The planned topics of the workshop are the analysis of quantum many-particle systems, stochastic methods like large deviations theory and interacting Brownian motions applied to such systems, and functional integral methods. We also aim to have dicsussions about connections between the different approaches. Talks start Thursday morning and terminate Saturday afternoon.

Speakers

Abdelmalek Abdesselam

Université Paris 13

Volker Bach

Johannes-Gutenberg-Universität, Mainz

Marek Biskup

UCLA, Los Angeles

Tony Dorlas

Dublin Institute for Advanced Studies of Theoretical Physics

Density functional theory is moving towards a more complicated version called Density-Matrix-Functional Theory, of which Hartree-Fock theory is an old example. A modification of Hartree-Fock theory due to Mueller is now being used by several researchers. The mathematical foundations (existence and uniqueness of energy minimizers, etc.) are, however, more complicated than for Hartree-Fock theory; in work with R. Frank, R. Seiringer and H. Siedentop we have been able to address the most important questions, but some interesting ones still remain.

I will describe a new general approach to the thermodynamic limit of charged systems, e.g. ordinary matter consisting of nuclei and electrons. As applications of this approach I will briefly sketch how to use it to show the existence of the thermodynamic limit in the case of dynamic nuclei and in the case of nuclei fixed in a crystalline ordering. The first case of dynamic nuclei was proved by Lieb and Lebowitz in 1972 and the latter case of fixed nuclei was settled by Fefferman in 1985 by a completely different method.

The time dependent Gross-Pitaevskii equation describes the dynamics of initially trapped Bose-Einstein condensates. We present a rigorous proof of this fact starting from a many-body bosonic Schroedinger equation with a short scale repulsive interaction in the dilute limit. Our proof shows the persistence of an explicit short scale correlation structure in the condensate. This is a joint work with B. Schlein and H.T. Yau.

We will report our recent rigorous non-perturbative construction of a massless discrete trajectory for Wilson's exact renormalization group: we do not use hierarchical approximations. The model is a three dimensional Euclidean field theory with a modified free propagator. The trajectory realizes the mean field to critical crossover from the ultraviolet Gaussian fixed point to an analogy recently constructed by Brydges, Mitter and Scoppola of the Wilson-Fisher nontrivial fixed point.

We discuss bounds on the free energy of homogeneous Bose and Fermi gases at non-zero temperature. In the dilute regime, the leading order correction compared to an ideal quantum gas depends on the particle interaction only through the scattering length. Moreover, in the Bose case it depends non-trivially on temperature through the critical density for Bose-Einstein condensation. Some of the key ingredients in the proof are the use of coherent states to deal with the condensation, as well as new correlation estimates relying on the monotonicity of the relative entropy under partial traces.

Functional integrals have long been used, formally, to provide intuition about the behaviour of quantum field theories. For the past several decades, they have also been used, rigorously, in the construction and analysis of those theories. I will talk about the rigorous derivation of some functional integral representations for the partition function and correlation functions of (cutoff) many Boson systems that provide a suitable starting point for their construction.

We develop a power series representation and estimates for an effective action of the form $$ \log\frac{\int{\rm e}^{\mathcal{A}(\phi,\psi)}{\rm d}\mu_r(\phi)}{\int{\rm e}^{\mathcal{A}(\phi,0)}{\rm d}\mu_r(\phi)}. $$ Here, $ \mathcal{A}(\phi,\psi) $ is an analytic function of the real fields $ \phi(x),\psi(x) $ indexed by $ x $ in a finite set $ X$, and $ {\rm d}\mu_r(\phi) $ is the product measure characterised by $$ \int f(\phi){\rm d}\mu_r(\phi)=\int f(\phi)\prod_{x\in X}\chi(|\phi(x)|\le r){\rm d}\phi(x). $$ Such effective interactions occur in the small field region for a renormlization group analysis. The customary way to analyse them is a cluster expansion, possibly preceded by a decoupling expansion. Using methods similar to a polymer expansion, we estimate the power series of effective interaction without introducing an artificial decomposition of the underlying space. This technique is illustrated by a model renormalization group flow motivated by the ultraviolet regime in many boson systems.

For a nonrelativistic atom coupled to the quantized radiation field, an algorithm for the construction of the ground state, the ground energy, and (Rayleigh) scattering matrix elements is derived. The algorithm allows to expand these quantities in explicit form in terms of bare quantities, to arbitrary accuracy, in powers of the finestructure constant $\alpha$. This is joint work with J. Fröhlich and A. Pizzo.

In bosonic many body systems described by mean-field Hamiltonians, the time evolution of factorized N body wave functions can be described in the limit of large N by the solution of a nonlinear Hartree equation. More precisely, one can prove that the one-particle density associated with the solution of the N body Schroedinger equation converges, as N tends to infinity, to the orthogonal projection onto the solution to the Hartree equation. Although the difference between the one-particle density and the orthogonal projection can be shown to be of order 1/N for short times, so far it has not been possible to extend this estimate on the error to long times. In this talk I am going to present some preliminary result in this direction obtained in collaboration with I. Rodnianski.

I will discuss a model of random permutations on a lattice where sites are preferably sent onto neighbours. The main question deals with the possible occurrence of infinite cycles, which should be related to Bose-Einstein condensation.
The random cycle model without interactions behaves very closely to the ideal Bose gas, both qualitatively and quantitatively. This was observed numerically. An interesting problem is to introduce interactions that mimic those of the Bose gas. This may shed light on how the critical temperature depends on the interactions.

We present a formulation of the theory of charges in equilibrium with the radiation field based on a joint functional representation, combining the Feynman-Kac-Ito path integral for the particles with the bosonic integral for photons. In this formalism cluster expansion techniques of classical statistical mechanics become operative.They provide an alternative to the usual Feynman diagrammatics in many-body problems, which is not perturbative with respect to the coupling constant. As an application we show that the effective Coulomb interaction between quantum charges is partially screened by thermalized photons at large distances as the result of a subtle conspiracy between matter and field Planck constants.

Starting from the many body Hamiltonian the leading order energy and density asymptotics for the ground state of a dilute, rotating Bose gas in an anharmonic trap is derived in the 'Thomas Fermi' (TF) limit when the Gross-Pitaevskii coupling parameter and/or the rotation velocity tend to infinity. Although the many-body wave function is expected to have a complicated phase, the leading order contribution to the energy can be computed by minimizing a simple functional of the density alone. This is joint work with J.B. Bru, M. Correggi and P. Pickl.

It has been known from the work of Berezin and Lieb from early 1970s that the free energy of quantum spin systems converges, in the limit of large spin, to the free energy of the corresponding classical model. Unfortunately, this alone does not give any information about phase transitions in the quantum system. I will show how one can enhance the Berezin-Lieb upper bound into an inequality for matrix elements (relative to the overcomplete basis of coherent states) which, for models that permit the use of chessboard estimates, allow proofs of phase transitions by direct comparison with the classical counterpart.
In some cases (anisotropic Heissenberg antiferromagnet) this offers an alternative to "exponential localization" developed by Frohlich and Lieb; in other cases (models with temperature driven transitions or highly degenerate ground states) this yields proofs of phase transitions that have not been accessible heretofore. Based on joint work with L. Chayes and S. Starr.

Participants

Abdelmalek Abdesselam

Université Paris 13

Stefan Adams

MPI MIS

Philipp Altrock

Universität Leipzig

Volker Bach

Johannes-Gutenberg-Universität

Marika Behnert

Universität Leipzig

Marek Biskup

UCLA

Jean-Bernard Bru

Universität Wien

Andrea Collevecchio

MPI MIS

Codina Cotar

University of British Columbia

Christoph Dehne

Universität Leipzig

Jean-Dominique Deuschel

TU Berlin

Nicolas Dirr

MPI MIS

Tony Dorlas

Dublin Institute for Advanced Studies of Theoretical Physics

Detlef Dürr

LMU München

Laszlo Erdös

LMU

Joel Feldman

University of British Columbia

Jürgen Gärtner

TU Berlin

Kay-Uwe Giering

Universität Leipzig

Sören Graupner

Universität Leipzig

Christoph Husemann

Universität Leipzig

Hans-Christoph Kaiser

WIAS

Horst Knörrer

ETH

Wolfgang König

Universität Leipzig

Roman Kotecký

Center for Theoretical Study

Micaela Krieger-Hauwede

MPI MIS

Boris Kruglikov

MPI MIS and University of Tromso

Elliott H. Lieb

Princeton University

Martin Lohmann

ETH

Stephan Luckhaus

Universität Leipzig

Fabricio Macia Lang

Universidad Complutense de Madrid

Philippe-André Martin

EPFL

Stefan Müller

MPI MIS

Walter Pedra

Universität Mainz

Peter Pickl

LMU München

Gerd Rudolph

Universität Leipzig

Manfred Salmhofer

Universität Leipzig

Benjamin Schlein

Harvard University

Robert Seiringer

Princeton University

Jan Philip Solovej

University of Copenhagen

Herbert Spohn

TU München

Jose Trashorras

Université Paris-Dauphine

Dimitrios Tsagkarogiannis

MPI MIS

Daniel Ueltschi

University of Warwick

Armin Uhlmann

Universität Leipzig

Rainer Verch

Universität Leipzig

Jakob Yngvason

Erwin Schroedinger Institut

Scientific Organizers

Stefan Adams

Max-Planck-Institut für Mathematik in den Naturwissenschaften