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Christian Wieczerkowski

Publications and source records attributed to Christian Wieczerkowski.

11 recordsLinked to original sources
arXiv1999

Positivity and Convergence in Fermionic Quantum Field Theory

We derive norm bounds that imply the convergence of perturbation theory in fermionic quantum field theory if the propagator is summable and has a finite Gram constant. These bounds are sufficient for an application in renormalization group studies. Our proof is conceptually simple and technically elementary; it clarifies how the applicability of Gram bounds with uniform constants is related to positivity properties of matrices associated to the procedure of taking connected parts of Gaussian convolutions. This positivity is preserved in the decouplings that also preserve stability in the case of two-body interactions.

math-ph↗
arXiv1998

Rigorous control of the non-perturbative corrections to the double expansion in $g$ and $g^2\ln(g)$ for the $ϕ^4_3$-trajectory in the hierarchical approximation

We study the renormalization invariant trajectory of the $ϕ^4$-perturbation of the free field fixed point in the hierarchical approximation. We parametrize it by a running $ϕ^4$-coupling $g$ with linear step $β$-function. We rigorously control the non-perturbative corrections to finite order approximants from double perturbation theory in $g$ and $g^2\ln(g)$. The construction uses a contraction mapping for the extended renormalization group composed of a hierarchical block spin transformation with a flow of $g$.

hep-lat↗
arXiv1998

Massless picture, massive picture, and symmetry in the Gaussian renormalization group

We consider renormalization groups of transformations composed of a Gaussian convolution and a field dilatation. As an example, we consider perturbations of a single component real Euclidean free field $ϕ$ with covariance $(-\bigtriangleup)^{-1+\fracε{2}}$. We show that the renormalization group admits two equivalent formulations called massless picture and massive picture respectively. We then show in the massive picture that the renormalization group has a symmetry. The symmetry consists of global scale transformations composed with certain Gaussian convolutions. We translate the symmetry back to the massless picture. The relation between the symmetry and the notion of an anomalous dimension is briefly discussed.

hep-th↗
arXiv1997

Running coupling expansion for the renormalized $ϕ^4_4$-trajectory from renormalization invariance

We formulate a renormalized running coupling expansion for the $β$--function and the potential of the renormalized $ϕ^4$--trajectory on four dimensional Euclidean space-time. Renormalization invariance is used as a first principle. No reference is made to bare quantities. The expansion is proved to be finite to all orders of perturbation theory. The proof includes a large momentum bound on the connected free propagator amputated vertices.

hep-th↗
arXiv1996

The renormalized $ϕ^4_4$-trajectory by perturbation theory in a running coupling II: the continuous renormalization group

The renormalized trajectory of massless $ϕ^4$-theory on four dimensional Euclidean space-time is investigated as a renormalization group invariant curve in the center manifold of the trivial fixed point, tangent to the $ϕ^4$-interaction. We use an exact functional differential equation for its dependence on the running $ϕ^4$-coupling. It is solved by means of perturbation theory. The expansion is proved to be finite to all orders. The proof includes a large momentum bound on amputated connected momentum space Green's functions.

hep-th↗
arXiv1996

Interpolation Parameter and Expansion for the Three Dimensional Non-Trivial Scalar Infrared Fixed Point

We compute the non--trivial infrared $ϕ^4_3$--fixed point by means of an interpolation expansion in fixed dimension. The expansion is formulated for an infinitesimal momentum space renormalization group. We choose a coordinate representation for the fixed point interaction in derivative expansion, and compute its coordinates to high orders by means of computer algebra. We compute the series for the critical exponent $ν$ up to order twenty five of interpolation expansion in this representation, and evaluate it using \pade, Borel--\pade, Borel--conformal--\pade, and Dlog--\pade resummation. The resummation returns $0.6262(13)$ as the value of $ν$.

hep-th↗
arXiv1994

Conformal blocks on elliptic curves and the Knizhnik--Zamolodchikov--Bernard equations

We give an explicit description of the vector bundle of WZW conformal blocks on elliptic curves with marked points as subbundle of a vector bundle of Weyl group invariant vector valued theta functions on a Cartan subalgebra. We give a partly conjectural characterization of this subbundle in terms of certain vanishing conditions on affine hyperplanes. In some cases, explicit calculation are possible and confirm the conjecture. The Friedan--Shenker flat connection is calculated, and it is shown that horizontal sections are solutions of Bernard's generalization of the Knizhnik--Zamolodchikov equation.

hep-th↗