Decay of correlation rate in the mean field limit of point vortices ensembles

We consider the Mean Field limit of Gibbsian ensembles of 2-dimensional (2D) point vortices on the torus. It is a classical result that in such limit correlations functions converge to 1, that is, point vortices decorrelate: We compute the rate at which this convergence takes place by means of Gaussian integration techniques, inspired by the correspondence between the 2D Coulomb gas and the Sine-Gordon Euclidean field theory.

The weakly interacting Bose gas at the critical temperature

Chapter 20 examines effects of weak repulsive interactions in a Bose–Einstein condensate and the transition from Bose–Einstein condensate to superfluid phase transition. Renormalization group methods are used and a universal amplitude is calculated by non–perturbative methods. After the discovery of the predicted Bose–Einstein condensation, which is a property of free bosons, an interesting issue was the effects of weak repulsive interactions. In this chapter, it is shown that, near the transition temperature, the initial non–relativistic field theory can be replaced by a relativistic effective Euclidean field theory known to describe a superfluid phase transition (a dimensional reduction). These theoretical considerations are illustrated by an evaluation of the universal variation of the transition temperature at weak coupling. For this purpose, the O(2) symmetry of the model is generalized to O(N) symmetry, and large N techniques are used.

CONSTRUCTIVE GEOMETRY AND THE PARALLEL POSTULATE

Euclidean geometry, as presented by Euclid, consists of straightedge-and-compass constructions and rigorous reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. This involves finding “uniform” constructions where normally a case distinction is used. For example, in finding a perpendicular to line L through point p, one usually uses two different constructions, “erecting” a perpendicular when p is on L, and “dropping” a perpendicular when p is not on L, but in constructive geometry, it must be done without a case distinction. Classically, the models of Euclidean (straightedge-and-compass) geometry are planes over Euclidean fields. We prove a similar theorem for constructive Euclidean geometry, by showing how to define addition and multiplication without a case distinction about the sign of the arguments. With intuitionistic logic, there are two possible definitions of Euclidean fields, which turn out to correspond to different versions of the parallel postulate.We consider three versions of Euclid’s parallel postulate. The two most important are Euclid’s own formulation in his Postulate 5, which says that under certain conditions two lines meet, and Playfair’s axiom (dating from 1795), which says there cannot be two distinct parallels to line L through the same point p. These differ in that Euclid 5 makes an existence assertion, while Playfair’s axiom does not. The third variant, which we call the strong parallel postulate, isolates the existence assertion from the geometry: it amounts to Playfair’s axiom plus the principle that two distinct lines that are not parallel do intersect. The first main result of this paper is that Euclid 5 suffices to define coordinates, addition, multiplication, and square roots geometrically.We completely settle the questions about implications between the three versions of the parallel postulate. The strong parallel postulate easily implies Euclid 5, and Euclid 5 also implies the strong parallel postulate, as a corollary of coordinatization and definability of arithmetic. We show that Playfair does not imply Euclid 5, and we also give some other independence results. Our independence proofs are given without discussing the exact choice of the other axioms of geometry; all we need is that one can interpret the geometric axioms in Euclidean field theory. The independence proofs use Kripke models of Euclidean field theories based on carefully constructed rings of real-valued functions. “Field elements” in these models are real-valued functions.

GENERALIZED SCHRÖDINGER EQUATION IN EUCLIDEAN FIELD THEORY

We investigate the idea of a "general boundary" formulation of quantum field theory in the context of the Euclidean free scalar field. We propose a precise definition for an evolution kernel that propagates the field through arbitrary space–time regions. We show that this kernel satisfies an evolution equation which governs its dependence on deformations of the boundary surface and generalizes the ordinary (Euclidean) Schrödinger equation. We also derive the classical counterpart of this equation, which is a Hamilton–Jacobi equation for general boundary surfaces.