Showing papers in "Journal of Mathematical Physics in 2005"
TL;DR: In this paper, the structure of the extended Clifford group is described and the action of the Clifford group operators on symmetric informationally complete-positive operator valued measures (or SIC-POVMs) covariant relative to the actions of the generalized Pauli group is investigated.
Abstract: We describe the structure of the extended Clifford group [defined to be the group consisting of all operators, unitary and antiunitary, which normalize the generalized Pauli group (or Weyl–Heisenberg group as it is often called)]. We also obtain a number of results concerning the structure of the Clifford group proper (i.e., the group consisting just of the unitary operators which normalize the generalized Pauli group). We then investigate the action of the extended Clifford group operators on symmetric informationally complete–positive operator valued measures (or SIC–POVMs) covariant relative to the action of the generalized Pauli group. We show that each of the fiducial vectors which has been constructed so far (including all the vectors constructed numerically by Renes et al.) is an eigenvector of one of a special class of order 3 Clifford unitaries. This suggests a strengthening of a conjecture of Zauner’s. We give a complete characterization of the orbits and stability groups in dimensions 2–7. Fina...
315 citations
TL;DR: In this article, two types of results are presented for distinguishing pure bipartite quantum states using local operations and classical communications, and upper bounds for the probability of error using LOCC taken over all sets of k orthogonal states in Cn⊗Cm.
Abstract: Two types of results are presented for distinguishing pure bipartite quantum states using local operations and classical communications. We examine sets of states that can be perfectly distinguished, in particular showing that any three orthogonal maximally entangled states in C3⊗C3 form such a set. In cases where orthogonal states cannot be distinguished, we obtain upper bounds for the probability of error using LOCC taken over all sets of k orthogonal states in Cn⊗Cm. In the process of proving these bounds, we identify some sets of orthogonal states for which perfect distinguishability is not possible.
161 citations
TL;DR: In this paper, it was shown that the spaces of truly second-, third-, fourth-, fifth-, sixth-order constants of the motion are of dimension 6, 4, 21, and 56, respectively.
Abstract: This paper is part of a series that lays the groundwork for a structure and classification theory of second-order superintegrable systems, both classical and quantum, in real or complex conformally flat spaces. Here we consider classical superintegrable systems with nondegenerate potentials in three dimensions. We show that there exists a standard structure for such systems, based on the algebra of 3×3 symmetric matrices, and that the quadratic algebra always closes at order 6. We show that the spaces of truly second-, third-, fourth-, and sixth-order constants of the motion are of dimension 6, 4, 21, and 56, respectively, and we construct explicit bases for the fourth- and sixth order constants in terms of products of the second order constants.
156 citations
TL;DR: In this article, the authors prove that any classical solution of the Camassa-Holm equation has compact support if its initial data has this property, and they prove that this property holds for all classical solutions of the equation.
Abstract: We prove that any classical solution of the Camassa–Holm equation will have compact support if its initial data has this property.
154 citations
TL;DR: In this article, a hierarchy of equations containing the short pulse equation, which describes the evolution of very short pulses in nonlinear media, and the elastic beam equation which describes nonlinear transverse oscillations of elastic beams under tension was studied.
Abstract: We study a new hierarchy of equations containing the short pulse equation, which describes the evolution of very short pulses in nonlinear media, and the elastic beam equation, which describes nonlinear transverse oscillations of elastic beams under tension. We show that the hierarchy of equations is integrable. We obtain the two compatible Hamiltonian structures. We construct an infinite series of both local and nonlocal conserved charges. A Lax description is presented for both systems. For the elastic beam equations we also obtain a nonstandard Lax representation.
146 citations
TL;DR: In this paper, the Stackel transform was used to derive a structure and classification theory of second order superintegrable systems in conformally flat spaces, and the underlying spaces were exactly those derived by Koenigs in his remarkable paper giving all 2D manifolds that admit at least three second order symmetries.
Abstract: This paper is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems, both classical and quantum, in conformally flat spaces. Here we study the Stackel transform (or coupling constant metamorphosis) as an invertible mapping between classical superintegrable systems on different spaces. Through the use of this tool we derive and classify for the first time all two-dimensional (2D) superintegrable systems. The underlying spaces are exactly those derived by Koenigs in his remarkable paper giving all 2D manifolds (with zero potential) that admit at least three second order symmetries. Our derivation is very simple and quite distinct. We also show that every superintegrable system is the Stackel transform of a superintegrable system on a constant curvature space.
138 citations
TL;DR: In this article, the authors generalize the de Finetti representation to the case of finite symmetric quantum states and show that, conditioned on the outcomes of an informationally complete measurement applied to a number of subsystems, the remaining subsystems are close to having product form.
Abstract: Consider a symmetric quantum state on an n-fold product space, that is, the state is invariant under permutations of the n subsystems. We show that, conditioned on the outcomes of an informationally complete measurement applied to a number of subsystems, the state in the remaining subsystems is close to having product form. This immediately generalizes the so-called de Finetti representation to the case of finite symmetric quantum states.
136 citations
TL;DR: In this article, a general geometric definition of asymptotic flatness at null infinity in d-dimensional general relativity (d even) within the framework of conformal infinity is given.
Abstract: We give a general geometric definition of asymptotic flatness at null infinity in d-dimensional general relativity (d even) within the framework of conformal infinity. Our definition is arrived at via an analysis of linear perturbations near null infinity and shown to be stable under such perturbations. The detailed falloff properties of the perturbations, as well as the gauge conditions that need to be imposed to make the perturbations regular at infinity, are qualitatively different in higher dimensions; in particular, the decay rate of a radiating solution at null infinity differs from that of a static solution in higher dimensions. The definition of asymptotic flatness in higher dimensions consequently also differs qualitatively from that in d=4. We then derive an expression for the generator conjugate to an asymptotic time translation symmetry for asymptotically flat space–times in d-dimensional general relativity (d even) within the Hamiltonian framework, making use especially of a formalism develop...
133 citations
TL;DR: In this paper, the authors evaluate the idea of Π-stability at the Landau-Ginzburg (LG) point in moduli space of compact Calabi-Yau manifolds, using matrix factorizations to B-model the topological D-brane category.
Abstract: We evaluate the ideas of Π-stability at the Landau-Ginzburg (LG) point in moduli space of compact Calabi-Yau manifolds, using matrix factorizations to B-model the topological D-brane category. The standard requirement of unitarity at the IR fixed point is argued to lead to a notion of “R-stability” for matrix factorizations of quasihomogeneous LG potentials. The D0-brane on the quintic at the Landau-Ginzburg point is not obviously unstable. Aiming to relate R-stability to a moduli space problem, we then study the action of the gauge group of similarity transformations on matrix factorizations. We define a naive moment maplike flow on the gauge orbits and use it to study boundary flows in several examples. Gauge transformations of nonzero degree play an interesting role for brane-antibrane annihilation. We also give a careful exposition of the grading of the Landau-Ginzburg category of B-branes, and prove an index theorem for matrix factorizations.
129 citations
TL;DR: In this article, the authors present new classes of exact solutions with non-commutative symmetries constructed in vacuum Einstein gravity (in general, with nonzero cosmological constant), five-dimensional (5D) gravity and (anti) de Sitter gauge gravity.
Abstract: We present new classes of exact solutions with noncommutative symmetries constructed in vacuum Einstein gravity (in general, with nonzero cosmological constant), five-dimensional (5D) gravity and (anti) de Sitter gauge gravity. Such solutions are generated by anholonomic frame transforms and parametrized by generic off-diagonal metrics. For certain particular cases, the new classes of metrics have explicit limits with Killing symmetries but, in general, they may be characterized by certain anholonomic noncommutative matrix geometries. We argue that different classes of noncommutative symmetries can be induced by exact solutions of the field equations in commutative gravity modeled by a corresponding moving real and complex frame geometry. We analyze two classes of black ellipsoid solutions (in the vacuum case and with cosmological constant) in four-dimensional gravity and construct the analytic extensions of metrics for certain classes of associated frames with complex valued coefficients. The third class of solutions describes 5D wormholes which can be extended to complex metrics in complex gravity models defined by noncommutative geometric structures. The anholonomic noncommutative symmetries of such objects are analyzed. We also present a descriptive account how the Einstein gravity can be related to gauge models of gravity and their noncommutative extensions and discuss such constructions in relation to the Seiberg–Witten map for the gauge gravity. Finally, we consider a formalism of vielbeins deformations subjected to noncommutative symmetries in order to generate solutions for noncommutative gravity models with Moyal (star) product.
128 citations
TL;DR: In this paper, the existence of a Lagrangian description for the second-order Riccati equation is analyzed and the results are applied to the study of two different nonlinear systems both related with the generalized Riemannian equation.
Abstract: The existence of a Lagrangian description for the second-order Riccati equation is analyzed and the results are applied to the study of two different nonlinear systems both related with the generalized Riccati equation. The Lagrangians are non-natural and the forces are not derivable from a potential. The constant value E of a preserved energy function can be used as an appropriate parameter for characterizing the behavior of the solutions of these two systems. In the second part the existence of two-dimensional versions endowed with superintegrability is proved. The explicit expressions of the additional integrals are obtained in both cases. Finally it is proved that the orbits of the second system, that represents a nonlinear oscillator, can be considered as nonlinear Lissajous figures
TL;DR: In this article, the main properties of the Kepler problem on spaces with curvature were studied, and the explicit expressions of the orbits were obtained by using two different methods, first by direct integration and second by obtaining the κ-dependent version of the Binet's equation.
Abstract: The Kepler problem is a dynamical system that is well defined not only on the Euclidean plane but also on the sphere and on the hyperbolic plane. First, the theory of central potentials on spaces of constant curvature is studied. All the mathematical expressions are presented using the curvature κ as a parameter, in such a way that they reduce to the appropriate property for the system on the sphere S2, or on the hyperbolic plane H2, when particularized for κ>0, or κ<0, respectively; in addition, the Euclidean case arises as the particular case κ=0. In the second part we study the main properties of the Kepler problem on spaces with curvature, we solve the equations and we obtain the explicit expressions of the orbits by using two different methods, first by direct integration and second by obtaining the κ-dependent version of the Binet’s equation. The final part of the paper, that has a more geometric character, is devoted to the study of the theory of conics on spaces of constant curvature.
TL;DR: In this paper, the complete tables of Clebsch-Gordan coefficients for a wide class of SO(10) SUSY grand unified theories (GUTs) are given.
Abstract: The complete tables of Clebsch–Gordan (CG) coefficients for a wide class of SO(10) SUSY grand unified theories (GUTs) are given. Explicit expressions of states of all corresponding multiplets under standard model gauge group G321=SU(3)C×SU(2)L×U(1)Y, necessary for evaluation of the CG coefficients are presented. The SUSY SO(10) GUT model considered here includes most of the Higgs irreducible representations usually used in the literature, 10, 45, 54, 120, 126, 126¯, and 210. Mass matrices of all G321 multiplets are found for the most general superpotential. These results are indispensable for the precision calculations of the gauge couplings unification and proton decay, etc.
TL;DR: By enlarging the associated matrix spectral problems, two specific classes of multicomponent integrable couplings of the physically important vector AKNS soliton equations are constructed, which can be linked to each other by a Backlund transformation.
Abstract: By enlarging the associated matrix spectral problems, two specific classes of multicomponent integrable couplings of the physically important vector AKNS soliton equations are constructed, which can be linked to each other by a Backlund transformation. The resulting two hierarchies of integrable couplings possess the enlarged zero curvature representations and recursion structures, and thus each system in the two hierarchies has infinitely many commuting symmetries.
TL;DR: In this paper, the phase transition structure of the Curie-Weiss-Potts spin model is analyzed using the theory of large deviations, which is equivalent to the Potts model on the complete graph on n vertices.
Abstract: Using the theory of large deviations, we analyze the phase transition structure of the Curie–Weiss–Potts spin model, which is a mean-field approximation to the nearest-neighbor Potts model. It is equivalent to the Potts model on the complete graph on n vertices. The analysis is carried out both for the canonical ensemble and the microcanonical ensemble. Besides giving explicit formulas for the microcanonical entropy and for the equilibrium macrostates with respect to the two ensembles, we analyze ensemble equivalence and nonequivalence at the level of equilibrium macrostates, relating these to concavity and support properties of the microcanonical entropy. The Curie–Weiss–Potts model is the first statistical mechanical model for which such a detailed and rigorous analysis has been carried out.
TL;DR: In this article, it is shown that if the original inequality defines a facet of the polytope of local joint outcome probabilities then the lifted one also defines a more complex polytoope.
Abstract: A Bell inequality defined for a specific experimental configuration can always be extended to a situation involving more observers, measurement settings, or measurement outcomes. In this article, such “liftings” of Bell inequalities are studied. It is shown that if the original inequality defines a facet of the polytope of local joint outcome probabilities then the lifted one also defines a facet of the more complex polytope.
TL;DR: In this paper, the asymptotic expansion for the trace of a spatially regularized heat operator LΘ(f)e−tΔΘ, where ΔΘ is a generalized Laplacian defined with Moyal products, is obtained.
Abstract: Extending a result of Vassilevich, we obtain the asymptotic expansion for the trace of a spatially regularized heat operator LΘ(f)e−tΔΘ, where ΔΘ is a generalized Laplacian defined with Moyal products and LΘ(f) is Moyal left multiplication. The Moyal planes corresponding to any skewsymmetric matrix Θ being spectral triples, the spectral action introduced in noncommutative geometry by Chamseddine and Connes is computed. This result generalizes the Connes–Lott action previously computed by Gayral for symplectic Θ.
TL;DR: In this article, the cleannes classification problem for number n of outcomes n⩽d (d dimension of the Hilbert space) is solved, and a set of either necessary or sufficient conditions for n>d, along with an iff condition for the case of informationally complete POVMs for n=d2.
Abstract: In quantum mechanics the statistics of the outcomes of a measuring apparatus is described by a positive operator valued measure (POVM). A quantum channel transforms POVMs into POVMs, generally irreversibly, thus losing some of the information retrieved from the measurement. This poses the problem of which POVMs are “undisturbed,” i.e., they are not irreversibly connected to another POVM. We will call such POVMs clean. In a sense, the clean POVMs would be “perfect,” since they would not have any additional “extrinsical” noise. Quite unexpectedly, it turns out that such a “cleanness” property is largely unrelated to the convex structure of POVMs, and there are clean POVMs that are not extremal and vice versa. In this article we solve the cleannes classification problem for number n of outcomes n⩽d (d dimension of the Hilbert space), and we provide a set of either necessary or sufficient conditions for n>d, along with an iff condition for the case of informationally complete POVMs for n=d2.
TL;DR: In this paper, it was shown that the solutions of the Lovelock equations with spherical, planar, or hyperbolic symmetry are locally isometric to the corresponding static Lovelocks black hole and that these solutions admit an additional Killing vector that can either be space-like or time-like.
Abstract: We show that the solutions of the Lovelock equations with spherical, planar, or hyperbolic symmetry are locally isometric to the corresponding static Lovelock black hole As a consequence, these solutions are locally static: they admit an additional Killing vector that can either be space-like or time-like, depending on the position This result also holds in the presence of an abelian gauge field, in which case the solutions are locally isometric to a charged static black hole
TL;DR: In this article, the authors investigated the problem of introducing consistent self-couplings in free theories for mixed tensor gauge fields whose symmetry properties are characterized by Young diagrams made of two columns of arbitrary (but different) lengths.
Abstract: We investigate the problem of introducing consistent self-couplings in free theories for mixed tensor gauge fields whose symmetry properties are characterized by Young diagrams made of two columns of arbitrary (but different) lengths. We prove that, in flat space, these theories admit no local, Poincare-invariant, smooth, self-interacting deformation with at most two derivatives in the Lagrangian. Relaxing the derivative and Lorentz-invariance assumptions, there still is no deformation that modifies the gauge algebra, and in most cases no deformation that alters the gauge transformations. Our approach is based on a Becchi-Rouet-Stora-Tyutin (BRST) -cohomology deformation procedure.
TL;DR: In this article, the authors review various methods for the numerical approximations of the Ginzburg-Landau models of superconductivity, and give particular attention to the different treatment of gauge invariance in both the finite element, finite difference, and finite volume settings.
Abstract: In this paper, we review various methods for the numerical approximations of the Ginzburg–Landau models of superconductivity. Particular attention is given to the different treatment of gauge invariance in both the finite element, finite difference, and finite volume settings. Representative theoretical results, typical numerical simulations, and computational challenges are presented. Generalizations to other relevant models are also discussed.
TL;DR: In this article, the authors introduced the notion of Gauss-Landau-Hall magnetic field on a Riemannian surface and showed that the corresponding Landau Hall problem is equivalent to the dynamics of a massive boson.
Abstract: We introduce the notion of Gauss-Landau-Hall magnetic field on a Riemannian surface. The corresponding Landau-Hall problem is shown to be equivalent to the dynamics of a massive boson. This allows one to view that problem as a globally stated, variational one. In this framework, flowlines appear as critical points of an action with density depending on the proper acceleration. Moreover, we can study global stability of flowlines. In this equivalence, the massless particle model corresponds with a limit case obtained when the force of the Gauss-Landau-Hall magnetic field increases arbitrarily. We also obtain properties related with the completeness of flowlines for general magnetic fields. The paper also contains results relative to the Landau-Hall problem associated with a uniform magnetic field. For example, we characterize those revolution surfaces whose parallels are all normal flowlines of a uniform magnetic field.
TL;DR: In this paper, the existence of standing wave solutions for a nonlinear Schrodinger equation on R3 under the influence of an external magnetic field B was proved for the case of a constant magnetic field having source in the potential A(x)=(b∕2)(−x2,x1,0) corresponding to the Lorentz gauge.
Abstract: We prove existence of standing wave solutions for a nonlinear Schrodinger equation on R3 under the influence of an external magnetic field B. In particular we deal with the physically meaningful case of a constant magnetic field B=(0,0,b) having source in the potential A(x)=(b∕2)(−x2,x1,0) corresponding to the Lorentz gauge.
TL;DR: In this paper, it was shown that the α2-dynamo of magnetohydrodynamics, the hydrodynamic Squire equation as well as an interpolation model of PT-symmetric quantum mechanics are closely related as spectral problems in Krein spaces.
Abstract: It is shown that the α2-dynamo of magnetohydrodynamics, the hydrodynamic Squire equation as well as an interpolation model of PT-symmetric quantum mechanics are closely related as spectral problems in Krein spaces. For the α2-dynamo and the PT-symmetric model the strong similarities are demonstrated with the help of a 2×2 operator matrix representation, whereas the Squire equation is reinterpreted as a rescaled and Wick-rotated PT-symmetric problem. Based on recent results on the Squire equation the spectrum of the PT-symmetric interpolation model is analyzed in detail and the Herbst limit is described as spectral singularity.
TL;DR: In this paper, a sharp upper bound on the quantum relative entropy in terms of the trace norm distance and the smallest eigenvalues of both states concerned is derived. But the result obtained here is more general than the corresponding one from our previous work.
Abstract: The quantum relative entropy is frequently used as a distance measure between two quantum states, and inequalities relating it to other distance measures are important mathematical tools in many areas of quantum information theory. We have derived many such inequalities in previous work. The present paper is a follow-up on this, and provides a sharp upper bound on the relative entropy in terms of the trace norm distance and of the smallest eigenvalues of both states concerned. The result obtained here is more general than the corresponding one from our previous work. As a corollary, we obtain a sharp upper bound on the regularised relative entropy introduced by Lendi, Farhadmotamed, and van Wonderen.
TL;DR: In this article, the Schrodinger equation was solved for a quark-antiquark system interacting via a Coulomb-plus-linear potential, and the wave functions were obtained as power series, with their coefficients given in terms of the combinatorics functions.
Abstract: We solve the Schrodinger equation for a quark–antiquark system interacting via a Coulomb-plus-linear potential, and obtain the wave functions as power series, with their coefficients given in terms of the combinatorics functions.
TL;DR: In this article, the equivalence of conservation laws with respect to Lie symmetry groups for fixed systems of differential equations was introduced and the notion of local dependence of potentials was defined.
Abstract: We introduce notions of equivalence of conservation laws with respect to Lie symmetry groups for fixed systems of differential equations and with respect to equivalence groups or sets of admissible transformations for classes of such systems. We also revise the notion of linear dependence of conservation laws and define the notion of local dependence of potentials. To construct conservation laws, we develop and apply the most direct method which is effective to use in the case of two independent variables. Admitting possibility of dependence of conserved vectors on a number of potentials, we generalize the iteration procedure proposed by Bluman and Doran-Wu for finding nonlocal (potential) conservation laws. As an example, we completely classify potential conservation laws (including arbitrary order local ones) of diffusion-convection equations with respect to the equivalence group and construct an exhaustive list of locally inequivalent potential systems corresponding to these equations.
TL;DR: In this paper, a hierarchy of equations is derived that links electron pairs, triplets, quadruplets, etc., which can then be used to derive more accurate approximations.
Abstract: Several explicit formulas for the kinetic energy of a many-electron system as a functional of the k-electron density are derived, with emphasis on the electron pair density. The emphasis is on general techniques for deriving approximate kinetic energy functionals and features generalized Weisacker bounds and methods using density-matrix reconstruction. Adapting results from statistical mechanics, a hierarchy of equations is derived that links electron pairs, triplets, quadruplets, etc.; this may be used to derive more accurate approximations. Several methods for defining the exact kinetic energy functional are presented, including the generalizations of the Levy and Lieb formulations of density-functional theory. Together with N-representability constraints on the k-density, this paper provides the basis for “generalized density functional theories” based on the electron pair density. There are also implications for conventional density-functional theory, notably regarding the development of more accurate...
TL;DR: In this article, the integrability of polynomial potentials with two degrees of freedom has been studied and the strongest necessary conditions for their integration have been obtained by a study of the differential Galois group of variational equations along straight line solutions.
Abstract: In this paper we study the integrability of natural Hamiltonian systems with a homogeneous polynomial potential. The strongest necessary conditions for their integrability in the Liouville sense have been obtained by a study of the differential Galois group of variational equations along straight line solutions. These particular solutions can be viewed as points of a projective space of dimension smaller by one than the number of degrees of freedom. We call them Darboux points. We analyze in detail the case of two degrees of freedom. We show that, except for a radial potential, the number of Darboux points is finite and it is not greater than the degree of the potential. Moreover, we analyze cases when the number of Darboux points is smaller than maximal. For two degrees of freedom the above-mentioned necessary condition for integrability can be expressed in terms of one nontrivial eigenvalue of the Hessian of potential calculated at a Darboux point. We prove that for a given potential these nontrivial eigenvalues calculated for all Darboux points cannot be arbitrary because they satisfy a certain relation which we give in an explicit form. We use this fact to strengthen maximally the necessary conditions for integrability and we show that in a generic case, for a given degree of the potential, there is only a finite number of potentials which satisfy these conditions. We also describe the nongeneric cases. As an example we give a full list of potentials of degree four satisfying these conditions. Then, investigating the differential Galois group of higher order variational equations, we prove that, except for one discrete family, among these potentials only those which are already known to be integrable are integrable. We check that a finite number of potentials from the exceptional discrete family are not integrable, and we conjecture that all of them are not integrable.
TL;DR: In this paper, the fundamental solution for time and space-fractional partial differential operator Dtλ+a2(−▵)γ∕2(λ,γ>0) is given in terms of the Fox's H-function.
Abstract: The fundamental solution for time- and space-fractional partial differential operator Dtλ+a2(−▵)γ∕2(λ,γ>0) is given in terms of the Fox’s H-function. Here the time-fractional derivative in the sense of generalized functions (distributions) Dtλ is defined by the convolution Dtλf(t)=Φ−λ(t)*f(t), where Φλ(t)=t+λ−1∕Γ(λ) and f(t)≡0 as t<0, and the fractional n-dimensional Laplace operator (−▵)γ∕2 is defined by its Fourier transform with respect to spatial variable F[(−▵)γ∕2g(x)]=∣ω∣γF[g(x)]. The solutions for initial value problems for time- and space-fractional partial differential equation in the sense of Caputo and Riemann–Liouville time-fractional derivatives, respectively, are obtained by the fundamental solution.