Constructive and Multiscale Methods in Quantum Theory

Titles and Abstracts

Abdelmalek Abdesselam  (University of Virginia at Charlottesville)

Clustering estimates for the truncated functions of an unbounded spin system

We will report on joint work with Aldo Procacci and Benedetto Scoppola regarding an unbounded spin system on the lattice given by a massive complex Gaussian measure perturbed by a quartic interaction. We obtained a unified treatment of clustering, i.e., volume independent l^1 type estimates on the truncated correlation functions for various regimes of the parameters. We will in particular focus on the near Gaussian regime which is the typical situation in quantum field theory. Our methods use a recent version of cluster expansion techniques from constructive field theory. Our result provides a proof in the Bosonic case of an important hypothesis used in recent work by Lukkarinen and Spohn on the kinetic limit of interating quantum fluids.

Stefan Adams  (University of Warwick)

Gradient fields with non-convex potentials and elasticity

ecently the study of gradients fields has attained a lot of attention because they are space-time analogy of Brownian motions, and are connected to the Schramm-Loewner evolution. The corresponding discrete versions arise in equilibrium statistical mechanics, e.g., as approximations of critical systems and as effective interface models. The latter models - seen as gradient fields - enable one to study effective descriptions of phase coexistence. Gradient fields have a continuous symmetry and coexistence of different phases breaks this symmetry. In the probabilistic setting gradient fields involve the study of strongly correlated random variables. As a result the asymptotic behavior (free energy, measures) depends on the boundary constraint (enforced tilt). Main challenge is the question of uniqueness of Gibbs measures and the strict convexity of the free energy (surface tension) for any non-convex interaction potential. We present in the talk the first break through for low temperatures using Gaussian measures and renormalization group techniques yielding an analysis in terms of dynamical systems. Our main input is a finite range decomposition for a family of Gaussian measures depending on non isotropic tuning parameters. We outline also the connection to the Cauchy-Born rule which states that the deformation on the atomistic level is locally given by an affine deformation at the boundary.

Work in cooperation with R. Kotecky and S. Müller.

David Brydges  (UBC, Vancouver)

Self-avoiding Walk and the Renormalisation Group

Self-avoiding walks on Z^d are simple-random walk paths without self-interstections. Self-avoiding walks of the same length are declared to be equally likely. The basic question is how far on average is their endpoint from the origin?

After a brief review of the current state of knowledge I will describe work in progress with Gordon Slade for the case d = 4 which is an application of the renormalisation group to a supersymmetric lattice field theory. Our immediate goal is to prove that the critical two-point function (Green function) for a spread-out model of self-avoiding walks on Z^d decays like |x|^{-2} at large distances, as it does for simple random walk.

Wojciech de Roeck  (ETH Zürich)

Irreversibility in quantum particle-field systems

I will discuss results and progress in studying irreversible aspects of the long-time behaviour of a quantum particle coupled to fields. The phenomena are: diffusion, decherence, friction and Cerenkov radiation. This is based on joint work with A. Pizzo and J. Fröhlich.

Joel Feldman  (UBC, Vancouver)

The Stationary Phase Approximation for Bose Systems

Jürg Fröhlich  (ETH Zürich)

Quantum Theory of Experiments

It is sketched what Quantum Mechanics (QM) predicts about possible events when experiments are performed to measure physical quantities. It is first clarified in which sense the predictions of QM are fundamentally probabilistic and under what circumstances deterministic predictions can be made. Subsequently, the conventional formalism for the calculation of "frequencies" (or probabilities) of "consistent histories" is reviewed. The role of decoherence in rendering histories of events "consistent" is explained. If time permits some remarks about "relativistic quantum theory" will be made. No claim of originality is made for anything discussed in this lecture.

Alessandro Giuliani   (Universita Roma III)

Periodic Striped Minimizers in a 2D Model for Martensitic Phase Transitions

We shall consider an effective 2D scalar model for the formation of mesoscopic domain patterns in martensitic shape-memory alloys at the interface between a region occupied by the parent (austenite) phase and a region occupied by the product (martensite) phase, which can occur in two variants (twins). The model, first proposed by Kohn and Müller (KM), depends on two material parameters, measuring the strength of the interaction between austenite and martensite, and the surface tension in the martensite. Depending on their relative values, the minimizers of the KM model are expected to display different qualitative features: either striped periodic order or self-similar branching. In this talk I will describe the proof that, in a suitable range of parameters, the minimizers, in fact, display periodic striped order. The proof is based on a combination of reflection positivity and Poincare'-type inequalities. The talk is based on joint work with S. Müller.

Marcel Griesemer   (Universität Stuttgart)

On the Minimal Energy of Translation Invariant N-Polaron Systems

For systems of N charged fermions (e.g. electrons) interacting with longitudinal optical quantized lattice vibrations of a polar crystal we derive upper and lower bounds on the minimal energy within the model of H. Fröhlich. The only parameters of this model, after removing the ultraviolet cutoff, are the constants U>0 and alpha>0 measuring the electron-electron and the electron-phonon coupling strengths. They are constrained by the condition sqrt{2} alpha U the phonon-mediated electron-electron attraction overcomes the Coulomb repulsion and E_N behaves like - N^{7/3}.

Razvan Gurau  (Perimeter Institute, Waterloo)

Two points of view on quantum field theory

This talk is divided into two parts. In the first part we give an introduction to a new point of view on nonperturbative quantum field theory. Our approach is based on the notion of tree, which appears to be the right compromise between graphs and functional integration. In the second part of this talk we will give an introduction to a new quantum field theory (called group field theory) which is the appropriate generalization of matrix models in higher dimmensions.

Christian Jäkel

The relativistic KMS condition for the thermal n-point functions of the P(phi)_2 model

Thermal quantum field theories are expected to obey the so-called relativistic KMS condition, which replaces both the relativistic spectrum condition familiar from Wightman quantum field theory and the traditional KMS condition that characterizes equilibrium states in quantum statistical mechanics. In a previous work (together with C. Gerard) it has been shown that the two-point function of the thermal P(phi)_2 model satisfies the relativistic KMS condition. Here we extend this result to general n-point functions. This is joint work with Florian Robl.

Jozsef Lörinczi

Feynman-Kac-type formulae for paths with jump discontinuities and QFT applications

Since the Feynman-Kac formula was introduced for studying Schrödinger operators, functional integration methods have been extended to include other classes of operators also covering QFT models, however, the analysis based on random processes having almost surely continuous paths remained an essential feature. In the mathematical physics literature there appear to be relatively few systematic attempts in going beyond continuous paths and replace them with paths allowing jump discontinuities.

In this talk I plan to present how more general Levy processes than Brownian motion are useful in a mathematical description of important features in quantum mechanics and QFT. First I explain how spin in the Pauli-Fierz model can be described by a joint Brownian motion and Poisson process. This leads, in particular, to a new energy comparison inequality. Then I address the functional integral representation of fractional Schrödinger operators and generalizations in terms of Bernstein functions of the Laplacian. This gives rise to subordinated Brownian motion, which may be seen as a random time change driven by another stochastic process. In particular, the case of relativistic Schrödinger operators with magnetic potential and spin will be covered (for which no Weyl-quantization will be needed), and new diamagnetic inequalities presented. Finally, I point out how this can be further applied to a rigorous analysis of ground states in the relativistic Nelson and Pauli-Fierz models.

Jani Lukkarinen  (Helsinki University)

Controlling a kinetic scaling limit of the nonlinear Schrödinger equation with random initial data

In this joint work with Herbert Spohn, we consider the decay of time-correlations of a complex field evolving under a discrete nonlinear Schrödinger equation, for weak nonlinearity. The initial date is random, distributed according to an equilibrium Gibbs measure, and we consider kinetic time-scales, O(lambda^(-2)), in the limit where the strength of the nonlinearity, lambda, is taken to zero. As we show in arXiv:0901.3283, the kinetic scaling limit can be controlled rigorously, using invariance properties and a diagrammic perturbation expansion. I discuss the classification of the diagrams in more detail.

Marco Merkli  (St. John's Memorial University)

A Resonance Approach to Evolution of Entanglement

Walter Pedra  (Universität Mainz)

Equilibrium States of Fermi Systems With Long Range Interactions

Benjamin Schlein  (Cambridge University)

Bulk Universality for Wigner Matrices

In this talk I am going to present recent results, obtained in collaboration with L. Erdos, J. Ramirez, and H.-T. Yau, concerning spectral properties of hermitian Wigner matrices. In particular, I am going to discuss a proof of the universality of Dyson's sine-kernel for the description of the local eigenvalue statistics, under appropriate assumptions on the probability density of the matrix entries.

Herbert Spohn  (TU München)

The kinetic scaling limit for weakly interacting quantum fluids

I report on joint work with J. Lukkarinen and provide an overview on the kinetic limit for weakly interacting quantum fluids. I stress the observation that on the diagrammatic level the difference between quantum fluids and the nonlinear Schroedinger equation with random initial data is marginal. For details see arXiv:0807.5072.