Showing papers by "Jürg Fröhlich published in 1981"

The Kosterlitz-Thouless transition in two-dimensional Abelian spin systems and the Coulomb gas

Abstract: We rigorously establish the existence of a Kosterlitz-Thouless transition in the rotator, the Villain, the solid-on-solid, and the ℤ n models, forn large enough, and in the Coulomb lattice gas, in two dimensions. Our proof is based on an inductive expansion of the Coulomb gas in the sine-Gordon representation, extending over all possible distance scales, which expresses that gas as a convex superposition of dilute gases of neutral molecules whose activities are small if β is sufficiently large. Such gases are known not to exhibit screening. Abelian spin systems are related to a Coulomb gas by means of a duality transformation.

On the absence of spontaneous symmetry breaking and of crystalline ordering in two-dimensional systems

Abstract: We develop a unified approach, based on Araki's relative entropy concept, to proving absence of spontaneous breaking of continuous, internal symmetries and translation invariance in two-dimensional statistical-mechanical systems. More precisely, we show that, under rather general assumptions on the interactions, all equilibrium states of a two-dimensional system have all the symmetries, compact internal and spatial, of the dynamics, except possibly rotation invariance. (Rotation invariance remains unbroken if connected correlations decay more rapidly than the inverse square distance.) We also prove that two-dimensional systems with a non-compact internal symmetry group, like anharmonic crystals, typically do not have Gibbs states.

On the statistical mechanics of classical Coulomb and dipole gases

Abstract: A detailed, rigorous study of the statistical mechanics-screening- and critical properties, phase diagrams, etc., of classical Coulomb monopole and dipole gases in two or more dimensions is presented. The statistical mechanics of the two-dimensionalXY and Villain models is reconsidered and related to the one of two-dimensional lattice Coulomb gases. At low temperatures and moderate densities those gases behave like dipole gases. The Kosterlitz-Thouless transition is analyzed in that perspective and characterized by an order parameter. Techniques useful for a proof of existence of such a transition in a two-dimensional hard-core Coulomb gas are developed and applied to the study of dipole gases.

On the Construction of Quantized Gauge Fields III. The Two-Dimensional Abelian Higgs Model Without Cutoffs

Abstract: In this paper the construction of the two-dimensional abelian Higgs model begun in two earlier articles is completed. First we show how to remove the remaining ultraviolet cutoff on the gauge field, then we construct the infinite volume limit and verify the axioms of Osterwalder and Schrader for the expectation values of gauge invariant local fields. Finally it is shown that an auxiliary gauge field mass that was introduced to avoid infrared problems can be safely removed.

Phase Diagrams and Critical Properties of (Classical) Coulomb Systems

Abstract: It seems good to start these lecture notes with the cautioning remark that, in a sense, their contents do not really fit into the topic “Rigorous Atomic and Molecular Physics”, although the results we shall discuss are certainly rigorous and do concern atoms, ions and molecules to some extent. Traditionally, the underlying theoretical framework for atomic and molecular physics is thought to be quantum mechanics, and the number of degrees of freedom of the atomic and molecular systems which are considered is finite, at least if the radiation field is neglected which is what workers in that field do almost always.

States of one-dimensional Coulomb systems as simple examples of θ vacua and confinement

Abstract: The one-dimensional Coulomb system is known to have equilibrium states with nonvanishing electric field. These states are shown here to be analogous, and related, to theθ vacua which have been discussed for gauge theories in two or more space-time dimensions. The system exhibits confinement of fractional charges, which we dicuss with the purpose of offering a simple example of theθ-vacua phenomenology. Precise relations and connections between one-dimensional Coulomb gases and two-dimensional Abelian gauge theories, and quantum-mechanical matter systems, are discussed.

Constructive and Euclidean Field Theory and Connections to Statistical Mechanics

Abstract: Outline of axiomatic relativistic quantum field theory R F Streater Stochastic quantisation: stabilising quantum models Z Haba Markovian representations of current algebras R F Streater Looking Back at Quantum Field Theory A S Wightman The renormalisation group approach to scaling in physics D J Wallace and R K P Zia The Skyrme model and strong interactions (On the 30th anniversary of the creation of the Skyrme model) V G Makhan'kov, Yu P Rybakov and Valerii I Sanyuk P. A. M. Dirac and the formation of the basic ideas of quantum field theory B V Medvedev and Dmitrii V Shirkov Constructive and Euclidean Field Theory and Connections to Statistical Mechanics