral Configurations and their Coxeter Geometries

AbstractWithin the theory of homogeneous coherent configurations, the dihedral configurations play the role which is played by the finite dihedral groups in the theory of finite groups. Imitating Tits’ construction of a geometry from a set of subgroups of a given group, we assign a geometry of rank 2 to each dihedral configuration, its ‘Coxeter geometry’. (Each finite generalized polygon is a Coxeter geometry in this sense.)Apart from general results on the relationship between dihedral configurations and their Coxeter geometries, we settle completely the (ordinary) representation theory of the dihedral configurations of rank 7. We obtain three major classes. The Coxeter geometries of the first class are exactly the non-symmetric 2-designs withI=1. The other two classes lead to questions which require a further combinatorial treatment

Trends and lines of development in scheme theory

AbstractThe concept of an association scheme is one of those mathematical concepts which were utilized as technical tools in various different mathematical areas for a long time before becoming the subject of a theory in their own right. The significance of symmetric schemes, for instance, in the design of (statistical) experiments was recognized as early as the first half of the last century. Coding theory has been associated with commutative schemes for more than three decades, and polynomial schemes have provided the language in which major topics in algebraic graph theory are communicated for about twenty years. The notion of a scheme itself, however–a notion which, if considered in its full generality, generalizes not only the notion of a group but also the notion of a Moore geometry and that of a building in the sense of Jacques Tits–has been considered as the subject of an abstract theory in itself only relatively recently.It is the purpose of this article to reflect on the lines of development, the Entwicklungslinien, along which abstract scheme theory has been developed so far and along which scheme theory might be developed in the future. The emphasis will be not so much on completeness as on an attempt to show exemplarily how naturally and organically the structure theory of association schemes arises from certain aspects in group theory

High-order renormalization of scalar quantum fields

Thema dieser Dissertation ist die Renormierung von perturbativer skalarer Quantenfeldtheorie bei großer Schleifenzahl. Der Hauptteil der Arbeit ist dem Einfluss von Renormierungsbedingungen auf renormierte Greenfunktionen gewidmet. Zunächst studieren wir Dyson-Schwinger-Gleichungen und die Renormierungsgruppe, inklusive der Gegenterme in dimensionaler Regularisierung. Anhand zahlreicher Beispiele illustrieren wir die verschiedenen Größen. Alsdann diskutieren wir, welche Freiheitsgrade ein Renormierungsschema hat und wie diese mit den Gegentermen und den renormierten Greenfunktionen zusammenhängen. Für ungekoppelte Dyson-Schwinger-Gleichungen stellen wir fest, dass alle Renormierungsschemata bis auf eine Verschiebung des Renormierungspunktes äquivalent sind. Die Verschiebung zwischen kinematischer Renormierung und Minimaler Subtraktion ist eine Funktion der Kopplung und des Regularisierungsparameters. Wir leiten eine neuartige Formel für den Fall einer linearen Dyson-Schwinger Gleichung vom Propagatortyp her, um die Verschiebung direkt aus der Mellintransformation des Integrationskerns zu berechnen. Schließlich berechnen wir obige Verschiebung störungstheoretisch für drei beispielhafte nichtlineare Dyson-Schwinger-Gleichungen und untersuchen das asymptotische Verhalten der Reihenkoeffizienten. Ein zweites Thema der vorliegenden Arbeit sind Diffeomorphismen der Feldvariable in einer Quantenfeldtheorie. Wir präsentieren eine Störungstheorie des Diffeomorphismusfeldes im Impulsraum und verifizieren, dass der Diffeomorphismus keinen Einfluss auf messbare Größen hat. Weiterhin untersuchen wir die Divergenzen des Diffeomorphismusfeldes und stellen fest, dass die Divergenzen Wardidentitäten erfüllen, die die Abwesenheit dieser Terme von der S-Matrix ausdrücken. Trotz der Wardidentitäten bleiben unendlich viele Divergenzen unbestimmt. Den Abschluss bildet ein Kommentar über die numerische Quadratur von Periodenintegralen.This thesis concerns the renormalization of perturbative quantum field theory. More precisely, we examine scalar quantum fields at high loop order. The bulk of the thesis is devoted to the influence of renormalization conditions on the renormalized Green functions. Firstly, we perform a detailed review of Dyson-Schwinger equations and the renormalization group, including the counterterms in dimensional regularization. Using numerous examples, we illustrate how the various quantities are computable in a concrete case and which relations they satisfy. Secondly, we discuss which degrees of freedom are present in a renormalization scheme, and how they are related to counterterms and renormalized Green functions. We establish that, in the case of an un-coupled Dyson-Schwinger equation, all renormalization schemes are equivalent up to a shift in the renormalization point. The shift between kinematic renormalization and Minimal Subtraction is a function of the coupling and the regularization parameter. We derive a novel formula for the case of a linear propagator-type Dyson-Schwinger equation to compute the shift directly from the Mellin transform of the kernel. Thirdly, we compute the shift perturbatively for three examples of non-linear Dyson-Schwinger equations and examine the asymptotic growth of series coefficients. A second, smaller topic of the present thesis are diffeomorphisms of the field variable in a quantum field theory. We present the perturbation theory of the diffeomorphism field in momentum space and find that the diffeomorphism has no influence on measurable quantities. Moreover, we study the divergences in the diffeomorphism field and establish that they satisfy Ward identities, which ensure their absence from the S-matrix. Nevertheless, the Ward identities leave infinitely many divergences unspecified and the diffeomorphism theory is perturbatively unrenormalizable. Finally, we remark on a third topic, the numerical quadrature of Feynman periods

Statistics of Feynman amplitudes in ϕ4\phi^4-theory

The amplitude of subdivergence-free logarithmically divergent Feynman graphs in $\phi^4$-theory in 4 spacetime dimensions is given by a single number, the Feynman period. We numerically compute the periods of 1.3 million non-isomorphic completed graphs, this represents more than 31 million graphs contributing to the beta function. Our data set includes all primitive graphs up to 13 loops, and non-complete samples up to 18 loops, with an accuracy of ca. 4 significant digits. We implement all known symmetries of the period in a new computer program and count them up to 14 loops. We discover some combinations of symmetries that had been overlooked earlier, resulting in an overall slightly lower count of independent graphs than previously assumed. Using the numerical data, we examine the distribution of Feynman periods. We confirm the leading asymptotic growth of the average period with growing loop order. At high loop order, a limiting distribution is reached for the amplitudes near the mean. We construct two different models to approximate this distribution. A small class of graphs, most notably the zigzags, grows significantly faster than the mean and causes the limiting distribution to have divergent moments even when normalized to unit mean. We examine the relation between the period and various properties of the underlying graphs. We confirm the strong correlation with the Hepp bound, the Martin invariant, and the number of 6-edge cuts. We find that, on average, the amplitude of planar graphs is significantly larger than that of non-planar graphs, irrespective of $O(N)$ symmetry. We estimate the primitive contribution to the 18-loop beta function of the $O(N)$-symmetric theory. We confirm that primitive graphs constitute a large part of the known asymptotics of the beta function in MS. However, we can not determine if they are, asymptotically, the only leading contribution.Comment: 59 pages, 70 figures, 17 table

Low temperature magnetic transitions of single crystal HoBi

We present resistivity, specific heat and magnetization measurements in high quality single crystals of HoBi, with a residual resistivity ratio of 126. We find, from the temperature and field dependence of the magnetization, an antiferromagnetic transition at 5.7 K, which evolves, under magnetic fields, into a series of up to five metamagnetic phases.Comment: 5 pages, 5 figure

Polymerized LB films imaged with a combined atomic force microscope-fluorescence microscope

The first results obtained with a new stand-alone atomic force microscope (AFM) integrated with a standard Zeiss optical fluorescence microscope are presented. The optical microscope allows location and selection of objects to be imaged with the high-resolution AFM. Furthermore, the combined microscope enables a direct comparison between features observed in the fluorescence microscope and those observed in the images obtained with the AFM, in air or under liquid. The cracks in polymerized Langmuir-Blodgett films of lO,l2-pentacosadiynoic acid as observed in the fluorescence microscope run parallel to one of the lattice directions of the crystal as revealed by molecular resolution images obtained with the AFM. The orientation of these cracks also coincides with the polarization direction of the fluorescent light, indicating that the cracks run along the polymer backbone. Ripple-like corrugations on a submicrometer scale have been observed, which may be due to mechanical stress created during the polymerization process