Taking the “corrector” into account, we obtain more accurate approximation for the resolvent in the (L2 → L2)-operator norm with the error estimate by C 2. Besides, we find approximation with corrector for the resolvent in the (L2 → H 1)-operator norm with the error estimate of order O( ). The obtained results are of new type in the homogenization theory.

The method is based on the abstract operator theory approach for selfadjoint operator families A(t ) admitting a factorization of the form A(t ) = X(t ) X(t ), X(t ) = X0 + t X1. It turns out that the homogenization procedure for the operator A is determined by the spectral characteristics of the periodic operator A near the bottom of the spectrum. Therefore, homogenization procedure can be treated as a threshold effect.

General results are applied to specific operators of mathematical physics: the acoustics operator, the operator of elasticity theory, the Maxwell operator. A special attention is paid to the operators of quantum mechanics, namely, to the Schrödinger operator, the magnetic Schrödinger operator (with sufficiently small magnetic potential), the two dimensional Pauli operator. The effective characteristics for these operators are studied. The effective matrix arising in the homogenization theory is closely related to the tensor of effective masses which is well known in quantum mechanics. It turns out that for the two dimensional periodic Pauli operator the tensor of effective masses is scalar, which attests some hidden symmetry.

The results were obtained in 2001–2006 jointly with M. Sh. Birman.

3.11 Quantum Field Theory

Organizer K. Fredenhagen (Hamburg)

3.11.1 Algebraic Aspects of Perturbative and Non-Perturbative QFT

Christoph Bergbauer

IHES bergbau@ihes.fr

We review the Connes-Kreimer approach to perturbative renormalization in terms of Hopf and Lie algebras of Feynman graphs which capture the combinatorial aspects of the renormalization procedure. The solution of the Bogoliubov recursion is essentially given by the antipode map of the Hopf algebra of graphs. Important properties can be traced back to 1-cocycles in the Hochschild cohomology of these Hopf algebras. At the same time these 1-cocycles provide the building blocks of DysonSchwinger equations and thus a link to non-perturbative results. We finally discuss new ideas on the structure and towards actual solutions of these Dyson-Schwinger equations.

3.11.2 Quantum Field Theory in Curved Space-Time

Stefan Hollands

University of Goettingen hollands@theorie.physik.uni-goettingen.de

The theory of quantum fields on a curved background is interesting both physically —describing effects such as the creation of primordial fluctuations, particle creation in the expansing universe, black-hole radiance—as well as mathematically, because it combines in an interesting way ideas from differential geometry, analysis, and quantum field theory.

I review recent developments in the field, emphasizing the role and construction the operator product expansion in curved spacetime. In particular, I will argue that properties such as associativity, general covariance, renormalization group flow/scaling, and spectral properties of the quantum field theory are encoded in the operator product expansion. I indicate how this tool may be used to analyze quantitatively dynamical processes in the Early Universe.

3.11.3String-Localized Quantum Fields, Modular Localization, and Gauge Theories

Jens Mund

Universidade Federal de Juiz de Fora mund@fisica.ufjf.br

The concept of modular localization introduced by Brunetti, Guido and Longo, and Schroer, can be used to construct quantum fields. It combines Wigner’s particle concept with the Tomita-Takesaki modular theory of operator algebras. I shall report on the construction of free fields which are localized in semi-infinite strings extending to spacelike infinity (joint work with B. Schroer and J. Yngvason). Particular applications are: The first local (in the above sense) construction of fields for Wigner’s massless “infinite spin” particles; Anyons in d = 2 + 1; String-localized vector/tensor potentials for Photons and Gravitons, respectively. Some ideas will be presented concerning the perturbative construction of gauge theories (and quantum gravity) completely within a Hilbert space, trading gauge dependence with dependence on the direction of the localization string.

3.11.4Quantization of the Teichmüller Spaces: Quantum Field Theoretical Applications

Joerg Teschner

DESY teschner@mail.desy.de

We will review the geometric interpretation of quantum Liouville theory as a quantum theory of spaces of Riemann surfaces. This interpretation can be used to establish the consistency of the bootstrap construction of Liouville theory in the presence of conformal boundary conditions. It also paves the way towards the study of Liouville theory on higher genus Riemann surfaces. If time permits we will outline a possible extension of this framework to more general conformal field theories.

3.12 2D Quantum Field Theory

Organizer J. Cardy (Oxford)

3.12.1 Lattice Supersymmetry From the Ground Up

Paul Fendley

University of Virginia fendley@rockpile.phys.virginia.edu

I discuss several models of itinerant fermions which exhibit explicit supersymmetry on the lattice. In 1 + 1 dimensions, one model gives a lattice regularization of the Thirring model, and shows how the combinatorial results of Stroganov et al. can be related to supersymmetry. In both 1 + 1 and 2 + 1 dimensions, we can find models with extensive ground-state entropy. Finally, I present results on a generalized Yangian-like symmetry algebra underlying some of these models.

3.12.2Analytical Solution for the Effective Charging Energy of the Single Electron Box

Sergei Lukyanov

Rutgers University sergei@physics.rutgers.edu

A single electron box is a low-capacitance metallic island, connected to an outside lead by a tunnel junction. Over the last decade, correct analytical expressions describing the single electron box in the limit of large tunneling conductance have been the subject of controversial debate. In this talk, we will discuss recent exact