Showing papers in "Communications in Mathematical Physics in 1981"

A new proof of the positive energy theorem

Abstract: A new proof is given of the positive energy theorem of classical general relativity. Also, a new proof is given that there are no asymptotically Euclidean gravitational instantons. (These theorems have been proved previously, by a different method, by Schoen and Yau.) The relevance of these results to the stability of Minkowski space is discussed.

Proof of the positive mass theorem. II

Abstract: The positive mass theorem states that for a nontrivial isolated physical system, the total energy, which includes contributions from both matter and gravitation is positive. This assertion was demonstrated in our previous paper in the important case when the space-time admits a maximal slice. Here this assumption is removed and the general theorem is demonstrated. Abstracts of the results of this paper appeared in [11] and [13].

Unitary representations of some infinite-dimensional groups

Abstract: We construct projective unitary representations of (a) Map(S1;G), the group of smooth maps from the circle into a compact Lie groupG, and (b) the group of diffeomorphisms of the circle. We show that a class of representations of Map(S1;T), whereT is a maximal torus ofG, can be extended to representations of Map(S1;G),

Absence of singular continuous spectrum for certain self-adjoint operators

Abstract: We give a sufficient condition for a self-adjoint operator to have the following properties in a neighborhood of a pointE of its spectrum: a) its point spectrum is finite; b) its singular continuous spectrum is empty; c) its resolvent satisfies a class of a priori estimates.

Geometrical structure and ultraviolet finiteness in the supersymmetric σ-model

Abstract: A complete geometrical classification of supersymmetric σ-models is given. Extended supersymmetry requires covariantly constant complex structures, and Kahler and hyperkahler manifolds play a special role. As an application of the classification, it is shown that a particular class of these models is on-shell ultraviolet finite to all orders in perturbation theory.

Absolutely continuous invariant measures for one-parameter families of one-dimensional maps

Abstract: Given a one-parameter familyf λ(x) of maps of the interval [0, 1], we consider the set of parameter values λ for whichf λ has an invariant measure absolutely continuous with respect to Lebesgue measure. We show that this set has positive measure, for two classes of maps: i)f λ(x)=λf(x) where 0<λ≦4 andf(x) is a functionC 3-near the quadratic mapx(1−x), and ii)f λ(x)=λf(x) (mod 1) wheref isC 3,f(0)=f(1)=0 andf has a unique nondegenerate critical point in [0, 1].

Schrödinger operators with magnetic fields. III. Atoms in homogeneous magnetic field

Abstract: We prove a large number of results about atoms in constant magnetic field including (i) Asymptotic formula for the ground state energy of Hydrogen in large field, (ii) Proof that the ground state of Hydrogen in an arbitrary constant field hasL z = 0 and of the monotonicity of the binding energy as a function ofB, (iii) Borel summability of Zeeman series in arbitrary atoms, (iv) Dilation analyticity for arbitrary atoms with infinite nuclear mass, and (v) Proof that every once negatively charged ion has infinitely many bound states in non-zero magnetic field with estimates of the binding energy for smallB and largeL z .

Statistical properties of Lorentz gas with periodic configuration of scatterers

Abstract: In our previous paper Markov partitions for some classes of dispersed billiards were constructed. Using these partitions we estimate the decay of velocity auto-correlation function and prove the central limit theorem of probability theory and Donsker's Invariance Principle for Lorentz Gas with periodic configuration of scatterers.

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.

Two-dimensional generalized Toda lattice

Abstract: The zero curvature representation is obtained for the two-dimensional generalized Toda lattices connected with semisimple Lie algebras. The reduction group and conservation laws are found and the mass spectrum is calculated.

A method of integration over matrix variables

Abstract: The integral over twon ×n hermitan matricesZ(g, c)=∫dAdBexp{−tr[A 2+B 2−2cAB+g/n(A 4+B 4)]} is evaluated in the limit of largen. For this purpose use is made of the theory of diffusion equation and that of orthogonal polynomials with a non-local weight. The above integral arises in the study of the planar approximation to quantum field theory.

Stability and isolation phenomena for Yang-Mills fields

Abstract: In this article a series of results concerning Yang-Mills fields over the euclidean sphere and other locally homogeneous spaces are proved using differential geometric methods One of our main results is to prove that any weakly stable Yang-Mills field overS4 with groupG=SU2, SU3 orU2 is either self-dual or anti-self-dual The analogous statement for SO4-bundles is also proved The other main series of results concerns gap-phenomena for Yang-Mills fields As a consequence of the non-linearity of the Yang-Mills equations, we can give explicitC0-neighbourhoods of the minimal Yang-Mills fields which contain no other Yang-Mills fields In this part of the study the nature of the groupG does not matter, neither is the dimension of the base manifold constrained to be four

Some twisted self-dual solutions for the Yang-Mills equations on a hypertorus

Abstract: TheSU(N) Yang-Mills equations are considered in a four-dimensional Euclidean box with periodic boundary conditions (hypertorus). Gauge-invariant twists can be introduced in these boundary conditions, to be labeled with integersn μν (= −n μν ), defined moduloN. The Pontryagin number in this space is often fractional. Whenever this number is zero there are solutions to the equationsG μν =0 HereG μν is the covariant curl. When this number is not zero we find a set of solutions to the equations $$G_{\mu u } = \tilde G_{\mu u } $$ , provided that the periodsa μ of the box satisfy certain relations.

Possible new strange attractors with spiral structure

Abstract: We define a class of three-dimensional differential equations which might present strange attractors of a new kind. This is illustrated by numerical observations on an explicit example.

The Inverse Scattering Method Approach to the Quantum Shabat-Mikhailov Model

Abstract: The Shabat-Mikhailov model is treated in the framework of the quantum inverse scattering method. The Baxter'sR-matrix for the model is calculated.

Tetrahedron equations and the relativistic S -matrix of straight-strings in 2+1-Dimensions

Abstract: The quantumS-matrix theory of straight-strings (infinite one-dimensional objects like straight domain walls) in 2+1-dimensions is considered. TheS-matrix is supposed to be “purely elastic” and factorized. The tetrahedron equations (which are the factorization conditions) are investigated for the special “two-colour” model. The relativistic three-stringS-matrix, which apparently satisfies this tetrahedron equation, is proposed.

Outer automorphisms and reduced crossed products of simple $C^*$-algebras

Abstract: Every outer automorphism of a separable simpleC*-Algebra is shown to have a pure state which is mapped into an inequivalent state under this automorphism. The reduced crossed product of a simpleC*-algebra by a discrete group of outer automorphisms is shown to be simple.

Cohomology and massless fields

Abstract: The geometry of twistors was first introduced in Penrose [28]. Since that time it has played a significant role in solutions of various problems in mathemetical physics of both a linear and nonlinear nature (cf. Penrose [29], Penrose [35], Ward [48], and Atiyah-Hitchin-Drinfeld-Manin [2], see Wells [52] for a recent survey of the topic with a more extensive bibliography). The major role it has played has been in setting up a general correspondence which translates certain important physical field equations in space-time into holomorphic structures on a related complex manifold known as twίstor space. The purpose of this paper is to give a rigorous discussion of this correspondence for the case of the linear massless free-field equations, including Maxwell's source-free equations, the wave equation, the Dirac-Weyl neutrino equations, and the linearized (weakfield limit of) Einstein's vacuum equations. These equations may also be analyzed from this point of view on a background provided by the nonlinear Yang-Mills or Einstein equations in the (anti-) self-dual case. The correspondence is effected by an integral-geometric transform, which transforms complex-analytic data on twistor space to solutions of the linear massless field equations, and is, in fact, a generalization of the classical Radon transform, which is discussed further below. The motivation for finding such a correspondence in general is that it forms an essential part of the \"twistor programme\" according to which one attempts to eliminate the equations of physics by deriving them from the rigidity of complex geometry and holomorphic functions (see, e.g. Penrose [38]). It is, in fact, rather remarkable the extent to which it is possible to achieve this. Success apparently comes about because in twistor-space descriptions the information is \"stored\" nonlocally. The (local) value of a field at a point in space-time depends upon the way that the holomorphic structure in the twistor-space is fitted together in the large. So sheaf cohomology and function theory of several complex variables turn out to be the appropriate tools in the twistor framework. It is hoped that, as part of the general twistor programme, some deeper insights may eventually be gained as to the interrelation between quantum mechanics or quantum field

The Thomas—Fermi—von Weizsäcker Theory of Atoms and Molecules

Abstract: We place the Thomas-Fermi-von Weizsacker model of atoms on a firm mathematical footing. We prove existence and uniqueness of solutions of the Thomas-Fermi-von Weizsacker equation as well as the fact that they minimize the Thomas-Fermi-von Weizsacker energy functional. Moreover, we prove the existence of binding for two very dissimilar atoms in the frame of this model.

On the bundle of connections and the gauge orbit manifold in Yang-Mills theory

Abstract: In an appropriate mathematical framework we supply a simple proof that the quotienting of the space of connections by the group of gauge transformations (in Yang-Mills theory) is aC ∞ principal fibration. The underlying quotient space, the gauge orbit space, is seen explicitly to be aC ∞ manifold modelled on a Hilbert space.

Small random perturbations of dynamical systems and the definition of attractors

Abstract: The “strange attractors” plotted by computers and seen in physical experiments do not necessarily have an open basin of attraction. In view of this we study a new definition of attractors based on ideas of Conley. We argue that the attractors observed in the presence of small random perturbations correspond to this new definition.

Symmetry and bifurcations of momentum mappings

Abstract: The zero set of a momentum mapping is shown to have a singularity at each point with symmetry. The zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function. The proof uses the Kuranishi theory of deformations. Among the applications, it is shown that the set of all solutions of the Yang-Mills equations on a Lorentz manifold has a singularity at any solution with symmetry, in the sense of a pure gauge symmetry. Similarly, the set of solutions of Einstein's equations has a singularity at any solution that has spacelike Killing fields, provided the spacetime has a compact Cauchy surface.

A Yang-Mills-Higgs monopole of charge $2$

Abstract: A new static, purely magnetic Yang-Mills-Higgs monopole solution is presented. It is axisymmetric and has a topological charge of 2; the charge is located at a single point.

Some relations between dimension and Lyapounov exponents

Abstract: We consider differentiable maps and compact invariant sets. We introduce dimensional quantities related to the ergodic invariant measures, and prove some simple relations.

The boost problem in general relativity

Abstract: We show that any asymptotically flat initial data for the Einstein field equations have a development which includes complete spacelike surfaces boosted relative to the initial surface. Furthermore, the asymptotic fall off is preserved along these boosted surfaces and there exists a global system of harmonic coordinates on such a development. We also extend former results on global solutions of the constraint equations. By virtue of this extension, the constraint and evolution parts of the problem fit together exactly. Several theorems are given which concern the behaviour in the large of general classes of linear and quasilinear differential systems. This paper contains in addition a systematic exposition of the functional spaces employed.

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.

New approach to the semiclassical limit of quantum mechanics. I. Multiple tunnelings in one dimension

Abstract: We propose a new approach for the estimate of the rate of degeneracy of the lowest eigenvalues of the Schrodinger operator in the presence of tunneling based on the theory of diffusion processes. Our method provides lower and upper bounds for the energy splittings and the rates of localization of the wave functions and enables us to discuss cases which, as far as we know, have never been treated rigorously in the literature. In particular we give an analysis of the effect on eigenvalues and eigenfunctions of localized deformations of 1) symmetric double well potentials 2) potentials periodic and symmetric over a finite interval. Theses situations are characterized by a remarkable dependence on such deformations. Our probabilistic techniques are inspired by the theory of small random perturbations of dynamical systems.

The local structure of the spectrum of the one-dimensional Schrödinger operator

Abstract: Let $$H_V = - \frac{{d^{\text{2}} }}{{dt^{\text{2}} }} + q(t,\omega )$$ be an one-dimensional random Schrodinger operator in ℒ2(−V,V) with the classical boundary conditions. The random potentialq(t, ω) has a formq(t, ω)=F(x t ), wherex t is a Brownian motion on the compact Riemannian manifoldK andF:K→R 1 is a smooth Morse function, $$\mathop {\min }\limits_K F = 0$$ . Let $$N_V (\Delta ) = \sum\limits_{Ei(V) \in \Delta } 1 $$ , where Δ∈(0, ∞),E i (V) are the eigenvalues ofH V . The main result (Theorem 1) of this paper is the following. IfV→∞,E 0>0,k∈Z + anda>0 (a is a fixed constant) then $$P\left\{ {N_V \left( {E_0 - \frac{a}{{2V}},E_0 + \frac{a}{{2V}}} \right) = k} \right\}\xrightarrow[{V \to \infty }]{}e^{ - an(E_0 )} (an(E_0 ))^k |k!,$$ wheren(E 0) is a limit state density ofH V ,V→∞. This theorem mean that there is no repulsion between energy levels of the operatorH V ,V→∞. The second result (Theorem 2) describes the phenomen of the repulsion of the corresponding wave functions.

Finite time analyticity for the two- and three-dimensional Kelvin-Helmholtz instability

Abstract: The well-posed property for the finite time vortex sheet problem with analytic initial data was first conjectured by Birkhoff in two dimensions and is shown here to hold both in two and three dimensions. Incompressible, inviscid and irrotational flow with a velocity jump across an interface is assumed. In two dimensions, global existence of a weak solution to the Euler equation with such initial conditions is established. In three dimensions, a Lagrangian representation of the vortex sheet analogous to the Birkhoff equation in two dimensions is presented.

The energy and the linear momentum of space-times in general relativity

Abstract: We extend our previous proof of the positive mass conjecture to allow a more general asymptotic condition proposed by York. Hence we are able to prove that for an isolated physical system, the energy momentum four vector is a future timelike vector unless the system is trivial. Furthermore, we allow singularities of the type of black holes.