Showing papers in "Journal of Mathematical Physics in 2008"
TL;DR: In this article, the surface charges and their algebra in interacting Lagrangian gauge field theories are constructed out of the underlying linearized theory using techniques from the variational calculus, and they are interpreted as a Pfaff system.
Abstract: Surface charges and their algebra in interacting Lagrangian gauge field theories are constructed out of the underlying linearized theory using techniques from the variational calculus. In the case of exact solutions and symmetries, the surface charges are interpreted as a Pfaff system. Integrability is governed by Frobenius’ theorem and the charges associated with the derived symmetry algebra are shown to vanish. In the asymptotic context, we provide a generalized covariant derivation of the result that the representation of the asymptotic symmetry algebra through charges may be centrally extended. Comparison with Hamiltonian and covariant phase space methods is made. All approaches are shown to agree for exact solutions and symmetries while there are differences in the asymptotic context.
350 citations
TL;DR: In this paper, the convergence conditions of quantum annealing to the target optimal state after an infinite-time evolution following the Schrodinger or stochastic (Monte Carlo) dynamics are presented.
Abstract: Quantum annealing is a generic name of quantum algorithms that use quantum-mechanical fluctuations to search for the solution of an optimization problem. It shares the basic idea with quantum adiabatic evolution studied actively in quantum computation. The present paper reviews the mathematical and theoretical foundations of quantum annealing. In particular, theorems are presented for convergence conditions of quantum annealing to the target optimal state after an infinite-time evolution following the Schrodinger or stochastic (Monte Carlo) dynamics. It is proved that the same asymptotic behavior of the control parameter guarantees convergence for both the Schrodinger dynamics and the stochastic dynamics in spite of the essential difference of these two types of dynamics. Also described are the prescriptions to reduce errors in the final approximate solution obtained after a long but finite dynamical evolution of quantum annealing. It is shown there that we can reduce errors significantly by an ingenious choice of annealing schedule (time dependence of the control parameter) without compromising computational complexity qualitatively. A review is given on the derivation of the convergence condition for classical simulated annealing from the view point of quantum adiabaticity using a classical-quantum mapping.
271 citations
TL;DR: In this paper, a multidimensional fractional action-like variational problem of the calculus of variations is studied. But this problem is not directly related to the one we consider in this paper.
Abstract: Fractional actionlike variational problems have recently gained importance in studying dynamics of nonconservative systems. In this note we address multidimensional fractional actionlike problems of the calculus of variations.
166 citations
TL;DR: A comprehensive review of what is currently known about the structure of degradable quantum channels is given, including a number of new results as well as alternate proofs of some known results.
Abstract: Degradable quantum channels are among the only channels whose quantum and private classical capacities are known. As such, determining the structure of these channels is a pressing open question in quantum information theory. We give a comprehensive review of what is currently known about the structure of degradable quantum channels, including a number of new results as well as alternate proofs of some known results. In the case of qubits, we provide a complete characterization of all degradable channels with two dimensional output, give a new proof that a qubit channel with two Kraus operators is either degradable or anti-degradable, and present a complete description of anti-degradable unital qubit channels with a new proof. For higher output dimensions we explore the relationship between the output and environment dimensions (dB and dE, respectively) of degradable channels. For several broad classes of channels we show that they can be modeled with an environment that is “small” in the sense of ΦC. Suc...
153 citations
TL;DR: In this article, the authors argue that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system, and that quantum physics arises when the topos is the category of sets.
Abstract: This paper is the first in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. In this paper, we discuss two different types of language that can be attached to a system S. The first is a propositional language PL(S); the second is a higher-order, typed language L(S). Both languages provide deductive systems with an intuitionistic logic. The reason for introducing PL(S) is that, as shown in Paper II of the series, it is the easiest way of understanding, and expanding on, the earlier work on topos theory and quantum physics. However, the main thrust o...
149 citations
TL;DR: In this paper, a topos representation of the propositional language, PL(S), for quantum theory is presented. But it is based on the notion of the spectral presheaf, which is a quantum analog of a state space.
Abstract: This paper is the second in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. In this paper, we study in depth the topos representation of the propositional language, PL(S), for the case of quantum theory. In doing so, we make a direct link with, and clarify, the earlier work on applying topos theory to quantum physics. The key step is a process we term “daseinisation” by which a projection operator is mapped to a subobject of the spectral presheaf—the topos quantum analog of a classical state space. In the second part of the paper, we change gear ...
125 citations
TL;DR: A supertrace identity on Lie superalgebras is established in this paper, which provides a tool for constructing super-Hierarchical structures of zero curvature equations associated with Lie super algaes.
Abstract: A supertrace identity on Lie superalgebras is established. It provides a tool for constructing super-Hamiltonian structures of zero curvature equations associated with Lie superalgebras. Applications in the case of the Lie superalgebra B(0,1) present super-Hamiltonian structures of a super-AKNS soliton hierarchy and a super-Dirac soliton hierarchy.
119 citations
TL;DR: In this paper, it is shown that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system, and that the relation between the topos representation for a composite system and the representations for its constituents is discussed.
Abstract: This paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics. The motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. The previous papers in this series are concerned with implementing this program for a single system. In the present paper, we turn to considering a collection of systems; in particular, we are interested in the relation between the topos representation for a composite system and the representations for its constituents. We also study this problem for the disjoint sum of two systems. Our approach to these matters is to construct a category of systems and to find a topos representation of the entire category.
115 citations
TL;DR: In this paper, a new integrable sixth-order nonlinear wave equation is discovered by means of the Painleve analysis, which is equivalent to the Korteweg-de Vries equation with a source.
Abstract: A new integrable sixth-order nonlinear wave equation is discovered by means of the Painleve analysis, which is equivalent to the Korteweg–de Vries equation with a source. A Lax representation and an auto-Backlund transformation are found for the new equation, and its traveling wave solutions and generalized symmetries are studied.
114 citations
TL;DR: In this paper, the phase-space formulation of non-commutative quantum mechanics in arbitrary dimension is studied. But the authors focus on the phase space formulation of quantum noncommutativity.
Abstract: We address the phase-space formulation of a noncommutative extension of quantum mechanics in arbitrary dimension, displaying both spatial and momentum noncommutativities. By resorting to a covariant generalization of the Weyl–Wigner transform and to the Darboux map, we construct an isomorphism between the operator and the phase-space representations of the extended Heisenberg algebra. This map provides a systematic approach to derive the entire structure of noncommutative quantum mechanics in phase space. We construct the extended star product and Moyal bracket and propose a general definition of noncommutative states. We study the dynamical and eigenvalue equations of the theory and prove that the entire formalism is independent of the particular choice of the Darboux map. Our approach unifies and generalizes all the previous proposals for the phase-space formulation of noncommutative quantum mechanics. For concreteness we rederive these proposals by restricting our formalism to some two-dimensional spaces.
113 citations
TL;DR: In this article, a transition operation matrix (TOM) is defined as a matrix whose entries are completely positive maps whose column sums form a quantum operation, and the discrete dynamics of the system is obtained by iterating the TOM E. In this case, there are two types of dynamics, a state dynamics and an operator dynamics.
Abstract: A new approach to quantum Markov chains is presented. We first define a transition operation matrix (TOM) as a matrix whose entries are completely positive maps whose column sums form a quantum operation. A quantum Markov chain is defined to be a pair (G,E) where G is a directed graph and E=[Eij] is a TOM whose entry Eij labels the edge from vertex j to vertex i. We think of the vertices of G as sites that a quantum system can occupy and Eij is the transition operation from site j to site i in one time step. The discrete dynamics of the system is obtained by iterating the TOM E. We next consider a special type of TOM called a transition effect matrix. In this case, there are two types of dynamics, a state dynamics and an operator dynamics. Although these two types are not identical, they are statistically equivalent. We next give examples that illustrate various properties of quantum Markov chains. We conclude by showing that our formalism generalizes the usual framework for quantum random walks.
TL;DR: This paper developed numerical methods for approximating Ricci flat metrics on Calabi-Yau hypersurfaces in projective spaces based on finding balanced metrics and builds on recent theoretical work by Donaldson.
Abstract: We develop numerical methods for approximating Ricci flat metrics on Calabi–Yau hypersurfaces in projective spaces. Our approach is based on finding balanced metrics and builds on recent theoretical work by Donaldson. We illustrate our methods in detail for a one parameter family of quintics. We also suggest several ways to extend our results.
TL;DR: An asymptotic bound for the error of state estimation is derived when the quantum correlation is allowed to use in the measuring apparatus and it is proven that this bound can be achieved in any statistical model in the qubit system.
Abstract: We derive an asymptotic bound for the error of state estimation when we are allowed to use the quantum correlation in the measuring apparatus. It is also proven that this bound can be achieved in any statistical model in the qubit system. Moreover, we show that this bound cannot be attained by any quantum measurement with no quantum correlation in the measuring apparatus except for several specific statistical models. That is, in such a statistical model, the quantum correlation can improve the accuracy of the estimation in an asymptotic setting.
TL;DR: In this paper, a method for constructing periodic solutions of the short pulse (SP) model equation that describes the propagation of ultrashort pulses in nonlinear media is presented. But the method is based on a hodograph transformation to convert the SP equation into the sine-Gordon (sG) equation.
Abstract: We develop a systematic procedure for constructing periodic solutions of the short pulse (SP) model equation that describes the propagation of ultrashort pulses in nonlinear media. We first summarize a novel exact method of solution that consists of a hodograph transformation to convert the SP equation into the sine-Gordon (sG) equation. We then exemplify some one- and two-phase periodic solutions of the sG equation for which the system of linear partial differential equations governing the inverse mapping can be integrated analytically to obtain periodic solutions of the SP equation in the form of the parametric representation. We obtain a new class of two-phase periodic solutions and investigate their properties in detail. Of particular interest is a nonsingular periodic solution that reduces to the breather solution in the long-wave limit. It may play an important role in studying the propagation of ultrashort pulses in a finite-length optical system.
TL;DR: Hayden et al. as mentioned in this paper considered the question of how large a subspace of a given bipartite quantum system can be when the subspace contains only highly entangled states.
Abstract: We consider the question of how large a subspace of a given bipartite quantum system can be when the subspace contains only highly entangled states This is motivated in part by results of Hayden et al [e-print arXiv:quant-ph∕0407049; Commun Math Phys, 265, 95 (2006)], which show that in large d×d-dimensional systems there exist random subspaces of dimension almost d2, all of whose states have entropy of entanglement at least logd−O(1) It is also a generalization of results on the dimension of completely entangled subspaces, which have connections with the construction of unextendible product bases Here we take as entanglement measure the Schmidt rank, and determine, for every pair of local dimensions dA and dB, and every r, the largest dimension of a subspace consisting only of entangled states of Schmidt rank r or larger This exact answer is a significant improvement on the best bounds that can be obtained using the random subspace techniques in Hayden et al We also determine the converse: the l
TL;DR: In this paper, a superintegrable Hamiltonian in three degrees of freedom, obtained as a reduction of pure Keplerian motion in six dimensions, was identified and a formulation of the system in action-angle variables was presented.
Abstract: We identify a new superintegrable Hamiltonian in three degrees of freedom, obtained as a reduction of pure Keplerian motion in six dimensions. The new Hamiltonian is a generalization of the Keplerian one, and has the familiar 1∕r potential with three barrier terms preventing the particle crossing the principal planes. In three degrees of freedom, there are five functionally independent integrals of motion, and all bound, classical trajectories are closed and strictly periodic. The generalization of the Laplace–Runge–Lenz vector is identified and shown to provide functionally independent isolating integrals. They are quartic in the momenta and do not arise from separability of the Hamilton–Jacobi equation. A formulation of the system in action-angle variables is presented.
TL;DR: In this article, a unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of Hermitian matrices of finite or infinite dimensions.
Abstract: A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of Hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schrodinger equations. The Hermitian matrices (factorizable Hamiltonians) are real symmetric tridiagonal (Jacobi) matrices corresponding to second order difference equations. By solving the eigenvalue problem in two different ways, the duality relation of the eigenpolynomials and their dual polynomials is explicitly established. Through the techniques of exact Heisenberg operator solution and shape invariance, various quantities, the two types of eigenvalues (the eigenvalues and the sinusoidal coordinates), the coefficients of the three term recurrence, the normalization measures and the normalisation constants, etc., are determined explicitly.
TL;DR: In this paper, a general limit curve theorem is formulated, which includes the case of converging curves with endpoints and the case in which the limit points assigned since the beginning are one, two, or at most denumerable.
Abstract: The subject of limit curve theorems in Lorentzian geometry is reviewed. A general limit curve theorem is formulated, which includes the case of converging curves with endpoints and the case in which the limit points assigned since the beginning are one, two, or at most denumerable. Some applications are considered. It is proved that in chronological spacetimes, strong causality is either everywhere verified or everywhere violated on maximizing lightlike segments with open domain. As a consequence, if in a chronological spacetime two distinct lightlike lines intersect each other then strong causality holds at their points. Finally, it is proved that two distinct components of the chronology violating set have disjoint closures or there is a lightlike line passing through each point of the intersection of the corresponding boundaries.
TL;DR: In this article, a geometric discretization of elasticity when the ambient space is Euclidean is presented, which is built on ideas from algebraic topology, exterior calculus, and the recent developments of discrete exterior calculus.
Abstract: This paper presents a geometric discretization of elasticity when the ambient space is Euclidean. This theory is built on ideas from algebraic topology, exterior calculus, and the recent developments of discrete exterior calculus. We first review some geometric ideas in continuum mechanics and show how constitutive equations of linearized elasticity, similar to those of electromagnetism, can be written in terms of a material Hodge star operator. In the discrete theory presented in this paper, instead of referring to continuum quantities, we postulate the existence of some discrete scalar-valued and vector-valued primal and dual differential forms on a discretized solid, which is assumed to be a triangulated domain. We find the discrete governing equations by requiring energy balance invariance under time-dependent rigid translations and rotations of the ambient space. There are several subtle differences between the discrete and continuous theories. For example, power of tractions in the discrete theory is written on a layer of cells with a nonzero volume. We obtain the compatibility equations of this discrete theory using tools from algebraic topology. We study a discrete Cosserat medium and obtain its governing equations. Finally, we study the geometric structure of linearized elasticity and write its governing equations in a matrix form. We show that, in addition to constitutive equations, balance of angular momentum is also metric dependent; all the other governing equations are topological.
TL;DR: In this article, the existence and uniqueness theorems for functional right-left delay and left-right advanced fractional functional differential equations with bounded delay and advance, respectively, are proved.
Abstract: The existence and uniqueness theorems for functional right-left delay and left-right advanced fractional functional differential equations with bounded delay and advance, respectively, are proved. The continuity with respect to the initial function for these equations is also proved under some Lipschitz kind conditions. The Q-operator is used to transform the delay-type equation to an advanced one and vice versa. An example is given to clarify the results.
TL;DR: In this article, it was shown that the closure of the Laplace operator Δ is self-adjoint in the Hilbert space l 2 (G0) and that Δ has a unique spectral resolution, determined by a projection valued measure on the Borel subsets of the infinite half-line.
Abstract: We study the operator theory associated with such infinite graphs G as occur in electrical networks, in fractals, in statistical mechanics, and even in internet search engines. Our emphasis is on the determination of spectral data for a natural Laplace operator associated with the graph in question. This operator Δ will depend not only on G but also on a prescribed positive real valued function c defined on the edges in G. In electrical network models, this function c will determine a conductance number for each edge. We show that the corresponding Laplace operator Δ is automatically essential self-adjoint. By this we mean that Δ is defined on the dense subspace D (of all the real valued functions on the set of vertices G0 with finite support) in the Hilbert space l2(G0). The conclusion is that the closure of the operator Δ is self-adjoint in l2(G0), and so, in particular, that it has a unique spectral resolution, determined by a projection valued measure on the Borel subsets of the infinite half-line. We...
TL;DR: In this paper, the derivatives of the confluent hypergeometric (Kummer) function F = F11(a,b,z) with respect to the parameter a or b are investigated and expressed in terms of generalizations of multivariable Kampe de Feriet functions.
Abstract: The derivatives to any order of the confluent hypergeometric (Kummer) function F=F11(a,b,z) with respect to the parameter a or b are investigated and expressed in terms of generalizations of multivariable Kampe de Feriet functions. Various properties (reduction formulas, recurrence relations, particular cases, and series and integral representations) of the defined hypergeometric functions are given. Finally, an application to the two-body Coulomb problem is presented: the derivatives of F with respect to a are used to write the scattering wave function as a power series of the Sommerfeld parameter.
TL;DR: In this article, the authors proposed a quantization of the massless vector field in the de Sitter (dS) space, where the field operator is defined with the help of coordinate-independent deSitter waves (the modes).
Abstract: We proceed to the quantization of the massless vector field in the de Sitter (dS) space. This work is the natural continuation of a previous article devoted to the quantization of the dS massive vector field [J. P. Gazeau and M. V. Takook, J. Math. Phys. 41, 5920 (2000); T. Garidi et al., ibid. 43, 6379 (2002).] The term “massless” is used by reference to conformal invariance and propagation on the dS lightcone whereas “massive” refers to those dS fields which unambiguously contract to Minkowskian massive fields at zero curvature. Due to the combined occurrences of gauge invariance and indefinite metric, the covariant quantization of the massless vector field requires an indecomposable representation of the de Sitter group. We work with the gauge fixing corresponding to the simplest Gupta–Bleuler structure. The field operator is defined with the help of coordinate-independent de Sitter waves (the modes). The latter are simple to manipulate and most adapted to group theoretical approaches. The physical sta...
TL;DR: In this article, an extended super-Schrodinger subalgebra of the superconformal algebra psu(2,2∣4) is presented, which contains 24 supercharges (i.e., 3/4 of the original supersymmetries) and the generators of so(6), as well as the generators from the original Schrodinger algebra.
Abstract: We discuss (extended) super-Schrodinger algebras obtained as subalgebras of the superconformal algebra psu(2,2∣4). The Schrodinger algebra with two spatial dimensions can be embedded into so(4,2). In the superconformal case the embedded algebra may be enhanced to the so-called super-Schrodinger algebra. In fact, we find an extended super-Schrodinger subalgebra of psu(2,2∣4). It contains 24 supercharges (i.e., 3/4 of the original supersymmetries) and the generators of so(6), as well as the generators of the original Schrodinger algebra. In particular, the 24 supercharges come from 16 rigid supersymmetries and half of 16 superconformal ones. Moreover, this superalgebra contains a smaller super-Schrodinger subalgebra, which is a supersymmetric extension of the original Schrodinger algebra and so(6) by eight supercharges (half of 16 rigid supersymmetries). It is still a subalgebra even if there are no so(6) generators. We also discuss super-Schrodinger subalgebras of the superconformal algebras, osp(8∣4) and ...
TL;DR: In this paper, it was shown that the formalism for Jacobi's last multiplier for a one-degree of freedom system extends naturally to systems of more than one degree of freedom thereby extending results of Whittaker dating from more than a century ago and Rao [Proceedings of the Benares Mathematical Society 2, 53 (1940)] dating from almost 70 years ago.
Abstract: We demonstrate that the formalism for the calculation of Jacobi’s last multiplier for a one degree of freedom system extends naturally to systems of more than one degree of freedom thereby extending results of Whittaker dating from more than a century ago and Rao [Proceedings of the Benares Mathematical Society 2, 53 (1940)] dating from almost 70 years ago. We illustrate the theory with an application taken from the theory of coupled oscillators. We indicate how many Lagrangians can be obtained for such a system.
TL;DR: In this article, the authors studied the potential of the caged anisotropic harmonic oscillator with rational frequency ratio (l:m:n) but additionally with barrier terms describing repulsive forces from the principal planes.
Abstract: We study the caged anisotropic harmonic oscillator, which is a new example of a superintegrable or accidentally degenerate Hamiltonian. The potential is that of the harmonic oscillator with rational frequency ratio (l:m:n) but additionally with barrier terms describing repulsive forces from the principal planes. This confines the classical motion to a sector bounded by the principal planes, or a cage. In three degrees of freedom, there are five isolating integrals of motion, ensuring that all bound trajectories are closed and strictly periodic. Three of the integrals are quadratic in the momenta, the remaining two are polynomials of order 2(l+m−1) and 2(l+n−1). In the quantum problem, the eigenstates are multiply degenerate, exhibiting l2m2n2 copies of the fundamental pattern of the symmetry group SU(3).
TL;DR: In this paper, various error exponents in a binary hypothesis testing problem were studied and extended to a setting where both the null hypothesis and the alternative hypothesis can be correlated states on a spin chain.
Abstract: We study various error exponents in a binary hypothesis testing problem and extend recent results on the quantum Chernoff and Hoeffding bounds for product states to a setting when both the null hypothesis and the alternative hypothesis can be correlated states on a spin chain. Our results apply to states satisfying a certain factorization property; typical examples are the global Gibbs states of translation-invariant finite-range interactions as well as certain finitely correlated states.
TL;DR: In this article, the one-dimensional eigenfunctions and the eigenvalues of massive spin-0 and spin-1 particles have been found by using the Duffin-Kemmer-Petiau equation.
Abstract: The one-dimensional eigenfunctions and the eigenvalues of massive spin-0 and spin-1 particles have been found by using the Duffin–Kemmer–Petiau equation. Following Greiner [Quantum Mechanics: An introduction, 4th ed. (Springer-verlag, Berlin, 2001)], we have shown that the eigensolutions in both cases are decoupled in two sets.
TL;DR: In this article, the authors model the behavior of a relativistic spherically symmetric shearing fluid undergoing gravitational collapse with heat flux and show that the governing equation for the gravitational behavior is a Riccati equation.
Abstract: We model the behavior of a relativistic spherically symmetric shearing fluid undergoing gravitational collapse with heat flux. It is demonstrated that the governing equation for the gravitational behavior is a Riccati equation. We show that the Riccati equation admits two classes of new solutions in closed form. We regain particular models, obtained in previous investigations, as special cases. A significant feature of our solutions is the general spatial dependence in the metric functions which allows for a wider study of the physical features of the model, such as the behavior of the causal temperature in inhomogeneous space-times.
TL;DR: In this paper, the non-holonomic Ricci flows of Riemannian metrics were modeled by imposing non-integrable (nonholonomic) constraints on the Ricci flow evolution, and the nonholonomic evolution equations were derived from Perelman's functionals.
Abstract: This is the second paper in a series of works devoted to nonholonomic Ricci flows. By imposing nonintegrable (nonholonomic) constraints on the Ricci flows of Riemannian metrics, we can model mutual transforms of generalized Finsler–Lagrange and Riemann geometries. We verify some assertions made in the first partner paper and develop a formal scheme in which the geometric constructions with Ricci flow evolution are elaborated for canonical nonlinear and linear connection structures. This scheme is applied to a study of Hamilton’s Ricci flows on nonholonomic manifolds and related Einstein spaces and Ricci solitons. The nonholonomic evolution equations are derived from Perelman’s functionals which are redefined in such a form that can be adapted to the nonlinear connection structure. Next, the statistical analogy for nonholonomic Ricci flows is formulated and the corresponding thermodynamical expressions are found for compact configurations. Finally, we analyze two physical applications, the nonholonomic Ricci flows associated with evolution models for solitonic pp-wave solutions of Einstein equations, and compute the Perelman’s entropy for regular Lagrange and analogous gravitational systems.