Axel Weber

Loop quantization as a continuum limit

We present an implementation of Wilsons renormalization group and a continuum limit tailored for loop quantization. The dynamics of loop-quantized theories is constructed as a continuum limit of the dynamics of effective theories. After presenting the general formalism we show as a first explicit example the 2D Ising field theory, an interacting relativistic quantum field theory with local degrees of freedom quantized by loop quantization techniques.

Epsilon Expansion for Infrared Yang-Mills theory in Landau Gauge

The study of the Dyson-Schwinger equations of Landau gauge Yang-Mills theory has revealed two types of solutions for the gluon and ghost propagators, with a scaling and a massive (decoupling) behavior in the extreme infrared, respectively. We show that both types of solutions are quantitatively reproduced by applying renormalization group equations of Callan-Symanzik type in an epsilon expansion to the infrared limit of Landau gauge Yang-Mills theory when a mass term for the gluons is added to the action. Only the decoupling solution corresponds to an infrared-stable fixed point in three and four space-time dimensions and is hence expected to be physically realized, in agreement with the results of recent lattice calculations.

Hamiltonian flow in Coulomb gauge Yang-Mills theory

We derive a new functional renormalization group equation for Hamiltonian Yang-Mills theory in Coulomb gauge. The flow equations for the static gluon and ghost propagators are solved under the assumption of ghost dominance within different diagrammatic approximations. The results are compared to those obtained in the variational approach and the reliability of the approximations is discussed.

Generalized Gell-Mann–Low theorem for relativistic bound states

The recently established generalized Gell-Mann\char21{}Low theorem is applied in lowest perturbative order to bound-state calculations in a simple scalar field theory with cubic couplings. The approach, via the generalized Gell-Mann\char21{}Low theorem retains, while being fully relativistic, many of the desirable features of the quantum mechanical approaches to bound states. In particular, no abnormal or unphysical solutions are found in the model under consideration. Both the nonrelativistic and one-body limits are straightforward and consistent. The results for the spectrum are compared to those of the Bethe-Salpeter equation (in the ladder approximation) and related equations.

Equal-time two-point correlation functions in Coulomb gauge Yang–Mills theory

We apply a new functional perturbative approach to the calculation of the equal- time two-point correlation functions and the potential between static color charges to one-loop order in Coulomb gauge Yang-Mills theory. The functional approach proceeds through a solution of the Schrodinger equation for the vacuum wave functional to order g 2 and derives the equal-time correlation functions from a functional integral representation via new diagrammatic rules. We show that the results coincide with those obtained from the usual Lagrangian functional integral approach, extract the beta function and determine the anomalous dimensions of the equal-time gluon and ghost two-point functions and the static potential under the assumption of multiplicative renormalizability to all orders.

Integral representations combining ladders and crossed-ladders

A bstractWe use the worldline formalism to derive integral representations for three classes of amplitudes in scalar field theory: (i) the scalar propagator exchanging N momenta with a scalar background field (ii) the “half-ladder” with N rungs in x-space (iii) the four-point ladder with N rungs in x-space as well as in (off-shell) momentum space. In each case we give a compact expression combining the N! Feynman diagrams contributing to the amplitude. As our main application, we reconsider the well-known case of two massive scalars interacting through the exchange of a massless scalar. Applying asymptotic estimates and a saddle-point approximation to the N-rung ladder plus crossed ladder diagrams, we derive a semi-analytic approximation formula for the lowest bound state mass in this model.

Bound States in Yukawa Theory

A generalization of the Gell-Mann–Low Theorem is applied to lowest nontrivial order to bound state calculations in Yukawa theory. We present the solution of the corresponding effective Schrödinger equation for two-fermion bound states with the exchange of a massless boson. The complete low-lying bound state spectrum is obtained for fine structure constants below one and different ratios of the constituent masses. The consistency of the nonrelativistic and one-body limits is explicitly verified.

Bloch–Wilson Hamiltonian and a Generalization of the Gell-Mann–Low Theorem

The effective Hamiltonian introduced many years ago by Bloch and generalized later by Wilson, appears to be the ideal starting point for Hamiltonian perturbation theory in quantum field theory. The present contribution derives the Bloch–Wilson Hamiltonian from a generalization of the Gell-Mann–Low theorem, thereby enabling a diagrammatic analysis of Hamiltonian perturbation theory in this approach.

The infrared fixed point of Landau gauge Yang-Mills theory: A renormalization group analysis

The infrared behavior of gluon and ghost propagators in Landau gauge Yang-Mills theory has been at the center of an intense debate over the last decade. Different solutions of the Dyson-Schwinger equations have a different behavior of the propagators in the infrared: in the so-called scaling solutions both propagators follow a power law, while in the decoupling solutions the gluon propagator shows a massive behavior. The latest lattice results favor the decoupling solutions. In this contribution, after giving a brief overview of the present status of analytical and semi-analytical approaches to the infrared regime of Landau gauge Yang-Mills theory, we will show how Callan-Symanzik renormalization group equations in an epsilon expansion reproduce both types of solutions and single out the decoupling solutions as the infrared-stable ones for space-time dimensions greater than two, in agreement with the lattice calculations.

MASS RENORMALIZATION IN A HAMILTONIAN APPROACH

We consider the energies of the one-particle states of relativistic quantum field theories in old-fashioned perturbation theory, and treat their regularization and renormalization from a strictly Hamiltonian perspective. We show that the one-loop diagrams lead to the renormalization of the mass for bosons and fermions identically to covariant perturbation theory, provided an appropriate regularization scheme is used. In particular, we show that a naive spatial momentum cutoff breaks covariance (in the limit where the cutoff is removed) in the case of the one-fermion states.