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Showing papers in "Communications in Mathematical Physics in 1979"


Journal ArticleDOI
TL;DR: In this paper, the authors show that positive solutions of second order elliptic equations are symmetric about the limiting plane, and that the solution is symmetric in bounded domains and in the entire space.
Abstract: We prove symmetry, and some related properties, of positive solutions of second order elliptic equations. Our methods employ various forms of the maximum principle, and a device of moving parallel planes to a critical position, and then showing that the solution is symmetric about the limiting plane. We treat solutions in bounded domains and in the entire space.

2,792 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the total mass associated with each asymptotic regime is non-negative with equality only if the space-time is flat, which is the assumption of the existence of a maximal spacelike hypersurface.
Abstract: LetM be a space-time whose local mass density is non-negative everywhere. Then we prove that the total mass ofM as viewed from spatial infinity (the ADM mass) must be positive unlessM is the flat Minkowski space-time. (So far we are making the reasonable assumption of the existence of a maximal spacelike hypersurface. We will treat this topic separately.) We can generalize our result to admit wormholes in the initial-data set. In fact, we show that the total mass associated with each asymptotic regime is non-negative with equality only if the space-time is flat.

1,473 citations


Journal ArticleDOI
TL;DR: In this article, it is shown that if a variational principle is invariant under some symmetry group G, then to test whether a symmetric field configuration ϕ is an extremal, it suffices to check the vanishing of the first variation of the action corresponding to variations ϕ + δϕ that are also symmetric.
Abstract: It is frequently explicitly or implicitly assumed that if a variational principle is invariant under some symmetry groupG, then to test whether a symmetric field configuration ϕ is an extremal, it suffices to check the vanishing of the first variation of the action corresponding to variations ϕ + δϕ that are also symmetric. We show by example that this is not valid in complete generality (and in certain cases its meaning may not even be clear), and on the other hand prove some theorems which validate its use under fairly general circumstances (in particular ifG is a group of Riemannian isometries, or if it is compact, or with some restrictions if it is semi-simple).

1,022 citations


Journal ArticleDOI
TL;DR: In this paper, the B-property for two-dimensional domains with focusing and neutral regular components is proved and some examples of three and more dimensional domains with billiards obeying this property are also considered.
Abstract: For billiards in two dimensional domains with boundaries containing only focusing and neutral regular components and satisfacting some geometrical conditionsB-property is proved. Some examples of three and more dimensional domains with billiards obeying this property are also considered.

574 citations


Journal ArticleDOI
TL;DR: In this article, the authors classified the action of one parameter isometry groups of Gravitational instantons, complete non-singular positive definite solutions of the Einstein equations with or without Λ term.
Abstract: We classify the action of one parameter isometry groups of Gravitational Instantons, complete non singular positive definite solutions of the Einstein equations with or without Λ term. The fixed points of the action are of 2-types, isolated points which we call “nuts” and 2-surfaces which we call “bolts”. We describe all known gravitational instantons and relate the numbers and types of the nuts and bolts occurring in them to their topological invariants. We perform a 3+1 decomposition of the field equations with respect to orbits of the isometry group and exhibit a certain duality between “electric” and “magnetic” aspects of gravity. We also obtain a formula for the gravitational action of the instantons in terms of the areas of the bolts and certain nut charges and potentials that we define. This formula can be interpreted thermodynamically in several ways.

556 citations


Journal ArticleDOI
TL;DR: In this paper, the iteration of maps of the interval which have negative Schwarzian derivative and one critical point is studied, and the maps in this class are classified up to topological equivalence.
Abstract: This paper studies the iteration of maps of the interval which have negative Schwarzian derivative and one critical point. The maps in this class are classified up to topological equivalence. The equivalence classes of maps which display sensitivity to initial conditions for large sets of initial conditions are characterized.

421 citations


Journal ArticleDOI
TL;DR: In this article, the authors cast quantum mechanics into a classical Hamiltonian form in terms of a symplectic structure, not on the Hilbert space of state-vectors but on the more physically relevant infinite-dimensional manifold of instantaneous pure states.
Abstract: Quantum mechanics is cast into a classical Hamiltonian form in terms of a symplectic structure, not on the Hilbert space of state-vectors but on the more physically relevant infinite-dimensional manifold of instantaneous pure states. This geometrical structure can accommodate generalizations of quantum mechanics, including the nonlinear relativistic models recently proposed. It is shown that any such generalization satisfying a few physically reasonable conditions would reduce to ordinary quantum mechanics for states that are “near” the vacuum. In particular the origin of complex structure is described.

321 citations


Journal ArticleDOI
TL;DR: The positive action conjecture requires that the action of any asymptotically Euclidean 4-dimensional Riemannian metric be positive, vanishing if and only if the space is flat as mentioned in this paper.
Abstract: The Positive Action conjecture requires that the action of any asymptotically Euclidean 4-dimensional Riemannian metric be positive, vanishing if and only if the space is flat. Because any Ricci flat, asymptotically Euclidean metric has zero action and is local extremum of the action which is a local minimum at flat space, the conjecture requires that there are no Ricci flat asymptotically Euclidean metrics other than flat space, which would establish that flat space is the only local minimum. We prove this for metrics onR 4 and a large class of more complicated topologies and for self-dual metrics. We show that ifR μ μ ≧0 there are no bound states of the Dirac equation and discuss the relevance to possible baryon non-conserving processes mediated by gravitational instantons. We conclude that these are forbidden in the lowest stationary phase approximation. We give a detailed discussion of instantons invariant under anSU(2) orSO(3) isometry group. We find all regular solutions, none of which is asymptotically Euclidean and all of which possess a further Killing vector. In an appendix we construct an approximate self-dual metric onK3 — the only simply connected compact manifold which admits a self-dual metric.

291 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied a forced, dissipative system of three ordinary differential equations and showed that prior to the onset of chaotic behavior there exists a preturbulent state where turbulent orbits exist but represent a set of measure zero of initial conditions.
Abstract: This paper studies a forced, dissipative system of three ordinary differential equations. The behavior of this system, first studied by Lorenz, has been interpreted as providing a mathematical mechanism for understanding turbulence. It is demonstrated that prior to the onset of chaotic behavior there exists a preturbulent state where turbulent orbits exist but represent a set of measure zero of initial conditions. The methodology of the paper is to postulate the short term behavior of the system, as observed numerically, to establish rigorously the behavior of particular orbits for all future time. Chaotic behavior first occurs when a parameter exceeds some critical value which is the first value for which the system possesses a homoclinic orbit. The arguments are similar to Smale's “horseshoe”.

270 citations


Journal ArticleDOI
TL;DR: In this paper, the classical field limit of non relativistic many-boson theories in space dimensionn≧3 was studied and the expected results were proved: when ħ tends to zero, the quantum theory tends in a suitable sense to the corresponding classical field theory, and the quantum fluctuations are governed by the equation obtained by linearizing the quantum evolution equation around the classical solution.
Abstract: We study the classical field limit of non relativistic many-boson theories in space dimensionn≧3, extending the results of a previous paper to more singular interactions. We prove the expected results: when ħ tends to zero, the quantum theory tends in a suitable sense to the corresponding classical field theory, and the quantum fluctuations are governed by the equation obtained by linearizing the quantum evolution equation around the classical solution. These results hold uniformly in time and therefore apply to scattering theory. The interactions considered here are so singular as to require a change of domain in order to define the generator of the evolution of the fluctuations, but sufficiently regular so that no energy renormalization is needed.

237 citations


Journal ArticleDOI
TL;DR: In this paper, the partition function of a degenerate quadratic functional is defined and studied, and it is shown that Ray-Singer invariants can be interpreted as partition functions of a non-quadratic functional.
Abstract: The partition function of a degenerate quadratic functional is defined and studied. It is shown that Ray-Singer invariants can be interpreted as partition functions of quadratic functionals. In the case of a degenerate non-quadratic functional the semiclassical approximation to the partition function is considered.

Journal ArticleDOI
TL;DR: In this article, the authors construct global solutions for quasilinear hyperbolic systems and study their asymptotic behaviors, including models of gas flows in a variable area duct and flows with a moving source.
Abstract: We construct global solutions for quasilinear hyperbolic systems and study their asymptotic behaviors. The systems include models of gas flows in a variable area duct and flows with a moving source. Our analysis is based on a numerical scheme which generalizes the Glimm scheme for hyperbolic conservation laws.

Journal ArticleDOI
TL;DR: In this paper, the authors investigated the global behavior of the quadratic diffeomorphism of the plane given by H(x,y)=(1+y−Ax2,Bx).
Abstract: We investigate the global behavior of the quadratic diffeomorphism of the plane given byH(x,y)=(1+y−Ax2,Bx). Numerical work by Henon, Curry, and Feit indicate that, for certain values of the parameters, this mapping admits a “strange attractor”. Here we show that, forA small enough, all points in the plane eventually move to infinity under iteration ofH. On the other hand, whenA is large enough, the nonwandering set ofH is topologically conjugate to the shift automorphism on two symbols.

Journal ArticleDOI
TL;DR: In this article, the authors derived an entropy production inequality which is used to prove global exponential decay of the free energy, and with its aid a uniform rate for strong convergence to equilibrium is proven, where the generators of the linear-lized flow at the vicinity of the equilibria are diagonalized.
Abstract: Chemical kinetics of a system of reacting polymers is modelled by an equation which shares certain properties with Boltzmann's equation. Being more tractable, however, this evolution may be of an illustrative value for the latter. The existence and uniqueness of solutions are analysed. We derive an entropy production inequality which is used to prove global exponential decay of the free energy. With its aid a uniform rate for strong convergence to equilibrium is proven. The generators of the linearlized flow at the vicinity of the equilibria are diagonalized.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the techniques of dilation analyticity still can be used to discuss the long-lived states (resonances) of a system described by a Hamiltonian of the formH=−Δ+ex1+V(x).
Abstract: The resolvent of the operatorH0(e, θ)=−Δe-20+ex1eθ is not analytic in θ for θ in a neighborhood of a real point, if the electric field e is non-zero. (One manifestation of this singular behavior is that for 0<|Im θ|<π/3,H0(e, θ) has no spectrum in the finite plane.) Nevertheless it is shown that the techniques of dilation analyticity still can be used to discuss the long-lived states (resonances) of a system described by a Hamiltonian of the formH=−Δ+ex1+V(x).

Journal ArticleDOI
TL;DR: In this paper, the counting problem for the vacuum diagrams of a φ4 theory was reduced to a moment problem and the generating function for the counting of diagrams on a torus with one hole was given.
Abstract: We reduce the counting problem for the vacuum diagrams of a φ4 theory to a moment problem. As a consequence we are able to give the generating function for the counting of diagrams on a torus with one hole, besides the known result for planar diagrams. The method can be extended to φn theory and also to the counting of diagrams on a torus with an arbitrary number of holes.

Journal ArticleDOI
TL;DR: In this article, the authors calculate the contribution of instanton fields to the partition function of ℂℙn−1 models in two dimensions and show that forn = 2, the pure instanton gas is infrared finite, infinitely dense and generates a mass dynamically.
Abstract: We calculate exactly the contribution of instanton fields to the partition function of ℂℙn−1 models in two dimensions. Forn=2, the pure instanton gas is infrared finite, infinitely dense and generates a mass dynamically. Forn≧3, the gas corresponds to a system with complicatedn-body interactions, whose properties are yet to be explored.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the same conclusion can still be obtained by applying the theorem mentioned to another transformation related to the original one by the method of inducing, when the hypotheses fail in certain ways.
Abstract: There is a theorem in ergodic theory which gives three conditions sufficient for a piecewise smooth mapping on the interval to admit a finite invariant ergodic measure equivalent to Lebesgue. When the hypotheses fail in certain ways, this work shows that the same conclusion can still be gotten by applying the theorem mentioned to another transformation related to the original one by the method of inducing.

Journal ArticleDOI
TL;DR: In this article, the authors studied SU(2) Yang-Mills theory from the canonical view-point, identifying the true configuration space as the base-space of a principal bundle with the gauge-group as structure group.
Abstract: We studySU(2) Yang-Mills theory onS3×ℝ from the canonical view-point. We use topological and differential geometric techniques, identifying the “true” configuration space as the base-space of a principal bundle with the gauge-group as structure group.

Journal ArticleDOI
TL;DR: The factorizedS-matrix with internal symmetry Z4 is constructed in two space-time dimensions and the two-particle amplitudes are obtained by means of solving the factorization, unitarity and analyticity equations as mentioned in this paper.
Abstract: The factorizedS-matrix with internal symmetryZ4 is constructed in two space-time dimensions. The two-particle amplitudes are obtained by means of solving the factorization, unitarity and analyticity equations. The solution of factorization equations can be expressed in terms of elliptic functions. TheS-matrix contains the resonance poles naturally. The simple formal relation between the general factorizedS-matrices and the Baxter-type lattice transfer matrices is found. In the sense of this relation theZ4-symmetricS-matrix corresponds to the Baxter transfer matrix itself.

Journal ArticleDOI
TL;DR: In this paper, the Lyapunov-Schmidt procedure is applied to the buckling of an Euler column and the potential energy function of the column can be found by solving the problem.
Abstract: Consider the familiar principle that typically (or generically) a system of m scalar equations in n variables where m>n has no solutions. This principle can be reformulated geometrically as follows. If S is a submanifold of a manifold X with codimension m (i.e. m = άimX — dimS) and iϊf:R-*X is a smooth mapping where m>n, then usually or generically Image /nS is empty. One of the basic tenets in the application of singularity theory is that this principle holds in a general way in function spaces. In the next few paragraphs we shall try to explain this more general situation as well as to explain its relevance to bifurcation problems. First we describe an example through which these ideas may be understood. Consider the buckling of an Euler column. Let λ denote the applied load and x denote the maximum deflection of the column. After an application of the Lyapunov-Schmidt procedure the potential energy function Ffor this system may be written as a function of x and λ alone and hence the steady-state configurations of the column may be found by solving

Journal ArticleDOI
TL;DR: In this paper, a quantum stochastic process (QSP) in discrete time capable of describing non-Markovian effects is introduced, which is based directly on the physically relevant correlation functions.
Abstract: A definition of a quantum stochastic process (QSP) in discrete time capable of describing non-Markovian effects is introduced. The formalism is based directly on the physically relevant correlation functions. The notion of complete positivity is used as the main mathematical tool. Two different but equivalent canonical representations of a QSP in terms of completely positive maps are derived. A quantum generalization of the Kolmogorov-Sinai entropy is proved to exist.

Journal ArticleDOI
TL;DR: In this paper, the number of instantons and zero fermion modes in the field of instanton was calculated and the quantum fluctuations of the instantons were studied. But the authors did not consider the quantum properties of the modes.
Abstract: The number of instantons and the number of zero fermion modes in the field of instanton are calculated. The quantum fluctuations of instantons are studied.

Journal ArticleDOI
TL;DR: In this paper, the compressible and heat-conductive Navier-Stokes equation obtained as the second approximation of the formal Chapman-Enskog expansion is investigated on its relations to the original nonlinear Boltzmann equation and also to the incompressible Navier Stokes equation.
Abstract: The compressible and heat-conductive Navier-Stokes equation obtained as the second approximation of the formal Chapman-Enskog expansion is investigated on its relations to the original nonlinear Boltzmann equation and also to the incompressible Navier-Stokes equation. The solutions of the Boltzmann equation and the incompressible Navier-Stokes equation for small initial data are proved to be asymptotically equivalent (mod decay ratet −5/4) ast→+∞ to that of the compressible Navier-Stokes equation for the corresponding initial data.

Journal ArticleDOI
TL;DR: In this paper, the singularity theory of mappings in the presence of a symmetry group was used to analyse the bifurcation equation obtained by the Lyapunov-Schmidt reduction applied to the Von Karman equations.
Abstract: We show that mode jumping in the buckling of a rectangular plate may be explained by a secondary bifurcation — as suggested by Bauer et al. [1] — when “clamped” boundary conditions on the vertical displacement function are assumed. In our analysis we use the singularity theory of mappings in the presence of a symmetry group to analyse the bifurcation equation obtained by the Lyapunov-Schmidt reduction applied to the Von Karman equations. Noteworthy is the fact that this explanation fails when the assumed boundary conditions are “simply supported”. Mode jumping in the presence of “clamped” boundary conditions was observed experimentally by Stein [9]; “simply supported” boundary conditions are frequently studied but are difficult — if not impossible — to realize physically. Thus, it is important to observe that the qualitative post-buckling behavior depends on which idealization for the boundary conditions one chooses.

Book ChapterDOI
TL;DR: In this paper, the authors show that the formal diffusion approximation for the motion of a particle in a random velocity field is valid under some specific conditions, and they show that this is the case for any particle.
Abstract: We show under some specific conditions that the formal diffusion approximation for the motion of a particle in a random velocity field is valid.

Journal ArticleDOI
TL;DR: In this paper, the true analogues of superspace and conformal superspace for spatially homogeneous cosmology are introduced and discussed in relation to the kinematics of the evolution of Cauchy data from a spatial homogeneous initial value surface using a spatially-homogeneous lapse function.
Abstract: The true analogues of superspace and conformal superspace for spatially homogeneous cosmology are introduced and discussed in relation to the kinematics of the evolution of Cauchy data from a spatially homogeneous initial value surface using a spatially homogeneous lapse function. Having fixed the slicing of spatially homogeneous spacetimes to be the natural one, an obvious restriction on the freedom of choice of the shift vector field occurs, and its relation to the three-dimensional diffeomorphism gauge group of the problem is explained. In this context the minimal distortion shift equation of Smarr and York naturally arises. Finally these ideas are used to simplify the dynamics.

Journal ArticleDOI
TL;DR: In this paper, a system of particles with central repulsive interactions, right and left velocities exist at each moment of time, including infinity, and the number of collisions in Sinai's billiard is finite.
Abstract: In an arbitrary system of particles with central repulsive interactions, right and left velocities exist at each moment of time, including infinity. An arbitrary system of particles with finite-range interactions splits into independent bounded clusters. The number of collisions in Sinai's billiard is finite.

Journal ArticleDOI
Abstract: Quantum field theory in curved spacetime is examined from the Euclidean approach, where one seeks to define the theory for metrics of positive (rather than Lorentzian) signature. Methods of functional analysis are used to give a proof of the heat kernel expansion for the Laplacian, which extends the well known result for compact manifolds to all non-compact manifolds for which the Laplacian and its powers are essentially self-adjoint on the initial domain of smooth functions of compact support. Using this result, precise prescriptions of the zeta-function, dimensional, and point-splitting type are given for renormalizing the action of a Klein-Gordon scalar field. These prescriptions are shown to be equivalent up to local curvature terms. It is also shown that for static spacetimes, the Euclidean prescription for defining the Feynman propagator agrees with the definition of Feyman propagator obtained by working directly on the spacetime.

Journal ArticleDOI
Leonard Gross1
TL;DR: In this paper, it was shown that the truncated pair correlation function decays in the same weighted summability sense as the potential, at high temperature, in statistical mechanical lattice models with many body potentials of finite or infinite range.
Abstract: In classical statistical mechanical lattice models with many body potentials of finite or infinite range and arbitrary spin it is shown that the truncated pair correlation function decays in the same weighted summability sense as the potential, at high temperature.