Showing papers in "Communications in Mathematical Physics in 2010"
TL;DR: In this article, the authors define a quantum walk as a time-homogeneous quantum process on a graph, which can be defined either in continuous or discrete time and can be obtained as the limit of a sequence of discrete-time random walks.
Abstract: Quantum walk is one of the main tools for quantum algorithms. Defined by analogy to classical random walk, a quantum walk is a time-homogeneous quantum process on a graph. Both random and quantum walks can be defined either in continuous or discrete time. But whereas a continuous-time random walk can be obtained as the limit of a sequence of discrete-time random walks, the two types of quantum walk appear fundamentally different, owing to the need for extra degrees of freedom in the discrete-time case.
395 citations
TL;DR: In this paper, a physical explanation of the Kontsevich-Soibelman wall-crossing formula for the BPS spectrum in Seiberg-Witten theories is given.
Abstract: We give a physical explanation of the Kontsevich-Soibelman wall-crossing formula for the BPS spectrum in Seiberg-Witten theories. In the process we give an exact description of the BPS instanton corrections to the hyperkahler metric of the moduli space of the theory on \({\mathbb R^3 \times S^1}\). The wall-crossing formula reduces to the statement that this metric is continuous. Our construction of the metric uses a four-dimensional analogue of the two-dimensional tt* equations.
373 citations
TL;DR: The universality of the eigenvalues of Wigner random matrices has been studied in this article, where the authors show that there is a significant bias in the Cauchy interlacing law near the edge of the spectrum, which allows one to continue ensuring the delocalization of eigenvectors.
Abstract: This is a continuation of our earlier paper (Tao and Vu, http://arxiv.org/abs/0908.1982v4[math.PR], 2010) on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in Tao and Vu (http://arxiv.org/abs/0908.1982v4[math.PR], 2010) from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results of Soshnikov (Commun Math Phys 207(3):697–733, 1999) for the largest eigenvalues, assuming moment conditions rather than symmetry conditions. The main new technical observation is that there is a significant bias in the Cauchy interlacing law near the edge of the spectrum which allows one to continue ensuring the delocalization of eigenvectors.
317 citations
CERN1
TL;DR: In this article, the authors proposed to combine the Wilson flow (which generates approximately trivializing maps for the Wilson gauge action) with the HMC simulation algorithm in order to improve the efficiency of lattice QCD simulations.
Abstract: In lattice gauge theory, there exist field transformations that map the theory to the trivial one, where the basic field variables are completely decoupled from one another. Such maps can be constructed systematically by integrating certain flow equations in field space. The construction is worked out in some detail and it is proposed to combine the Wilson flow (which generates approximately trivializing maps for the Wilson gauge action) with the HMC simulation algorithm in order to improve the efficiency of lattice QCD simulations.
253 citations
TL;DR: In this paper, the authors consider the time evolution of a system of identical bosons whose interaction potential is rescaled by N−1 and derive bounds on the rate of convergence of the quantum N-body dynamics to the Hartree equation.
Abstract: We consider the time evolution of a system of N identical bosons whose interaction potential is rescaled by N−1. We choose the initial wave function to describe a condensate in which all particles are in the same one-particle state. It is well known that in the mean-field limit N → ∞ the quantum N-body dynamics is governed by the nonlinear Hartree equation. Using a nonperturbative method, we extend previous results on the mean-field limit in two directions. First, we allow a large class of singular interaction potentials as well as strong, possibly time-dependent external potentials. Second, we derive bounds on the rate of convergence of the quantum N-body dynamics to the Hartree dynamics.
232 citations
TL;DR: In this paper, a new nonlinear Schrodinger equation was derived that describes a second-order correction to the usual tensor product approximation for the Hamiltonian evolution of a many-particle system in Bose-Einstein condensation.
Abstract: Inspired by the works of Rodnianski and Schlein [31] and Wu [34,35], we derive a new nonlinear Schrodinger equation that describes a second-order correction to the usual tensor product (mean-field) approximation for the Hamiltonian evolution of a many-particle system in Bose-Einstein condensation. We show that our new equation, if it has solutions with appropriate smoothness and decay properties, implies a new Fock space estimate. We also show that for an interaction potential $${v(x)= \epsilon \chi(x) |x|^{-1}}$$
, where $${\epsilon}$$
is sufficiently small and $${\chi \in C_0^{\infty}}$$
even, our program can be easily implemented locally in time. We leave global in time issues, more singular potentials and sophisticated estimates for a subsequent part (Part II) of this paper.
197 citations
TL;DR: In this paper, a Sacks-Uhlenbeck/Struwe type global regularity result for wave-maps was proved for general compact target manifolds, where the wave-map can be decomposed into compact target manifold.
Abstract: In this article we prove a Sacks-Uhlenbeck/Struwe type global regularity result for wave-maps $${\Phi:\mathbb{R}^{2+1} \to\mathcal{M} }$$
into general compact target manifolds $${\mathcal{M} }$$
.
179 citations
TL;DR: In this paper, dispersive properties for the wave equation in the Schwarzschild space-time were studied and a local energy estimate was obtained, which was then used to establish global-in-time Strichartz estimates.
Abstract: We study dispersive properties for the wave equation in the Schwarzschild space-time. The first result we obtain is a local energy estimate. This is then used, following the spirit of [29], to establish global-in-time Strichartz estimates. A considerable part of the paper is devoted to a precise analysis of solutions near the trapping region, namely the photon sphere.
163 citations
TL;DR: In this article, the existence of an Operator Product Expansion (OPE) is elevated to a fundamental status, and all of the properties of the quantum field theory are determined by its OPE.
Abstract: The usual formulations of quantum field theory in Minkowski spacetime make crucial use of features—such as Poincare invariance and the existence of a preferred vacuum state—that are very special to Minkowski spacetime. In order to generalize the formulation of quantum field theory to arbitrary globally hyperbolic curved spacetimes, it is essential that the theory be formulated in an entirely local and covariant manner, without assuming the presence of a preferred state. We propose a new framework for quantum field theory, in which the existence of an Operator Product Expansion (OPE) is elevated to a fundamental status, and, in essence, all of the properties of the quantum field theory are determined by its OPE. We provide general axioms for the OPE coefficients of a quantum field theory. These include a local and covariance assumption (implying that the quantum field theory is constructed in a local and covariant manner from the spacetime metric and other background structure, such as time and space orientations), a microlocal spectrum condition, an “associativity” condition, and the requirement that the coefficient of the identity in the OPE of the product of a field with its adjoint have positive scaling degree. We prove curved spacetime versions of the spin-statistics theorem and the PCT theorem. Some potentially significant further implications of our new viewpoint on quantum field theory are discussed.
160 citations
TL;DR: In this article, the authors consider large data Wave-Maps from a compact Riemannian manifold and prove that regularity and dispersive bounds persist as long as a certain type of bulk (non-dispersive) concentration is absent.
Abstract: In this article we consider large data Wave-Maps from \({\mathbb R^{2+1}}\) into a compact Riemannian manifold \({(\mathcal{M},g)}\), and we prove that regularity and dispersive bounds persist as long as a certain type of bulk (non-dispersive) concentration is absent. This is a companion to our concurrent article [21], which together with the present work establishes a full regularity theory for large data Wave-Maps.
150 citations
TL;DR: In this paper, the incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions was studied and it was rigorously shown that the weak solutions of these equations converged to the strong solution of the viscous or inviscid incompressibly magnetodynamic equations as long as the latter exists both for well-prepared initial data and general initial data.
Abstract: This paper is concerned with the incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions. It is rigorously shown that the weak solutions of the compressible magnetohydrodynamic equations converge to the strong solution of the viscous or inviscid incompressible magnetohydrodynamic equations as long as the latter exists both for the well-prepared initial data and general initial data. Furthermore, the convergence rates are also obtained in the case of the well-prepared initial data.
TL;DR: For general asymptotically subadditive potentials, this paper established variational relations between the topological entropy of the level sets of Lyapunov exponents, measure-theoretic entropies and topological pressures.
Abstract: For general asymptotically sub-additive potentials (resp. asymptotically additive potentials) on general topological dynamical systems, we establish some variational relations between the topological entropy of the level sets of Lyapunov exponents, measure-theoretic entropies and topological pressures in this general situation. Most of our results are obtained without the assumption of the existence of unique equilibrium measures or the differentiability of pressure functions. Some examples are constructed to illustrate the irregularity and the complexity of multifractal behaviors in the sub-additive case and in the case that the entropy map is not upper-semi continuous.
TL;DR: In this paper, the stability of the equilibrium states and the rate of convergence of solutions towards them for the continuous kinetic version of the Cucker-Smale flocking in presence of diffusion whose strength depends on the density were studied.
Abstract: We study the stability of the equilibrium states and the rate of convergence of solutions towards them for the continuous kinetic version of the Cucker-Smale flocking in presence of diffusion whose strength depends on the density. This kinetic equation describes the collective behavior of an ensemble of organisms, animals or devices which are forced to adapt their velocities according to a certain rule implying a final configuration in which the ensemble flies at the mean velocity of the initial configuration. Our analysis takes advantage both from the fact that the global equilibrium is a Maxwellian distribution function, and, on the contrary to what happens in the Cucker-Smale model (IEEE Trans Autom Control 52:852–862, 2007), the interaction potential is an integrable function. Precise conditions which guarantee polynomial rates of convergence towards the global equilibrium are found.
TL;DR: In this paper, the authors discuss some examples of abelian gauged linear sigma models realizing twisted derived equivalences between non-birational spaces, and realizing geometries in novel fashions.
Abstract: In this paper we discuss some examples of abelian gauged linear sigma models realizing twisted derived equivalences between non-birational spaces, and realizing geometries in novel fashions. Examples of gauged linear sigma models with non-birational Kahler phases are a relatively new phenomenon. Most of our examples involve gauged linear sigma models for complete intersections of quadric hypersurfaces, though we also discuss some more general cases and their interpretation. We also propose a more general understanding of the relationship between Kahler phases of gauged linear sigma models, namely that they are related by (and realize) Kuznetsov’s ‘homological projective duality.’ Along the way, we shall see how ‘noncommutative spaces’ (in Kontsevich’s sense) are realized physically in gauged linear sigma models, providing examples of new types of conformal field theories. Throughout, the physical realization of stacks plays a key role in interpreting physical structures appearing in GLSMs, and we find that stacks are implicitly much more common in GLSMs than previously realized.
TL;DR: In this paper, a Poisson structure on the space of twisted polygons is found and integrability of the pentagram map relative to this structure is shown to be completely integrable.
Abstract: The pentagram map is a projectively natural transformation defined on (twisted) polygons. A twisted polygon is a map from \({\mathbb Z}\) into \({{\mathbb{RP}}^2}\) that is periodic modulo a projective transformation called the monodromy. We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation. A research announcement of this work appeared in [16].
TL;DR: In this article, the authors generalize the classical study of generalized Lax pairs, the related $${\mathcal O}$$ -operators and the (modified) classical Yang-Baxter equation by introducing the concepts of nonabelian generalized R-matrix ansatz and related double Lie algebra structures.
Abstract: We generalize the classical study of (generalized) Lax pairs, the related $${\mathcal O}$$
-operators and the (modified) classical Yang-Baxter equation by introducing the concepts of nonabelian generalized Lax pairs, extended $${\mathcal O}$$
-operators and the extended classical Yang-Baxter equation. We study in this context the nonabelian generalized r-matrix ansatz and the related double Lie algebra structures. Relationship between extended $${\mathcal O}$$
-operators and the extended classical Yang-Baxter equation is established, especially for self-dual Lie algebras. This relationship allows us to obtain an explicit description of the Manin triples for a new class of Lie bialgebras. Furthermore, we show that a natural structure of PostLie algebra is behind $${\mathcal O}$$
-operators and fits in a setup of the triple Lie algebra that produces self-dual nonabelian generalized Lax pairs.
TL;DR: In this paper, a Lie 2-algebra of observables is proposed to describe the dynamics of a classical bosonic string in the presence of a B field, where the B field affects the 2-plectic structure for the string.
Abstract: A Lie 2-algebra is a ‘categorified’ version of a Lie algebra: that is, a category equipped with structures analogous to those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry describes an n-dimensional field theory using a phase space that is an ‘n-plectic manifold’: a finite-dimensional manifold equipped with a closed nondegenerate (n + 1)-form. Here we consider the case n = 2. For any 2-plectic manifold, we construct a Lie 2-algebra of observables. We then explain how this Lie 2-algebra can be used to describe the dynamics of a classical bosonic string. Just as the presence of an electromagnetic field affects the symplectic structure for a charged point particle, the presence of a B field affects the 2-plectic structure for the string.
TL;DR: In this article, a priori estimates for the three-dimensional compressible Euler equations with moving physical vacuum boundary were obtained, with an equation of state given by p(ρ) = Cγ ρ γ for γ> 1.
Abstract: We prove a priori estimates for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) = Cγ ρ γ for γ> 1. The vacuum condition necessitates the vanishing of the pressure, and hence density, on the dynamic boundary, which creates a degenerate and characteristic hyperbolic free-boundary system to which standard methods of symmetr- izable hyperbolic equations cannot be applied.
TL;DR: In this article, it was shown that the data-to-solution map is not uniformly continuous in Sobolev H for any > 0 if the domain is the (flat) torus.
Abstract: We show that continuous dependence on initial data of solutions to the Euler equations of incompressible hydrodynamics is optimal More precisely, we prove that the data-to-solution map is not uniformly continuous in Sobolev H
s
(Ω) topology for any $${s \in \mathbb{R}}$$
if the domain Ω is the (flat) torus $${\mathbb{T}^n=\mathbb{R}^n/2\pi\mathbb{Z}^n}$$
and for any s > 0 if the domain is the whole space $${\mathbb{R}^n}$$
TL;DR: This work presents a generalization of quantum Stein’s Lemma to the situation in which the alternative hypothesis is formed by a family of states, which implies that whenever a multipartite state can be asymptotically converted into another entangled state by local operations and classical communication, the rate of conversion must be non-zero.
Abstract: Given many independent and identically-distributed (i.i.d.) copies of a quantum system described either by the state ρ or σ (called null and alternative hypotheses, respectively), what is the optimal measurement to learn the identity of the true state? In asymmetric hypothesis testing one is interested in minimizing the probability of mistakenly identifying ρ instead of σ, while requiring that the probability that σ is identified in the place of ρ is bounded by a small fixed number. Quantum Stein’s Lemma identifies the asymptotic exponential rate at which the specified error probability tends to zero as the quantum relative entropy of ρ and σ.
TL;DR: In this article, the free energy of the directed polymer in a random environment model in dimension 1 + 1 and 1 + 2 was studied and it was shown that very strong disorder holds at all temperatures.
Abstract: We study the free energy of the directed polymer in a random environment model in dimension 1 + 1 and 1 + 2. For dimension one, we improve the statement of Comets and Vargas in [8] concerning very strong disorder by giving sharp estimates on the free energy at high temperature. In dimension two, we prove that very strong disorder holds at all temperatures, thus solving a long standing conjecture in the field.
TL;DR: In this article, the authors studied the length distribution of the nodal lines of random spherical harmonics and showed that the expected length should be of order n, due to the natural scaling.
Abstract: Using the multiplicities of the Laplace eigenspace on the sphere (the space of spherical harmonics) we endow the space with Gaussian probability measure. This induces a notion of random Gaussian spherical harmonics of degree n having Laplace eigenvalue E = n(n + 1). We study the length distribution of the nodal lines of random spherical harmonics. It is known that the expected length is of order n. It is natural to conjecture that the variance should be of order n, due to the natural scaling. Our principal result is that, due to an unexpected cancelation, the variance of the nodal length of random spherical harmonics is of order log n. This behaviour is consistent with the one predicted by Berry for nodal lines on chaotic billiards (Random Wave Model). In addition we find that a similar result is applicable for “generic” linear statistics of the nodal lines.
TL;DR: In this article, a family of equations defined on the space of tensor densities of weight λ on the circle and introduced two integrable PDEs were studied. But the integrability of the Burgers equation was not discussed.
Abstract: We study a family of equations defined on the space of tensor densities of weight λ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct “peakon” and “multi-peakon” solutions for all λ ≠ 0, 1, and “shock-peakons” for λ = 3. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold’s approach to Euler equations on Lie groups.
TL;DR: In this paper, a direct connection between refined topological vertex and sl(N ) homological invariants of the Hopf link is established, which includes Khovanov-Rozansky invariants as a special case.
Abstract: We establish a direct map between refined topological vertex and sl(N ) homological invariants of the of Hopf link, which include Khovanov-Rozansky homol- ogy as a special case. This relation provides an exact answer for homological invariants of the Hopf link, whose components are colored by arbitrary representations of sl(N ). At present, the mathematical formulation of such homological invariants is available only for the fundamental representation (the Khovanov-Rozansky theory) and the rela- tion with the refined topological vertex should be useful for categorizing quantum group invariants associated with other representations (R1, R2). Our result is a first direct veri- fication of a series of conjectures which identifies link homologies with the Hilbert space of BPS states in the presence of branes, where the physical interpretation of gradings is in terms of charges of the branes ending on Lagrangian branes.
TL;DR: In this article, a lattice field model for real symmetric band matrices is studied, and it is shown that in three or more dimensions the model has a "diffusive" phase at low temperatures.
Abstract: We study a lattice field model which qualitatively reflects the phenomenon of Anderson localization and delocalization for real symmetric band matrices. In this statistical mechanics model, the field takes values in a supermanifold based on the hyperbolic plane. Correlations in this model may be described in terms of a random walk in a highly correlated random environment. We prove that in three or more dimensions the model has a ‘diffusive’ phase at low temperatures. Localization is expected at high temperatures. Our analysis uses estimates on non-uniformly elliptic Green’s functions and a family of Ward identities coming from internal supersymmetry.
TL;DR: In this paper, the controllability of the Bloch equation for an ensemble of non interacting half-spins, inastaticmagneticfield, withdispersion in the Larmorf frequency was investigated.
Abstract: We study the controllability of the Bloch equation, for an ensemble of non interactinghalf-spins,inastaticmagneticfield,withdispersionintheLarmorfrequency. This system may be seen as a prototype for infinite dimensional bilinear systems with continuous spectrum, whose controllability is not well understood. We provide several mathematical answers, with discrimination between approximate and exact controlla- bility, and between finite time or infinite time controllability: this system is not exactly controllable in finite time T with bounded controls in L 2 (0, T ), but it is approximately controllable in L ∞ in finite time with unbounded controls in L ∞ ((0,+∞)). Moreover, we propose explicit controls realizing the asymptotic exact controllability to a uniform state of spin +1/ 2o r−1/2.
TL;DR: In this article, a nonlinear pseudodifferential equation describing the dynamics of dislocations in crystals is studied and the long time asymptotics of solutions are described by the self-similar profiles.
Abstract: We study a nonlinear pseudodifferential equation describing the dynamics of dislocations in crystals. The long time asymptotics of solutions is described by the self-similar profiles.
TL;DR: The SU(3) theory with six quarks and the SU(2) gauge theory with one pair of quarks coupled to the superconformal theory with E 6 flavor symmetry were shown to be S-dual in this article.
Abstract: We demonstrate the agreement between the Higgs branches of two \({\mathcal{N}=2}\) theories proposed by Argyres and Seiberg to be S-dual, namely the SU(3) gauge theory with six quarks, and the SU(2) gauge theory with one pair of quarks coupled to the superconformal theory with E 6 flavor symmetry. In mathematical terms, we demonstrate the equivalence between a hyperkahler quotient of a linear space and another hyperkahler quotient involving the minimal nilpotent orbit of E 6, modulo the identification of the twistor lines.
TL;DR: In this paper, a graphical tool was developed to compute the expected moments of the output of a random quantum channel from the perspective of the random matrix point of view, which is similar to the one presented in this paper.
Abstract: This paper is the first of a series where we study quantum channels from the random matrix point of view. We develop a graphical tool that allows us to compute the expected moments of the output of a random quantum channel.
TL;DR: In this paper, it was shown that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order Θ(n) when observables with n possible outcomes are used.
Abstract: In this work we show that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order \({{\rm \Omega} \left(\frac{\sqrt{n}}{\log^2n} \right)}\) when observables with n possible outcomes are used. A central tool in the analysis is a close relation between this problem and operator space theory and, in particular, the very recent noncommutative Lp embedding theory.