Showing papers in "Journal of Mathematical Physics in 2009"

On integrability of the Yang–Baxter σ-model

Abstract: We prove that the recently introduced Yang–Baxter σ-model can be considered as an integrable deformation of the principal chiral model. We find also an explicit one-to-one map transforming every solution of the principal chiral model into a solution of the deformed model. With the help of this map, the standard procedure of the dressing of the principal chiral solutions can be directly transferred into the deformed Yang–Baxter context.

The (G′/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics

Abstract: I the present paper, we construct the traveling wave solutions involving parameters of the combined Korteweg-de Vries–modified Korteweg-de Vries equation, the reaction-diffusion equation, the compound KdV–Burgers equation, and the generalized shallow water wave equation by using a new approach, namely, the (G′/G)-expansion method, where G=G(ξ) satisfies a second order linear ordinary differential equation. When the parameters take special values, the solitary waves are derived from the traveling waves. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions.

Asymptotic analysis of the Engle–Pereira–Rovelli–Livine four-simplex amplitude

Abstract: The semiclassical limit of a four-simplex amplitude for a spin foam quantum gravity model with an Immirzi parameter is studied. If the boundary state represents a nondegenerate four-simplex geometry, the asymptotic formula contains the Regge action for general relativity. A canonical choice of phase for the boundary state is introduced and is shown to be necessary to obtain the results.

Strongly interacting bumps for the Schrödinger–Newton equations

Abstract: We study concentrated bound states of the Schrodinger–Newton equations h2Δψ−E(x)ψ+Uψ=0, ψ>0, x∊R3; ΔU+12|ψ|2=0, x∊R3; ψ(x)→0, U(x)→0 as |x|→∞. Moroz et al. [“An analytical approach to the Schrodinger-Newton equations,” Nonlinearity 12, 201 (1999)] proved the existence and uniqueness of ground states of Δψ−ψ+Uψ=0, ψ>0, x∊R3; ΔU+12|ψ|2=0, x∊R3; ψ(x)→0, U(x)→0 as |x|→∞. We first prove that the linearized operator around the unique ground state radial solution (ψ0,U0) with ψ0(r)=(Ae−r/r)(1+o(1)) as r=|x|→∞, U0(r)=(B/r)(1+o(1)) as r=|x|→∞ for some A,B>0 has a kernel whose dimension is exactly 3 (corresponding to the translational modes). Using this result we further show that if for some positive integer K the points Pi∊R3, i=1,2…,K, with Pi≠Pj for i≠j are all local minimum or local maximum or nondegenerate critical points of E(P), then for h small enough there exist solutions of the Schrodinger–Newton equations with K bumps which concentrate at Pi. We also prove that given a local maximum point P0 of E(P) the...

A commutative algebra on degenerate CP1 and Macdonald polynomials

Abstract: We introduce a unital associative algebra A associated with degenerate CP1. We show that A is a commutative algebra and whose Poincare series is given by the number of partitions. Thereby, we can regard A as a smooth degeneration limit of the elliptic algebra introduced by Feigin and Odesskii [Int. Math. Res. Notices 11, 531 (1997)]. Then we study the commutative family of the Macdonald difference operators acting on the space of symmetric functions. A canonical basis is proposed for this family by using A and the Heisenberg representation of the commutative family studied by Shiraishi [ Commun. Math. Phys. 263, 439 (2006)]. It is found that the Ding–Iohara algebra [Lett. Math. Phys. 41, 183 (1997)] provides us with an algebraic framework for the free field construction. An elliptic deformation of our construction is discussed, showing connections with the Drinfeld quasi-Hopf twisting [Leningrad Math. J. 1, 1419 (1990)] in the sence of Babelon-Bernard–Billey [Phys. Lett. B. 375, 89 (1996)], the Ruijsenaar...

A classification of near-horizon geometries of extremal vacuum black holes

Abstract: We consider the near-horizon geometries of extremal, rotating black hole solutions of the vacuum Einstein equations, including a negative cosmological constant, in four and five dimensions. We assume the existence of one rotational symmetry in four dimensions (4D), two commuting rotational symmetries in five dimensions (5D), and in both cases nontoroidal horizon topology. In 4D we determine the most general near-horizon geometry of such a black hole and prove it is the same as the near-horizon limit of the extremal Kerr-AdS4 black hole. In 5D, without a cosmological constant, we determine all possible near-horizon geometries of such black holes. We prove that the only possibilities are one family with a topologically S1×S2 horizon and two distinct families with topologically S3 horizons. The S1×S2 family contains the near-horizon limit of the boosted extremal Kerr string and the extremal vacuum black ring. The first topologically spherical case is identical to the near-horizon limit of two different black...

Generalization of n-ary Nambu algebras and beyond

Abstract: The aim of this paper is to introduce n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type including n-ary Nambu algebras, n-ary Nambu-Lie algebras and n-ary Lie algebras, and n-ary algebras of associative type including n-ary totally associative and n-ary partially associative algebras. We provide examples of the new structures and present some properties and construction theorems. We describe the general method allowing one to obtain an n-ary Hom-algebra structure starting from an n-ary algebra and an n-ary algebra endomorphism. Several examples are derived using this process. Also we initiate investigation of classification problems for algebraic structures introduced in the article and describe all ternary three-dimensional Hom-Nambu-Lie structures with diagonal homomorphism.

Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics II:Painleve transcendent potentials

Abstract: We consider a superintegrable quantum potential in two-dimensional Euclidean space with a second and a third order integral of motion. The potential is written in terms of the fourth Painleve transcendent. We construct for this system a cubic algebra of integrals of motion. The algebra is realized in terms of parafermionic operators and we present Fock-type representations which yield the corresponding energy spectra. We also discuss this potential from the point of view of higher order supersymmetric quantum mechanics and obtain ground state wave functions.

Adiabatic approximation with exponential accuracy for many-body systems and quantum computation

Abstract: We derive a version of the adiabatic theorem that is especially suited for applications in adiabatic quantum computation, where it is reasonable to assume that the adiabatic interpolation between the initial and final Hamiltonians is controllable. Assuming that the Hamiltonian is analytic in a finite strip around the real-time axis, that some number of its time derivatives vanish at the initial and final times, and that the target adiabatic eigenstate is nondegenerate and separated by a gap from the rest of the spectrum, we show that one can obtain an error between the final adiabatic eigenstate and the actual time-evolved state which is exponentially small in the evolution time, where this time itself scales as the square of the norm of the time derivative of the Hamiltonian divided by the cube of the minimal gap.

Cones of positive maps and their duality relations

Abstract: The structure of cones of positive and k-positive maps acting on a finite-dimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive and k-superpositive maps. We characterize k-positive and k-superpositive maps with regard to their properties under taking compositions. A number of results obtained for maps are also rephrased for the corresponding cones of block positive, k-block positive, separable, and k-entangled operators due to the Jamiolkowski–Choi isomorphism. Generalizations to a situation where no such simple isomorphism is available are also made, employing the idea of mapping cones. As a side result to our discussion, we show that extreme entanglement witnesses, which are optimal, should be of special interest in entanglement studies.

On a regularity criterion for the Navier–Stokes equations involving gradient of one velocity component

Abstract: We improve the regularity criterion for the incompressible Navier–Stokes equations in the full three-dimensional space involving the gradient of one velocity component. The method is based on recent results of Cao and Titi [see “Regularity criteria for the three dimensional Navier–Stokes equations,” Indiana Univ. Math. J. 57, 2643 (2008)] and Kukavica and Ziane [see “Navier-Stokes equations with regularity in one direction,” J. Math. Phys. 48, 065203 (2007)]. In particular, for s∊[2,3], we get that the solution is regular if ∇u3∊Lt(0,T;Ls(R3)), 2/t+3/s≤2312.

Quantum geometry from phase space reduction

Abstract: In this work, we give an explicit isomorphism between the usual spin network basis and the direct quantization of the reduced phase space of tetrahedra. The main outcome is a formula that describes the space of SU(2) invariant states by an integral over coherent states satisfying the closure constraint exactly or, equivalently, as an integral over the space of classical tetrahedra. This provides an explicit realization of theorems by Guillemin–Sternberg and Hall that describe the commutation of quantization and reduction. In the final part of the paper, we use our result to express the Freidel–Krasnov spin foam model as an integral over classical tetrahedra, and the asymptotics of the vertex amplitude is determined.

Multi-instantons and multicuts

Abstract: We discuss various aspects of multi-instanton configurations in generic multicut matrix models. Explicit formulas are presented in the two-cut case and, in particular, we obtain general formulas for multi-instanton amplitudes in the one-cut matrix model case as a degeneration of the two-cut case. These formulas show that the instanton gas is ultradilute due to the repulsion among the matrix model eigenvalues. We exemplify and test our general results in the cubic matrix model, where multi-instanton amplitudes can be also computed with orthogonal polynomials. As an application, we derive general expressions for multi-instanton contributions in two-dimensional quantum gravity, verifying them by computing the instanton corrections to the string equation. The resulting amplitudes can be interpreted as regularized partition functions for multiple ZZ-branes, which take into full account their backreaction on the target geometry. Finally, we also derive structural properties of the trans-series solution to the Painleve I equation.

Are the spectra of geometrical operators in Loop Quantum Gravity really discrete

Abstract: One of the celebrated results of Loop Quantum Gravity (LQG) is the discreteness of the spectrum of geometrical operators such as length, area and volume operators. This is an indication that Planck scale geometry in LQG is discontinuous rather than smooth. However, there is no rigorous proof thereof at present, because the afore mentioned operators are not gauge invariant, they do not commute with the quantum constraints. The relational formalism in the incarnation of Rovelli’s partial and complete observables provides a possible mechanism for turning a non gauge invariant operator into a gauge invariant one. In this paper we investigate whether the spectrum of such a physical, that is gauge invariant, observable can be predicted from the spectrum of the corresponding gauge variant observables. We will not do this in full LQG but rather consider much simpler examples where field theoretical complications are absent. We find, even in those simpler cases, that kinematical discreteness of the spectrum does not necessarily survive at the gauge invariant level. Whether or not this happens depends crucially on how the gauge invariant completion is performed. This indicates that “fundamental discreteness at Planck scale in LQG” is far from established. To prove it, one must provide the detailed construction of gauge invariant versions of geometrical operators.

The flavor group Δ(6n2)

Abstract: Many non-Abelian finite subgroups of SU(3) have been used to explain the flavor structure of the standard model. In order to systematize and classify successful models, a detailed knowledge of their mathematical structure is necessary. In this paper, we shall therefore look closely at the series of finite non-Abelian groups known as Δ(6n2), its smallest members being S3 (n=1) and S4 (n=2). For arbitrary n, we determine the conjugacy classes, the irreducible representations, the Kronecker products, as well as the Clebsch–Gordan coefficients.

An efficient algorithm for computing the Baker–Campbell–Hausdorff series and some of its applications

Abstract: We provide a new algorithm for generating the Baker–Campbell–Hausdorff (BCH) series Z=log(eXeY) in an arbitrary generalized Hall basis of the free Lie algebra L(X,Y) generated by X and Y. It is based on the close relationship of L(X,Y) with a Lie algebraic structure of labeled rooted trees. With this algorithm, the computation of the BCH series up to degree of 20 [111 013 independent elements in L(X,Y)] takes less than 15min on a personal computer and requires 1.5Gbytes of memory. We also address the issue of the convergence of the series, providing an optimal convergence domain when X and Y are real or complex matrices.

On polynomial invariants of several qubits

Abstract: It is a recent observation that entanglement classification for qubits is closely related to local SL(2,C)-invariants including the invariance under qubit permutations [Dur, et al., Phys. Rev. A 62, 062314 (2000); Osterloh, A. and Siewert, J., Phys. Rev. A 72, 012337 (2005); O. Chterental and D. Ž. Ðokovic, Linear Algebra Research Advances (Nova Science, Hauppauge, N.Y., 2007), Chap. 4, p. 133], which has been termed SL∗ invariance. In order to single out the SL∗ invariants, we analyze the SL(2,C)-invariants of four (five) qubits and decompose them into irreducible modules for the symmetric group S4 (S5) of qubit permutations. A classifying set of measures of genuine multipartite entanglement is given by the ideal of the algebra of SL∗-invariants vanishing on arbitrary product states. We find that low degree homogeneous components of this ideal can be constructed in full by using the approach introduced by Osterloh and Siewert [Phys. Rev. A 72, 012337 (2005); Int. J. Quant. Inf. 4, 531 (2006)]. Our analys...

On the thermodynamic limit of form factors in the massless XXZ Heisenberg chain

Abstract: We consider the problem of computing form factors of the massless XXZ Heisenberg spin-1/2 chain in a magnetic field in the (thermodynamic) limit where the size M of the chain becomes large. For that purpose, we take the particular example of the matrix element of the operator σz between the ground state and an excited state with one particle and one hole located at the opposite ends of the Fermi interval (umklapp-type term). We exhibit its power-law decrease in terms of the size of the chain M and compute the corresponding exponent and amplitude. As a consequence, we show that this form factor is directly related to the amplitude of the leading oscillating term in the long-distance asymptotic expansion of the correlation function ⟨σ1zσm+1z⟩.

Generalized relative entropies and the capacity of classical-quantum channels

Abstract: We provide lower and upper bounds on the information transmission capacity of one single use of a classical-quantum channel. The lower bound is expressed in terms of the Hoeffding capacity, which we define similarly to the Holevo capacity but replacing the relative entropy with the Hoeffding distance. Similarly, our upper bound is in terms of a quantity obtained by replacing the relative entropy with the recently introduced max-relative entropy in the definition of the divergence radius of a channel.

Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. I. Rational function potentials

Abstract: We consider a superintegrable Hamiltonian system in a two-dimensional space with a scalar potential that allows one quadratic and one cubic integrals of motion. We construct the most general cubic algebra and we present specific realizations. We use them to calculate the energy spectrum. All classical and quantum superintegrable potentials separable in Cartesian coordinates with a third order integral are known. The general formalism is applied to quantum reducible and irreducible rational potentials separable in Cartesian coordinates in E2. We also discuss these potentials from the point of view of supersymmetric and PT-symmetric quantum mechanics.

On some properties of orthogonal Weingarten functions

Abstract: We give a Fourier-type formula for computing the orthogonal Weingarten formula. The Weingarten calculus was introduced as a systematic method to compute integrals of polynomials with respect to Haar measure over classical groups. Although a Fourier-type formula was known in the unitary case, the orthogonal counterpart was not known. It relies on the Jack polynomial generalization of both Schur and zonal polynomials. This formula substantially reduces the complexity involved in the computation of Weingarten formulas. We also describe a few more new properties of the Weingarten formula, state a conjecture, and give a table of values.

On staggered indecomposable Virasoro modules

Abstract: In this article, certain indecomposable Virasoro modules are studied. Specifically, the Virasoro mode L0 is assumed to be nondiagonalizable, possessing Jordan blocks of rank 2. Moreover, the module is further assumed to have a highest weight submodule, the “left module,” and that the quotient by this submodule yields another highest weight module, the “right module.” Such modules, which have been called staggered, have appeared repeatedly in the logarithmic conformal field theory literature, but their theory has not been explored in full generality. Here, such a theory is developed for the Virasoro algebra using rather elementary techniques. The focus centers on two different but related questions typically encountered in practical studies: How can one identify a given staggered module, and how can one demonstrate the existence of a proposed staggered module. Given just the values of the highest weights of the left and right modules, themselves subject to simple necessary conditions, invariants are defined which together with the knowledge of the left and right modules uniquely identify a staggered module. The possible values of these invariants form a vector space of dimension 0, 1, or 2, and the structures of the left and right modules limit the isomorphism classes of the corresponding staggered modules to an affine subspace (possibly empty). The number of invariants and affine restrictions is purely determined by the structures of the left and right modules. Moreover, in order to facilitate applications, the expressions for the invariants and restrictions are given by formulas as explicit as possible (they generally rely on expressions for Virasoro singular vectors). Finally, the text is liberally peppered throughout with examples illustrating the general concepts. These have been carefully chosen for their physical relevance or for the novel features they exhibit.

Distinguished quantum states in a class of cosmological spacetimes and their Hadamard property

Abstract: In a recent paper, we proved that a large class of spacetimes, not necessarily homogeneous or isotropous and relevant at a cosmological level, possesses a preferred codimension 1 submanifold, i.e., the past cosmological horizon, on which it is possible to encode the information of a scalar field theory living in the bulk. Such bulk-to-boundary reconstruction procedure entails the identification of a preferred quasifree algebraic state for the bulk theory, enjoying remarkable properties concerning invariance under isometries (if any) of the bulk and energy positivity and reducing to well-known vacua in standard situations. In this paper, specializing to open Friedmann–Robertson–Walker models, we extend previously obtained results and we prove that the preferred state is of Hadamard form, hence the backreaction on the metric is finite and the state can be used as a starting point for renormalization procedures. Such state could play a distinguished role in the discussion of the evolution of scalar fluctuations of the metric, an analysis often performed in the development of any model describing the dynamic of an early Universe which undergoes an inflationary phase of rapid expansion in the past.

Correlation functions of integrable models: A description of the ABACUS algorithm

Abstract: Recent developments in the theory of integrable models have provided the means of calculating dynamical correlation functions of some important observables in systems such as Heisenberg spin chains and one-dimensional atomic gases. This article explicitly describes how such calculations are generally implemented in the ABACUS C++ library, emphasizing the universality in treatment of different cases coming as a consequence of unifying features within the Bethe ansatz.

Octonic representation of electromagnetic field equations

Abstract: In this paper we represent eight-component values “octons,” generating associative noncommutative algebra. It is shown that the electromagnetic field in a vacuum can be described by a generalized octonic equation, which leads both to the wave equations for potentials and fields and to the system of Maxwell’s equations. The octonic algebra allows one to perform compact combined calculations simultaneously with scalars, vectors, pseudoscalars, and pseudovectors. Examples of such calculations are demonstrated by deriving the relations for energy, momentum, and Lorentz invariants of the electromagnetic field.

Analysis of generalized Forchheimer flows of compressible fluids in porous media

Abstract: This work is focused on the analysis of nonlinear flows of slightly compressible fluids in porous media not adequately described by Darcy’s law We study a class of generalized nonlinear momentum equations which covers all three well-known Forchheimer equations, the so-called two-term, power, and three-term laws The generalized Forchheimer equation is inverted to a nonlinear Darcy equation with implicit permeability tensor depending on the pressure gradient This results in a degenerate parabolic equation for the pressure Two classes of boundary conditions are considered, given pressure and given total flux In both cases they are allowed to be unbounded in time The uniqueness, Lyapunov and asymptotic stabilities, and other long-time dynamical features of the corresponding initial boundary value problems are analyzed The results obtained in this paper have clear hydrodynamic interpretations and can be used for quantitative evaluation of engineering parameters Some numerical simulations are also included

Approximate ternary Jordan derivations on Banach ternary algebras

Abstract: Let A be a Banach ternary algebra over a scalar field R or C and X be a ternary Banach A-module. A linear mapping D:(A,[ ]A)→(X,[ ]X) is called a ternary Jordan derivation if D([xxx]A)=[D(x)xx]X+[xD(x)x]X+[xxD(x)]X for all x∊A. In this paper, we investigate ternary Jordan derivations on Banach ternary algebras, associated with the following functional equation f((x+y+z)/4)+f((3x−y−4z)/4)+f((4x+3z)/4)=2f(x). Moreover, we prove the generalized Ulam–Hyers stability of ternary Jordan derivations on Banach ternary algebras.

Periods and Feynman integrals

Abstract: We consider multiloop integrals in dimensional regularization and the corresponding Laurent series. We study the integral in the Euclidean region and where all ratios of invariants and masses have rational values. We prove that in this case all coefficients of the Laurent series are periods.

Spectral functionals, nonholonomic Dirac operators, and noncommutative Ricci flows

Abstract: We formulate a noncommutative generalization of the Ricci flow theory in the framework of spectral action approach to noncommutative geometry. Grisha Perelman’s functionals are generated as commutative versions of certain spectral functionals defined by nonholonomic Dirac operators and corresponding spectral triples. We derive the formulas for spectral averaged energy and entropy functionals and state the conditions when such values describe (non)holonomic Riemannian configurations.

Abstract cluster expansion with applications to statistical mechanical systems

Abstract: We formulate a general setting for the cluster expansion method and we discuss sufficient criteria for its convergence. We apply the results to systems of classical and quantum particles with stable interactions.