showing 18 related works from this author
On Overlapping Divergences
1998
Using set-theoretic considerations, we show that the forest formula for overlapping divergences comes from the Hopf algebra of rooted trees.
One loop integrals revisited
1992
We present a new calculation of the well-known one-loop two-point scalar and tensor functions. We also present a systematic reduction to a certain class of functions which minimizes the effort for calculating tensor integrals drastically. We avoid standard techniques such as Feynman parametrization and Wick rotation.
The master two-loop two-point function. The general case
1991
Abstract We present a new calculation of the two-loop two-point function. Avoiding standard techniques such as Feynman parametrization and Wick rotation we end up with a simple double integral representation valid for arbitrary mass-cases. Numerical and analytical checks confirm our result.
A practicableγ 5-scheme in dimensional regularization
1992
We present a new simpleγ5 regularization scheme. We discuss its use in the standard radiative correction calculations including the anomaly contributions. The new scheme features an anticommutingγ5 which leads to great simplifications in practical calculations. We carefully discuss the underlying mathematics of ourγ5-scheme which is formulated in terms of simple projection operations.
Comment on “Topological invariants, instantons, and the chiral anomaly on spaces with torsion”
1999
In Riemann-Cartan spacetimes with torsion only its axial covector piece $A$ couples to massive Dirac fields. Using renormalization group arguments, we show that besides the familiar Riemannian term only the Pontrjagin type four-form $dA\wedge dA$ does arise additionally in the chiral anomaly, but not the Nieh-Yan term $d^\star A$, as has been claimed in a recent paper [PRD 55, 7580 (1997)].
Feynman diagrams as a weight system: four-loop test of a four-term relation
1996
At four loops there first occurs a test of the four-term relation derived by the second author in the course of investigating whether counterterms from subdivergence-free diagrams form a weight system. This test relates counterterms in a four-dimensional field theory with Yukawa and $\phi^4$ interactions, where no such relation was previously suspected. Using integration by parts, we reduce each counterterm to massless two-loop two-point integrals. The four-term relation is verified, with $ = 0 - 3\zeta_3 + 6\zeta_3 - 3\zeta_3 = 0$, demonstrating non-trivial cancellation of the trefoil knot and thus supporting the emerging connection between knots and counterterms, via transcendental number…
Hopf algebras, renormalization and noncommutative geometry
1998
We explore the relation between the Hopf algebra associated to the renormalization of QFT and the Hopf algebra associated to the NCG computations of transverse index theory for foliations.
Locality, QED and classical electrodynamics
1998
We report on some conceptual changes in our present understanding of Quantum Field Theory and muse about possible consequences for the understanding of $v>c$ signals.
Renormalization and Knot Theory
1997
We investigate to what extent renormalization can be understood as an algebraic manipulation on concatenated one-loop integrals. We find that the resulting algebra indicates a useful connection to knot theory as well as number theory and report on recent results in support of this connection.
Perturbative quantum field theory
2000
pQFT In this chapter we repeat the main steps towards a derivation of the Feynman rules, following the well-known path of canonical quantization. This is standard material, and readers who are not acquainted with such topics are referred to [Bjorken and Drell 1965, Bogoliubov and Shirkov 1980, Itzykson and Zuber 1980, Kaku 1993, Weinberg 1995, Peskin and Schroeder 1995, Teller 1997]. We hope that the short summary given here, similar to that in [Kreimer 1997a], is helpful for readers who want to refresh their memory. Having introduced Feynman rules, we next introduce Schwinger–Dyson equations as a motivation for the introduction of Z -factors. We remark on dimensional regularization and giv…
The next-to-ladder approximation for linear Dyson–Schwinger equations
2007
We solve the linear Dyson Schwinger equation for a massless vertex in Yukawa theory, iterating the first two primitive graphs.
The γ5-problem and anomalies — A Clifford algebra approach
1990
Abstract It is shown that a strong correspondence between noncyclicity and anomalies exists. This allows, by fundamental properties of Clifford algebras, to build a simple and consistent scheme for treating γ 5 without using ( d −4)-dimensional objects
The two-loop three-point functions. General massive cases
1992
Abstract We present a calculation of the two-loop three-point scalar functions for the two overlapping topologies. These are the master functions for the ladder and the crossed ladder graphs. We also present a method for the extraction of possible (on-shell) mass singularities.
Beyond the triangle and uniqueness relations: non-zeta counterterms at large $N$ from positive knots
1997
Counterterms that are not reducible to ζn are generated by 3F2 hypergeometric series arising from diagrams for which triangle and uniqueness relations furnish insufficient data. Irreducible double sums, corresponding to the torus knots (4, 3) = 819 and (5, 3) = 10124, are found in anomalous dimensions at O(1/N 3) in the large-N limit, which we compute analytically up to terms of level 11, corresponding to 11 loops for 4-dimensional field theories and 12 loops for 2-dimensional theories. High-precision numerical results are obtained up to 24 loops and used in Pade resummations of e-expansions, which are compared with analytical results in 3 dimensions. The O(1/N 3) results entail knots gener…
CHIRAL ANOMALY IN ASHTEKAR'S APPROACH TO CANONICAL GRAVITY
1998
The Dirac equation in Riemann–Cartan spacetimes with torsion is reconsidered. As is well-known, only the axial covector torsion A, a one-form, couples to massive Dirac fields. Using diagrammatic techniques, we show that besides the familiar Riemannian term only the Pontrjagin type four-form dA ∧ dA does arise additionally in the chiral anomaly, but not the Nieh–Yan term d* A, as has been claimed recently. Implications for cosmic strings in Einstein–Cartan theory as well as for Ashtekar's canonical approach to quantum gravity are discussed.
The methods of XLOOPS An introduction to parallel space techniques
1997
Abstract The package XLOOPS presented in this workshop relies on the application of parallel space techniques. We introduce these techniques covering the following topics: • - The generation of integral representations for massive two-loop diagrams. • - The treatment of tensor structures. • - The handling of the γ-algebra in this scheme. • - The separation of UV and IR divergences in analytic expressions. We present two-loop examples taken from Standard Model calculations.
Weight Systems from Feynman Diagrams
1996
We find that the overall UV divergences of a renormalizable field theory with trivalent vertices fulfil a four-term relation. They thus come close to establish a weight system. This provides a first explanation of the recent successful association of renormalization theory with knot theory.
Chen’s iterated integral represents the operator product expansion
1999
The recently discovered formalism underlying renormalization theory, the Hopf algebra of rooted trees, allows to generalize Chen’s lemma. In its generalized form it describes the change of a scale in Green functions, and hence relates to the operator product expansion. Hand in hand with this generalization goes the generalization of the ordinary factorial n! to the tree factorial t. Various identities on tree-factorials are derived which clarify the relation between Connes-Moscovici weights and Quantum Field Theory.