# Showing papers in "Communications in Mathematical Physics in 1986"

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TL;DR: In this article, it was shown that the global charges of a gauge theory may yield a nontrivial central extension of the asymptotic symmetry algebra already at the classical level.

Abstract: It is shown that the global charges of a gauge theory may yield a nontrivial central extension of the asymptotic symmetry algebra already at the classical level. This is done by studying three dimensional gravity with a negative cosmological constant. The asymptotic symmetry group in that case is eitherR×SO(2) or the pseudo-conformal group in two dimensions, depending on the boundary conditions adopted at spatial infinity. In the latter situation, a nontrivial central charge appears in the algebra of the canonical generators, which turns out to be just the Virasoro central charge.

3,072 citations

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TL;DR: The low-lying energy values associated to energy eigenstates describing two stable particles enclosed in a (space-like) box of sizeL are shown to be expandable in an asymptotic power series of 1/L as mentioned in this paper.

Abstract: The low-lying energy values associated to energy eigenstates describing two stable particles enclosed in a (space-like) box of sizeL are shown to be expandable in an asymptotic power series of 1/L The coefficients in these expansions are related to the appropriate elastic scattering amplitude in a simple and apparently universal manner At low energies, the scattering amplitude can thus be determined, if an accurate calculation of two-particle energy values is possible (by numerical simulation, for example)

1,060 citations

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TL;DR: In this paper, it was shown that a method previously given for constructing representations of the Virasoro algebra out of representations of affine Kac-Moody algebras yields the full discrete series of highest weight irreducible representations.

Abstract: It is shown that a method previously given for constructing representations of the Virasoro algebra out of representations of affine Kac-Moody algebras yields the full discrete series of highest weight irreducible representations of the Virasoro algebra. The corresponding method for the super-Virasoro algebras (i.e. the Neveu-Schwarz and Ramond algebras) is described in detail and shown to yield the full discrete series of irreducible highest weight representations.

1,013 citations

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TL;DR: In this paper, a functional central limit theorem for additive functionals of stationary reversible ergodic Markov chains was proved under virtually no assumptions other than the necessary ones, and they used these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk.

Abstract: We prove a functional central limit theorem for additive functionals of stationary reversible ergodic Markov chains under virtually no assumptions other than the necessary ones. We use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk.

909 citations

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Kyoto University

^{1}TL;DR: In this paper, the explicit form of the quantumR matrix in the fundamental representation for the generalized Toda system associated with non-exceptional affine Lie algebras is given.

Abstract: We report the explicit form of the quantumR matrix in the fundamental representation for the generalized Toda system associated with non-exceptional affine Lie algebras.

868 citations

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TL;DR: In this paper, a detailed proof of this basic relation is given for all orders of perturbation theory in arbitrary massive quantum field theories, which follows from an apparently universal asymptotic formula, which relates Δm to certain forward elastic scattering amplitudes.

Abstract: Due to polarization effects, the massM of a stable particle in a quantum field theory enclosed in a large (space-like) box of sizeL and periodic boundary conditions in general differs from its infinite volume valuem. AsL increases, the finite size mass shift Δm =M−m goes to zero exponentially with a rate, which depends on the particle considered and on the spectrum of light particles in the theory. This behaviour follows from an apparently universal asymptotic formula, already presented earlier, which relates Δm to certain forward elastic scattering amplitudes. A detailed proof of this basic relation is given here to all orders of perturbation theory in arbitrary massive quantum field theories.

764 citations

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TL;DR: In this article, the authors evaluate Polyakov's path integral for the sum over all closed surfaces with the topology of a torus, in the critical dimensiond = 26, and apply it to the partition function and cosmological constant of the free bosonic string, and to tachyon scattering amplitudes.

Abstract: We evaluate Polyakov's path integral for the sum over all closed surfaces with the topology of a torus, in the critical dimensiond = 26. The result is applied to the partition function and cosmological constant of the free bosonic string, and to tachyon scattering amplitudes.

562 citations

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TL;DR: In this paper, it was shown that the affine Lie algebra A1(1) has a natural family πμ, υ,v of Fock representations on the space C[xi,yj;i ∈ ℤ andj ∈ C2], parametrized by (μ,v) ∈C2, and that the parameter spaceC2 forms a double cover of the weight spaceCΛ0⊕C −1 with singularities at linear forms of level −2.

Abstract: The aim of this note is to show that the affine Lie algebraA1(1) has a natural family πμ, υ,v of Fock representations on the spaceC[xi,yj;i ∈ ℤ andj ∈ ℕ], parametrized by (μ,v) ∈C2. By corresponding the highest weightΛμ, υ of πμ, υ to each (μ,ν), the parameter spaceC2 forms a double cover of the weight spaceCΛ0⊕C −1 with singularities at linear forms of level −2; this number is (−1)-times the dual Coxeter number. Our results contain explicit realizations of irreducible non-integrable highest wieghtA1(1)-modules for generic (μ,v).

514 citations

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TL;DR: In this article, the problem of finding the minimum energy of a point singularity when the location and topological degree of the singularities are prescribed is solved, and it is shown that the only case in which g(x/|x|) minimizes the energy isg=const org(x)=±Rx withR a rotation.

Abstract: Two problems concerning maps ϕ with point singularities from a domain Ω C ℝ3 toS
2 are solved The first is to determine the minimum energy of ϕ when the location and topological degree of the singularities are prescribed In the second problem Ω is the unit ball and ϕ=g is given on ∂Ω; we show that the only cases in whichg(x/|x|) minimizes the energy isg=const org(x)=±Rx withR a rotation Extensions of these problems are also solved, eg points are replaced by “holes,” ℝ3,S
2 is replaced by ℝ
N
,S
N−1 or by ℝ
N
, ℝP
N−1, the latter being appropriate for the theory of liquid crystals

508 citations

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TL;DR: In this article, the potentials for the Schrodinger equation (−∂¯¯¯¯ 2 + V(x))∅=k 2 ∅ such that some family of eigenfunctions ∅ satisfies a differential equation in the spectral parameter k of the formB(k, ∂¯¯¯¯ k )o=Θ(x)o.

Abstract: We determine all the potentialsV(x) for the Schrodinger equation (−∂
2
+V(x))∅=k2∅ such that some family of eigenfunctions ∅ satisfies a differential equation in the spectral parameterk of the formB(k, ∂
k
)o=Θ(x)o. For each suchV(x) we determine the algebra of all possible operatorsB and the corresponding functions Θ(x)

495 citations

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TL;DR: In this paper, the results obtained in [BF1] were generalized to the case of Dirac operators and the curvature of the unitary connection on the determinant bundle was analyzed.

Abstract: In this paper we specialize the results obtained in [BF1] to the case of a family of Dirac operators. We first calculate the curvature of the unitary connection on the determinant bundle which we introduced in [BF1].

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TL;DR: In this article, it was shown that initial data sufficiently close to De-Sitter data develop into solutions of Einstein's equations Ric[g]=Λg with positive cosmological constant Λ, which are asymptotically simple in the past as well as in the future, whence null geodesically complete.

Abstract: It is demonstrated that initial data sufficiently close to De-Sitter data develop into solutions of Einstein's equations Ric[g]=Λg with positive cosmological constant Λ, which are asymptotically simple in the past as well as in the future, whence null geodesically complete Furthermore it is shown that hyperboloidal initial data (describing hypersurfaces which intersect future null infinity in a space-like two-sphere), which are sufficiently close to Minkowskian hyperboloidal data, develop into future asymptotically simple whence null geodesically future complete solutions of Einstein's equations Ric[g]=0, for which future null infinity forms a regular cone with vertexi+ that represents future time-like infinity

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TL;DR: In this paper, it was shown that the spin structure dependence of the chiral Dirac determinant on a Riemann surface is given by RiemANN's theta function, which was used to investigate the modular invariance of multiloop heterotic string amplitudes.

Abstract: We use Quillen's theorem and algebraic geometry to investigate the modular transformation properties of some quantities of interest in string theory. In particular, we show that the spin structure dependence of the chiral Dirac determinant on a Riemann surface is given by Riemann's theta function. We use this result to investigate the modular invariance of multiloop heterotic string amplitudes.

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TL;DR: In this article, it was shown that the order parameter for a massless Dirac spinor is nonzero, and a coordinate-independent formula for the bispinor was given.

Abstract: The two-point function for spinors on maximally symmetric four-dimensional spaces is obtained in terms of intrinsic geometric objects. In the massless case, Weyl spinors in anti de Sitter space can not satisfy boundary conditions appropriate to the supersymmetric models. This is because these boundary conditions break chiral symmetry, which is proven by showing that the “order parameter”\(\left\langle {\bar \psi \psi } \right\rangle \) for a massless Dirac spinor is nonzero. We also give a coordinate-independent formula for the bispinor\(S(x)\bar S(x')\) introduced by Breitenlohner and Freedman [1], and establish the precise connection between our results and those of Burges, Davis, Freedman and Gibbons [2].

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TL;DR: In this article, a new direct proof of convergence of cluster expansions for polymer models is given in an abstract setting, without relying on Kirkwood-Salsburg type equations or "combinatorics of trees".

Abstract: A new direct proof of convergence of cluster expansions for polymer (contour) models is given in an abstract setting. It does not rely on Kirkwood-Salsburg type equations or “combinatorics of trees.” A distinctive feature is that, at all steps, the considered clusters contain every polymer at most once.

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TL;DR: In this paper, the existence and partial regularity of boundary-value problems for the static theory of liquid crystals is established and some related problems involving magnetic or electric fields are also discussed.

Abstract: We establish the existence and partial regularity for solutions of some boundary-value problems for the static theory of liquid crystals. Some related problems involving magnetic or electric fields are also discussed.

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TL;DR: In this paper, the Radon-Nikodym cocycle condition was formulated in terms of generalized conditional expectations and the relative Hamiltonian, and the main result was that a von Neumann algebra is weakly sufficient for a pair (φ, ω) of states if the relative entropy of φ and ω coincides with their restrictions.

Abstract: A subalgebraM
0 of a von Neumann algebraM is called weakly sufficient with respect to a pair (φ,ω) of states if the relative entropy of φ and ω coincides with the relative entropy of their restrictions toM
0. The main result says thatM
0 is weakly sufficient for (φ,ω) if and only ifM
0 contains the Radon-Nikodym cocycle [Dφ,Dω]
t
. Other conditions are formulated in terms of generalized conditional expectations and the relative Hamiltonian.

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TL;DR: In this article, the Quillen metric on the determinant bundle associated with a family of elliptic first order differential operators is constructed, and a unitary connection on λ is introduced to calculate its curvature.

Abstract: In this paper, we construct the Quillen metric on the determinant bundle associated with a family of elliptic first order differential operators. We also introduce a unitary connection on λ and calculate its curvature. Our results will be applied to the case of Dirac operators in a forthcoming paper.

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TL;DR: In this paper, a short-time existence theorem for the Euler equations for nonisentropic compressible fluid flow in a bounded domain was proven for incompressible flow.

Abstract: A short-time existence theorem is proven for the Euler equations for nonisentropic compressible fluid flow in a bounded domain, and solutions with low Mach number and almost incompressible initial data are shown to be close to corresponding solutions of the equations for incompressible flow.

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TL;DR: In this paper, the authors consider the equations which describe the motion of a viscous compressible fluid, taking into consideration the case of inflow and/or outflow through the boundary.

Abstract: We consider the equations which describe the motion of a viscous compressible fluid, taking into consideration the case of inflow and/or outflow through the boundary. By means of some a priori estimates we prove the existence of a global (in time) solution. Moreover, as a consequence of a stability result, we show that there exist a periodic solution and a stationary solution.

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TL;DR: In this article, the existence and uniqueness results for solutions to the Becker-Doring cluster equations are established for solutions with rapidly decaying initial data, and the density ϱ is shown to be a conserved quantity.

Abstract: Existence and uniqueness results are established for solutions to the Becker-Doring cluster equations. The density ϱ is shown to be a conserved quantity. Under hypotheses applying to a model of a quenched binary alloy the asymptotic behaviour of solutions with rapidly decaying initial data is determined. Denoting the set of equilibrium solutions byc
(ϱ), 0 ≦ ϱ ≦ ϱ
s
, the principal result is that if the initial density ϱ0 ≦ ϱ
s
then the solution converges strongly toc
(ϱo), while if ϱ0 > ϱ
s
the solution converges weak* toc
(ϱs). In the latter case the excess density ϱ0–ϱ
s
corresponds to the formation of larger and larger clusters, i.e. condensation. The main tools for studying the asymptotic behaviour are the use of a Lyapunov function with desirable continuity properties, obtained from a known Lyapunov function by the addition of a special multiple of the density, and a maximum principle for solutions.

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TL;DR: In this paper, a lower bound for the square of the eigenvalues of the Dirac operator by the smallest eigenvalue of the conformal Laplacian (the Yamabe operator) was given.

Abstract: On a Riemannian spin manifold, we give a lower bound for the square of the eigenvalues of the Dirac operator by the smallest eigenvalue of the conformal Laplacian (the Yamabe operator). We prove, in the limiting case, that the eigenspinor field is a killing spinor, i.e., parallel with respect to a natural connection. In particular, if the scalar curvature is positive, the eigenspinor field is annihilated by harmonic forms and the metric is Einstein.

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TL;DR: In this article, the authors studied the global initial value problem for the spherically symmetric case with a massless scalar field as the material model and proved the local, in retarded time, existence and global uniqueness of classical solutions.

Abstract: In this paper we begin the study of the global initial value problem for Einstein's equations in the spherically symmetric case with a massless scalar field as the material model. We reduce the problem to a single nonlinear evolution equation. Taking as initial hypersurface a future light cone with vertex at the center of symmetry, we prove, the local, in retarded time, existence and global uniqueness of classical solutions. We also prove that if the initial data is sufficiently small there exists a global classical solution which disperses in the infinite future.

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TL;DR: In this article, a physically motivated condition on the energy-level density of well-localized states is proposed and discussed, and it is shown that any model satisfying this condition obeys a strong form of the principle of causal (statistical) independence, which manifests itself in a specific algebraic structure of the local algebras.

Abstract: Within the general framework of local quantum field theory a physically motivated condition on the energy-level density of well-localized states is proposed and discussed. It is shown that any model satisfying this condition obeys a strong form of the principle of causal (statistical) independence, which manifests itself in a specific algebraic structure of the local algebras (“split property”). It is also shown that the proposed condition holds in a free field theory.

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TL;DR: In this paper, a large class of billiards with convex pieces of the boundary which have nonvanishing Lyapunov exponents was introduced, where the exponents have non-vanishing exponents.

Abstract: We introduce a large class of billiards with convex pieces of the boundary which have nonvanishing Lyapunov exponents.

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TL;DR: In this article, the existence, uniqueness, and long time behavior of spin processing in the magnetic field created by their closest neighbours were investigated, and the results concerning existence and uniqueness were presented.

Abstract: The continuum limit of a cubic latice of classical spins processing in the magnetic field created by their closest neighbours is considered. Results concerning existence, uniqueness and (for initially small spin deviation) long time behaviour, are presented.

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TL;DR: In this article, the determinants of Laplacians on tensors and spinors of arbitrary weights on compact hyperbolic Riemann surfaces are computed in terms of values of Selberg zeta functions at half integer points.

Abstract: Determinants of Laplacians on tensors and spinors of arbitrary weights on compact hyperbolic Riemann surfaces are computed in terms of values of Selberg zeta functions at half integer points.

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Academia Sinica

^{1}TL;DR: In this article, the authors show that the equation Δu+|u|4/(n−2)εu=0 on Ω n ≥ 3 has infinitely many distinct solutions with finite energy and which change sign.

Abstract: Forn≧3, the equation Δu+|u|4/(n−2)
u=0 on ℝ
n
has infinitely many distinct solutions with finite energy and which change sign.

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TL;DR: In this article, it was shown that the breakdown of smooth solutions to the 3-D incompressible slightly viscous Navier-Stokes equations cannot occur without breakdown in the corresponding solution of the Euler (ideal fluid) equation.

Abstract: One of the central problems in the mathematical theory of turbulence is that of breakdown of smooth (indefinitely differentiable) solutions to the equations of motion. In 1934 J. Leray advanced the idea that turbulence may be related to the spontaneous appearance of singularities in solutions of the 3—D incompressible Navier-Stokes equations. The problem is still open. We show in this report that breakdown of smooth solutions to the 3—D incompressible slightly viscous (i.e. corresponding to high Reynolds numbers, or “highly turbulent”) Navier-Stokes equations cannot occur without breakdown in the corresponding solution of the incompressible Euler (ideal fluid) equation. We prove then that solutions of distorted Euler equations, which are equations closely related to the Euler equations for short term intervals, do breakdown.

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TL;DR: In this article, it was shown that no percolation is possible if for short bonds Kx,y≦p = 1, and the occupation probability of a bond −Kx,Y, has a slow power decay as a function of the bond's length.

Abstract: We consider one dimensional percolation models for which the occupation probability of a bond −Kx,y, has a slow power decay as a function of the bond's length. For independent models — and with suitable reformulations also for more general classes of models, it is shown that: i) no percolation is possible if for short bondsKx,y≦p =1. This dichotomy resembles one for the magnetization in 1/|x−y|2 Ising models which was first proposed by Thouless and further supported by the renormalization group flow equations of Anderson, Yuval, and Hamann. The proofs of the above percolation phenomena involve (rigorous) renormalization type arguments of a different sort.