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Showing papers in "Journal of Mathematical Physics in 2000"


Journal ArticleDOI
TL;DR: In this paper, the authors present a review of the results obtained up to this point and some problems for the future will be discussed at the end of the book, including some problems that are closely connected with the modular theory and that should be treated in the future.
Abstract: In the book of Haag [Local Quantum Physics (Springer Verlag, Berlin, 1992)] about local quantum field theory the main results are obtained by the older methods of C * - and W * -algebra theory. A great advance, especially in the theory of W * -algebras, is due to Tomita’s discovery of the theory of modular Hilbert algebras [Quasi-standard von Neumann algebras, Preprint (1967)]. Because of the abstract nature of the underlying concepts, this theory became (except for some sporadic results) a technique for quantum field theory only in the beginning of the nineties. In this review the results obtained up to this point will be collected and some problems for the future will be discussed at the end. In the first section the technical tools will be presented. Then in the second section two concepts, the half-sided translations and the half-sided modular inclusions, will be explained. These concepts have revolutionized the handling of quantum field theory. Examples for which the modular groups are explicitly known are presented in the third section. One of the important results of the new theory is the proof of the PCT theorem in the theory of local observables. Questions connected with the proof are discussed in Sec. IV. Section V deals with the structure of local algebras and with questions connected with symmetry groups. In Sec. VI a theory of tensor product decompositions will be presented. In the last section problems that are closely connected with the modular theory and that should be treated in the future will be discussed.

227 citations


Journal ArticleDOI
TL;DR: In this article, discrete approaches to gravity, both classical and quantum, are reviewed, with emphasis on the method using piecewise-linear spaces, and progress in generalizing these models to four dimensions is discussed, as is the relationship of these models in both three and four dimensions to topological theories.
Abstract: Discrete approaches to gravity, both classical and quantum, are reviewed briefly, with emphasis on the method using piecewise-linear spaces. Models of three-dimensional quantum gravity involving 6j-symbols are then described, and progress in generalizing these models to four dimensions is discussed, as is the relationship of these models in both three and four dimensions to topological theories. Finally, the repercussions of the generalizations are explored for the original formulation of discrete gravity using edge-length variables.

224 citations


Journal ArticleDOI
TL;DR: In this article, the authors present examples that motivate a need for such a theory, give plausibility arguments for the existence of a theory of positively dependent events, outline a few possible directions a theory might take, and state a number of specific conjectures which pertain to the examples and to a wish list of theorems.
Abstract: The FKG theorem says that the positive lattice condition, an easily checkable hypothesis which holds for many natural families of events, implies positive association, a very useful property. Thus there is a natural and useful theory of positively dependent events. There is, as yet, no corresponding theory of negatively dependent events. There is, however, a need for such a theory. This paper, unfortunately, contains no substantial theorems. Its purpose is to present examples that motivate a need for such a theory, give plausibility arguments for the existence of such a theory, outline a few possible directions such a theory might take, and state a number of specific conjectures which pertain to the examples and to a wish list of theorems.

213 citations


Journal ArticleDOI
TL;DR: In this paper, the singularity of the eigenfunctions in the Lax representation of soliton equation with self-consistent sources (SESCS) is treated to determine the evolution of scattering data.
Abstract: In contrast with the soliton equations, the evolution of the eigenfunctions in the Lax representation of soliton equation with self-consistent sources (SESCS) possesses singularity. We present a general method to treat the singularity to determine the evolution of scattering data. The AKNS hierarchy with self-consistent sources, the MKdV hierarchy with self-consistent sources, the nonlinear Schrodinger equation hierarchy with self-consistent sources, the Kaup–Newell hierarchy with self-consistent sources and the derivative nonlinear Schrodinger equation hierarchy with self-consistent sources are integrated directly by using the inverse scattering method. The N soliton solutions for some SESCS are presented. It is shown that the insertion of a source may cause the variation of the velocity of soliton. This approach can be applied to all other (1+1)-dimensional soliton hierarchies.

204 citations


Journal ArticleDOI
TL;DR: In this paper, the problem of sampling uniformly at random from the set of proper k-colorings of a graph with maximum degree Δ was considered and a simple Markov chain was designed that converges in O(nk log n) time to the desired distribution when k>116Δ.
Abstract: We consider the problem of sampling uniformly at random from the set of proper k-colorings of a graph with maximum degree Δ. Our main result is the design of a simple Markov chain that converges in O(nk log n) time to the desired distribution when k>116Δ.

204 citations


Journal ArticleDOI
TL;DR: In this article, an atom with finitely many energy levels in contact with a heat bath consisting of photons (blackbody radiation) at a temperature T > 0 was studied, and it was shown that an arbitrary initial state that is normal with respect to the equilibrium state of the uncoupled system at temperature T converges to an equilibrium state at the same temperature, as time tends to +∞ (return to equilibrium).
Abstract: We study an atom with finitely many energy levels in contact with a heat bath consisting of photons (blackbody radiation) at a temperature T>0. The dynamics of this system is described by a Liouville operator, or thermal Hamiltonian, which is the sum of an atomic Liouville operator, of a Liouville operator describing the dynamics of a free, massless Bose field, and a local operator describing the interactions between the atom and the heat bath. We show that an arbitrary initial state that is normal with respect to the equilibrium state of the uncoupled system at temperature T converges to an equilibrium state of the coupled system at the same temperature, as time tends to +∞ (return to equilibrium).

202 citations


Journal ArticleDOI
TL;DR: In this paper, a polynomial equation for geometries on the four-sphere with fixed volume is given, where the Dirac operator D is played by an idempotent e, playing the role of the instanton, and the expectation is the projection on the commutant of the algebra of 4 by 4 matrices.
Abstract: We give a survey of selected topics in noncommutative geometry, with some emphasis on those directly related to physics, including our recent work with Dirk Kreimer on renormalization and the Riemann–Hilbert problem. We discuss at length two issues. The first is the relevance of the paradigm of geometric space, based on spectral considerations, which is central in the theory. As a simple illustration of the spectral formulation of geometry in the ordinary commutative case, we give a polynomial equation for geometries on the four-sphere with fixed volume. The equation involves an idempotent e, playing the role of the instanton, and the Dirac operator D. It is of the form 〈(e−12)[D,e]4〉=γ5 and determines both the sphere and all its metrics with fixed volume form. The expectation 〈x〉 is the projection on the commutant of the algebra of 4 by 4 matrices. We also show, using the noncommutative analog of the Polyakov action, how to obtain the noncommutative metric (in spectral form) on the noncommutative tori fr...

198 citations


Journal ArticleDOI
TL;DR: In this paper, a selective review of the mathematical theory of the Schrodinger equation for N-body Hamiltonians is presented, focusing on the interplay between the spectral theory of Hamiltonians and the space-time and phase-space analysis of bound states and scattering states.
Abstract: This selective review is written as an introduction to the mathematical theory of the Schrodinger equation for N particles. Characteristic for these systems are the cluster properties of the potential in configuration space, which are expressed in a simple geometric language. The methods developed over the last 40 years to deal with this primary aspect are described by giving full proofs of a number of basic and by now classical results. The central theme is the interplay between the spectral theory of N-body Hamiltonians and the space–time and phase-space analysis of bound states and scattering states.

177 citations


Journal ArticleDOI
TL;DR: Quantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalization of symmetry groups for certain integrable systems, and on the other as part of a generalisation of geometry itself powerful enough to make sense in the quantum domain this paper.
Abstract: Quantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalization of symmetry groups for certain integrable systems, and on the other as part of a generalization of geometry itself powerful enough to make sense in the quantum domain. Just as the last century saw the birth of classical geometry, so the present century sees at its end the birth of this quantum or noncommutative geometry, both as an elegant mathematical reality and in the form of the first theoretical predictions for Planck-scale physics via ongoing astronomical measurements. Noncommutativity of space–time, in particular, amounts to a postulated new force or physical effect called cogravity.

175 citations


Journal ArticleDOI
TL;DR: In this article, a spectral problem and the associated Gerdjikov-Ivanov (GI) hierarchy of nonlinear evolution equations is presented, and an explicit N-fold Darboux transformation for the GI equation is constructed with the help of a gauge transformation of spectral problems and a reduction technique.
Abstract: A spectral problem and the associated Gerdjikov–Ivanov (GI) hierarchy of nonlinear evolution equations is presented. As a reduction, the well-known GI equation of derivative nonlinear Schrodinger equations is obtained. It is shown that the GI hierarchy is integrable in a Liouville sense and possesses bi-Hamiltonian structure. Moreover, the spectral problem can be nonlinearized as a finite dimensional completely integrable system under the Bargmann constraint between the potentials and the eigenfunctions. In particular, an explicit N-fold Darboux transformation for the GI equation is constructed with the help of a gauge transformation of spectral problems and a reduction technique. Some explicit solitonlike solutions of the GI equation are given by applying its Darboux transformation.

163 citations


Journal ArticleDOI
TL;DR: In this article, a new method for studying boundary value problems for linear and for integrable nonlinear partial differential equations (PDE) in two dimensions is presented, which provides a unification as well as a significant extension of the following three seemingly different topics: (a) the classical integral transform method for solving linear PDEs and several of its variations such as the Wiener-Hopf technique.
Abstract: A new method for studying boundary value problems for linear and for integrable nonlinear partial differential equations (PDE’s) in two dimensions is reviewed. This method provides a unification as well as a significant extension of the following three seemingly different topics: (a) The classical integral transform method for solving linear PDE’s and several of its variations such as the Wiener–Hopf technique. (b) The integral representation of the solution of linear PDE’s in terms of the Ehrenpreis fundamental principle. (c) The inverse spectral (scattering) method for solving the initial value problem for nonlinear integrable evolution equations. The detailed implementation of the method is presented for: (a) An arbitrary linear dispersive evolution equation on the half line. (b) The nonlinear Schrodinger equation on the half line. (c) The Laplace, Helmholtz and modified Helmholtz equations in an arbitrary convex polygon. In addition, several other applications are briefly considered. The possible extension of this method to multidimensions is also discussed.

Journal ArticleDOI
TL;DR: This work presents a definition of entropy production rate for classes of deterministic and stochastic dynamics, motivated by recent work on the Gallavotti–Cohen (local) fluctuation theorem.
Abstract: We present a definition of entropy production rate for classes of deterministic and stochastic dynamics. The point of departure is a Gibbsian representation of the steady state path space measure for which “the density” is determined with respect to the time-reversed process. The Gibbs formalism is used as a unifying algorithm capable of incorporating basic properties of entropy production in nonequilibrium systems. Our definition is motivated by recent work on the Gallavotti–Cohen (local) fluctuation theorem and it is illustrated via a number of examples.

Journal ArticleDOI
TL;DR: In this paper, a review of the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics is presented, with a focus on nonlinearity.
Abstract: This paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics.

Journal ArticleDOI
TL;DR: Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincare, and others as discussed by the authors.
Abstract: Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincare, and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as electromagnetism, fluid mechanics, plasma physics, solid mechanics as well as quantum mechanics, and relativistic theories, including gravity. Symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to subtle symmetries underlying integrable systems. Reduction theory concerns the removal of symmetries and their associated conservation laws. Variational principles, along with symplectic and Poisson geometry, provide fundamental tools for this endeavor. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many physical theories, including new variational and Poisson structures, to stability theory, integrable systems, as well as geometric phases. This paper surveys progress in selected topics in reduction theory, especially those of the last few decades as well as presenting new results on non-Abelian Routh reduction. We develop the geometry of the associated Lagrange–Routh equations in some detail. The paper puts the new results in the general context of reduction theory and discusses some future directions.

Journal ArticleDOI
TL;DR: In this article, the authors survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph, and most attention is devoted to transitive graphs and trees.
Abstract: We survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph. Most attention is devoted to transitive graphs and trees.

Journal ArticleDOI
TL;DR: In this article, a canonical form for pure states of a general multipartite system, in which the constraints on the coordinates (with respect to a factorizable orthonormal basis) are simply that certain ones vanish and certain others are real.
Abstract: We find a canonical form for pure states of a general multipartite system, in which the constraints on the coordinates (with respect to a factorizable orthonormal basis) are simply that certain ones vanish and certain others are real. For identical particles they are invariant under permutations of the particles. As an application, we find the dimension of the generic local equivalence class.

Journal ArticleDOI
TL;DR: In this paper, the authors developed the general geometric theory of transformations of conjugate nets and their discrete analogs: the multidimensional quadrilateral lattices, i.e., lattices x:ZN→RM, N⩽M, whose elementary quadrilaterals are planar.
Abstract: Motivated by the classical studies on transformations of conjugate nets, we develop the general geometric theory of transformations of their discrete analogs: the multidimensional quadrilateral lattices, ie, lattices x:ZN→RM, N⩽M, whose elementary quadrilaterals are planar Our investigation is based on the discrete analog of the theory of the rectilinear congruences, which we also present in detail We study, in particular, the discrete analogs of the Laplace, Combescure, Levy, radial, and fundamental transformations and their interrelations The composition of these transformations and their permutability is also investigated from a geometric point of view The deep connections between “transformations” and “discretizations” is also investigated for quadrilateral lattices We finally interpret these results within the ∂ formalism

Journal ArticleDOI
TL;DR: This work uses the comparison technique of Diaconis and Saloff-Coste to show that several of the natural single-point update algorithms are efficient, and relates the mixing rate of these algorithms to the corresponding nonlocal algorithms which have already been analyzed.
Abstract: A popular technique for studying random properties of a combinatorial set is to design a Markov chain Monte Carlo algorithm. For many problems there are natural Markov chains connecting the set of allowable configurations which are based on local moves, or “Glauber dynamics.” Typically these single-site update algorithms are difficult to analyze, so often the Markov chain is modified to update several sites simultaneously. Recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems. In this work we use the comparison technique of Diaconis and Saloff-Coste to show that several of the natural single-point update algorithms are efficient. The strategy is to relate the mixing rate of these algorithms to the corresponding nonlocal algorithms which have already been analyzed. This allows us to give polynomial time bounds for single-point update algorithms for problems such as generating planar tilings and random triangulations of convex polygons. We also survey several other comparison techniques, along with specific applications, which have been used in the context of estimating mixing rates of Markov chains.

Journal ArticleDOI
TL;DR: The rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade as discussed by the authors, and the main results that have been obtained, both in two and higher dimensions, can be found in this paper.
Abstract: The rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade. We review here the main results that have been obtained, both in two and higher dimensions. In particular, we describe how the phenomenological Wulff and Winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases. We focus on the main conceptual issues and describe the central ideas of the existing approaches.

Journal ArticleDOI
TL;DR: A group classification of invariant difference models, i.e., difference equations and meshes, is presented and it is shown that the discrete model can be invariant under Lie groups of dimension 0⩽n ⩽6.
Abstract: A group classification of invariant difference models, i.e., difference equations and meshes, is presented. In the continuous limit the results go over into Lie’s classification of second-order ordinary differential equations. The discrete model is a three point one and we show that it can be invariant under Lie groups of dimension 0⩽n⩽6.

Journal ArticleDOI
TL;DR: This is an invited survey on the relation between the partition function of the Potts model and the Tutte polynomial and highlights the connections with Abelian sandpiles, counting problems on random graphs, error correcting codes, and the EhrhartPolynomial of a zonotope.
Abstract: This is an invited survey on the relation between the partition function of the Potts model and the Tutte polynomial. On the assumption that the Potts model is more familiar we have concentrated on the latter and its interpretations. In particular we highlight the connections with Abelian sandpiles, counting problems on random graphs, error correcting codes, and the Ehrhart polynomial of a zonotope. Where possible we use the mean field and square lattice as illustrations. We also discuss in some detail the complexity issues involved.

Journal ArticleDOI
TL;DR: In this paper, the Schwinger model on a non-commutative sphere has been described and the model is quantized, and an exact expression for the chiral anomaly is found.
Abstract: We describe scalar and spinor fields on a noncommutative sphere starting from canonical realizations of the enveloping algebra A=U(u(2)). The gauge extension of a free spinor model, the Schwinger model on a noncommutative sphere, is defined and the model is quantized. The noncommutative version of the model contains only a finite number of dynamical modes and is nonperturbatively UV regular. An exact expression for the chiral anomaly is found. In the commutative limit the standard formula is recovered.

Journal ArticleDOI
TL;DR: In this article, the authors derived analytic expressions for three flavor neutrino oscillations in the presence of matter in the plane wave approximation using the Cayley-Hamilton formalism.
Abstract: We derive analytic expressions for three flavor neutrino oscillations in the presence of matter in the plane wave approximation using the Cayley–Hamilton formalism. Especially, we calculate the time evolution operator in both flavor and mass bases. Furthermore, we find the transition probabilities, matter mass squared differences, and matter mixing angles all expressed in terms of the vacuum mass squared differences, the vacuum mixing angles, and the matter density. The conditions for resonance in the presence of matter are also studied in some examples.

Journal ArticleDOI
TL;DR: In this article, a conformal N=4 extension of AM−1 Calogero models is presented, which for generic values of the coupling constant are not SU(1, 1|2) superconformal.
Abstract: We study N=4 supersymmetric quantum-mechanical many-body systems with M bosonic and 4M fermionic degrees of freedom. We also investigate the further restrictions of conformal and superconformal invariance. In particular, we construct conformal N=4 extensions of the AM−1 Calogero models, which for generic values of the coupling constant are not SU(1,1|2) superconformal. This class of models is also extended to arbitrary (even) N. We give both Hamiltonian and (classical) Lagrangian formulations. In the latter case, we use both component and N=4 superfield formulations.

Journal ArticleDOI
TL;DR: In this paper, the authors analyze how exactly one can associate combinatorial four-manifolds with the Feynman diagrams of certain tensor theories, and show that it is possible to encode all possible space-times of a suitable field theory in a four-dimensional manifold.
Abstract: The problem of constructing a quantum theory of gravity has been tackled with very different strategies, most of which rely on the interplay between ideas from physics and from advanced mathematics. On the mathematical side, a central role is played by combinatorial topology, often used to recover the space–time manifold from the other structures involved. An extremely attractive possibility is that of encoding all possible space–times as specific Feynman diagrams of a suitable field theory. In this work we analyze how exactly one can associate combinatorial four-manifolds with the Feynman diagrams of certain tensor theories.

Journal ArticleDOI
TL;DR: Inverse problems are those where a set of measured results is analyzed in order to get as much information as possible on a "model" which is proposed to represent a system in the real world as mentioned in this paper.
Abstract: Inverse problems are those where a set of measured results is analyzed in order to get as much information as possible on a “model” which is proposed to represent a system in the real world. Exact inverse problems are related to most parts of mathematics. Applied inverse problems are the keys to other sciences. Hence the field, which is very wealthy, yields the best example of interdisciplinary research but it has nevertheless a strong individuality. The obtained results and explored directions of the 20th century are sketched in this review, with attempts to predict their evolution.

Journal ArticleDOI
TL;DR: For independent nearest-neighbor bond percolation on Zd with d ≥ 6, it was shown in this paper that the incipient infinite cluster's two-point function and threepoint function converge to those of integrated super-Brownian excursion (ISE) in the scaling limit.
Abstract: For independent nearest-neighbor bond percolation on Zd with d≫6, we prove that the incipient infinite cluster’s two-point function and three-point function converge to those of integrated super-Brownian excursion (ISE) in the scaling limit. The proof is based on an extension of the new expansion for percolation derived in a previous paper, and involves treating the magnetic field as a complex variable. A special case of our result for the two-point function implies that the probability that the cluster of the origin consists of n sites, at the critical point, is given by a multiple of n−3/2, plus an error term of order n−3/2−e with e>0. This is a strong version of the statement that the critical exponent δ is given by δ=2.

Journal ArticleDOI
TL;DR: In this paper, the Clarkson and Kruskal (CK) direct method is modified to get the similarity and conditional similarity reductions of a (2+1) dimensional KdV-type equation.
Abstract: To get the similarity solutions of a nonlinear physical equation, one may use the classical Lie group approach, nonclassical Lie group approach and the Clarkson and Kruskal (CK) direct method. In this paper the direct method is modified to get the similarity and conditional similarity reductions of a (2+1) dimensional KdV-type equation. Ten types of usual similarity reductions [including the (1+1)-dimensional shallow water wave equation and the variable KdV equation] and six types of conditional similarity reductions of the (2+1)-dimensional KdV equation are obtained. Some special solutions of the conditional similarity reduction equations are found to show the nontriviality of the conditional similarity reduction approach. The conditional similarity solutions cannot be obtained by using the nonclassical Lie group approach in its present form. How to modify the nonclassical Lie group approach to obtain the conditional similarity solutions is still open.

Journal ArticleDOI
TL;DR: In this article, it was shown that for any two commuting von Neumann algebras of infinite type, the open set of Bell correlated states for the two algesbras is norm dense.
Abstract: We prove that for any two commuting von Neumann algebras of infinite type, the open set of Bell correlated states for the two algebras is norm dense. We then apply this result to algebraic quantum field theory—where all local algebras are of infinite type—in order to show that for any two spacelike separated regions, there is an open dense set of field states that dictate Bell correlations between the regions. We also show that any vector state cyclic for one of a pair of commuting non-Abelian von Neumann algebras is entangled (i.e., nonseparable) across the algebras—from which it follows that every field state with bounded energy is entangled across any two spacelike separated regions.

Journal ArticleDOI
TL;DR: In this paper, a generalized Lane-Emden equation with indices (α,β,ν,n) is discussed, which reduces to the Lane-emden equation proper for α = 2, β = 1, ν = 1.
Abstract: A generalized Lane–Emden equation with indices (α,β,ν,n) is discussed, which reduces to the Lane–Emden equation proper for α=2, β=1, ν=1. General properties of the set of solutions of this equation are derived, and exact solutions are given. These include a singular solution without free integration constant for arbitrary n and for particular relations between n, ν, and α. Among the two-parameter solutions nonequivalent families of solutions of the same equation are obtained.