Showing papers in "Journal of Mathematical Physics in 2003"

Integrable hydrodynamic chains

Abstract: A new approach for derivation of Benney-type moment chains and integrable hydrodynamic type systems is presented. New integrable hydrodynamic chains are constructed; all their hydrodynamical reductions are described and integrated. New (2+1) integrable hydrodynamic type systems are found.

Pseudo-Hermiticity and Generalized PT- and CPT-Symmetries

Abstract: We study certain linear and antilinear symmetry generators and involution operators associated with pseudo-Hermitian Hamiltonians and show that the theory of pseudo-Hermitian operators provides a simple explanation for the recent results of Bender, Brody and Jones (quant-ph/0208076) on the CPT-symmetry of a class of PT-symmetric non-Hermitian Hamiltonians. We present a natural extension of these results to the class of diagonalizable pseudo-Hermitian Hamiltonians H with a discrete spectrum. In particular, we introduce generalized parity (P), time-reversal (T), and charge-conjugation (C) operators and establish the PT- and CPT-invariance of H.

The n-Component KP Hierarchy and Representation Theory

Abstract: It is the aim of the present article to give all formulations of the n-component KP hierarchy and clarify connections between them. The generalization to the n-component KP hierarchy is important because it contains many of the most popular systems of soliton equations, like the Davey–Stewartson system (for n=2), the two-dimensional Toda lattice (for n=2), the n-wave system (for n⩾3) and the Darboux–Egoroff system. It also allows us to construct natural generalizations to the Davey–Stewartson and Toda lattice systems. Of course, the inclusion of all these systems in the n-component KP hierarchy allows us to construct their solutions by making use of vertex operators.

Extended multilinear variable separation approach and multivalued localized excitations for some (2+1)-dimensional integrable systems

Abstract: The multilinear variable separation approach and the related “universal” formula have been applied to many (2+1)-dimensional nonlinear systems Starting from the universal formula, abundant (2+1)-dimensional localized excitations have been found In this paper, the universal formula is extended in two different ways One is obtained for the modified Nizhnik–Novikov–Veselov equation such that two universal terms can be combined linearly and this type of extension is also valid for the (2+1)-dimensional symmetric sine-Gordon system The other is for the dispersive long wave equation, the Broer–Kaup–Kupershmidt system, the higher order Broer–Kaup–Kupershmidt system, and the Burgers system where arbitrary number of variable separated functions can be involved Because of the existence of the arbitrary functions in both the original universal formula and its extended forms, the multivalued functions can be used to construct a new type of localized excitations, folded solitary waves (FSWs) and foldons The FSWs

Superintegrable systems in Darboux spaces

Abstract: Almost all research on superintegrable potentials concerns spaces of constant curvature. In this paper we find by exhaustive calculation, all superintegrable potentials in the four Darboux spaces of revolution that have at least two integrals of motion quadratic in the momenta, in addition to the Hamiltonian. These are two-dimensional spaces of nonconstant curvature. It turns out that all of these potentials are equivalent to superintegrable potentials in complex Euclidean 2-space or on the complex 2-sphere, via “coupling constant metamorphosis” (or equivalently, via Stackel multiplier transformations). We present a table of the results.

A new loop algebra and a corresponding integrable hierarchy, as well as its integrable coupling

Abstract: A type of new interesting loop algebra GM (M=1,2,…) with a simple commutation operation just like that in the loop algebra A1 is constructed. With the help of the loop algebra GM, a new multicomponent integrable system, M-AKNS-KN hierarchy, is worked out. As reduction cases, the M-AKNS hierarchy and M-KN hierarchy are engendered, respectively. In addition, the system 1-AKNS-KN, which is a reduced case of the M-AKNS-KN hierarchy above, is a unified expressing integrable model of the AKNS hierarchy and the KN hierarchy. Obviously, the M-AKNS-KN hierarchy is again a united expressing integrable model of the multicomponent AKNS hierarchy (M-AKNS) and the multicomponent KN hierarchy(M-KN). This article provides a simple method for obtaining multicomponent integrable hierarchies of soliton equations. Finally, we work out an integrable coupling of the M-AKNS-KN hierarchy.

Robust procedures for converting among Lindblad, Kraus and matrix representations of quantum dynamical semigroups

Abstract: Given a quantum dynamical semigroup expressed as an exponential superoperator acting on a space of N-dimensional density operators, eigenvalue methods are presented by which canonical Kraus and Lindblad operator sum representations can be computed. These methods provide a mathematical basis on which to develop novel algorithms for quantum process tomography—the statistical estimation of superoperators and their generators—from a wide variety of experimental data. Theoretical arguments and numerical simulations are presented which imply that these algorithms will be quite robust in the presence of random errors in the data.

Controllability properties for finite dimensional quantum Markovian master equations

Abstract: Various notions from geometric control theory are used to characterize the behavior of the Markovian master equation for N-level quantum mechanical systems driven by unitary control and to describe the structure of the sets of reachable states. It is shown that the system can be accessible but neither small-time controllable nor controllable in finite time. In particular, if the generators of quantum dynamical semigroups are unital, then the reachable sets admit easy characterizations as they monotonically grow in time. The two level case is treated in detail.

Products and ratios of characteristic polynomials of random Hermitian matrices

Abstract: We present new and streamlined proofs of various formulas for products and ratios of characteristic polynomials of random Hermitian matrices that have appeared recently in the literature.

Monopoles, affine algebras and the gluino condensate

Abstract: We examine the low-energy dynamics of four-dimensional supersymmetric gauge theories and calculate the values of the gluino condensate for all simple gauge groups By initially compactifying the theory on a cylinder we are able to perform calculations in a controlled weakly coupled way for a small radius The dominant contributions to the path integral on the cylinder arise from magnetic monopoles which play the role of instanton constituents We find that the semi-classically generated superpotential of the theory is the affine Toda potential for an associated twisted affine algebra We determine the supersymmetric vacua and calculate the values of the gluino condensate The number of supersymmetric vacua is equal to c2, the dual Coxeter number, and in each vacuum the monopoles carry a fraction 1/c2 of topological charge As the results are independent of the radius of the circle, they are also valid in the strong coupling regime where the theory becomes decompactified For gauge groups SU(N), SO(N) and

Spin networks for noncompact groups

Abstract: Spin networks are a natural generalization of Wilson loop functionals. They have been extensively studied in the case where the gauge group is compact and it has been shown that they naturally form a basis of gauge invariant observables. Physically the restriction to compact gauge groups is enough for the study of Yang–Mills theories, however it is well known that noncompact groups naturally arise as internal gauge groups for Lorentzian gravity models. In this context, a proper construction of gauge invariant observables is needed. The purpose of the present work is to define the notion of spin network states for noncompact groups. We first build, by a careful gauge fixing procedure, a natural measure and a Hilbert space structure on the space of gauge invariant graph connections. Spin networks are then defined as generalized eigenvectors of a complete set of hermitic commuting operators. We show how the delicate issue of taking the quotient of a space by noncompact groups can be address in term of algebraic geometry. We finally construct the full Hilbert space containing all spin network states. Having in mind applications to gravity, we illustrate our results for the groups SL(2,R) and SL(2,C).

A quantum weak energy inequality for spin-one fields in curved space–time

Abstract: Quantum weak energy inequalities (QWEI) provide state-independent lower bounds on averages of the renormalized energy density of a quantum field. We derive QWEIs for the electromagnetic and massive spin-one fields in globally hyperbolic space–times whose Cauchy surfaces are compact and have trivial first homology group. These inequalities provide lower bounds on weighted averages of the renormalized energy density as “measured” along an arbitrary timelike trajectory, and are valid for arbitrary Hadamard states of the spin-one fields. The QWEI bound takes a particularly simple form for averaging along static trajectories in ultrastatic space–times; as specific examples we consider Minkowski space (in which case the topological restrictions may be dispensed with) and the static Einstein universe. A significant part of the paper is devoted to the definition and properties of Hadamard states of spin-one fields in curved space–times, particularly with regard to their microlocal behavior.

Wigner–Yanase information on quantum state space: The geometric approach

Abstract: In the search of appropriate Riemannian metrics on quantum state space, the concept of statistical monotonicity, or contraction under coarse graining, has been proposed by Chentsov. The metrics with this property have been classified by Petz. All the elements of this family of geometries can be seen as quantum analogs of Fisher information. Although there exists a number of general theorems shedding light on this subject, many natural questions, also stemming from applications, are still open. In this paper we discuss a particular member of the family, the Wigner–Yanase information. Using a well-known approach that mimics the classical pull-back approach to Fisher information, we are able to give explicit formulas for the geodesic distance, the geodesic path, the sectional and scalar curvatures associated to Wigner–Yanase information. Moreover, we show that this is the only monotone metric for which such an approach is possible.

Choi’s proof as a recipe for quantum process tomography

Abstract: Quantum process tomography is a procedure by which an unknown quantum operation can be fully experimentally characterized. We reinterpret Choi’s proof [Linear Algebr. Appl. 10, 285 (1975)] of the fact that any completely positive linear map has a Kraus representation as a method for quantum process tomography. The analysis for obtaining the Kraus operators is extremely simple. We discuss the systems in which this tomography method is particularly suitable.

Integrable systems and reductions of the self-dual Yang–Mills equations

Abstract: Many integrable equations are known to be reductions of the self-dual Yang–Mills equations. This article discusses some of the well known reductions including the standard soliton equations, the classical Painleve equations and integrable generalizations of the Darboux–Halphen system and Chazy equations. The Chazy equation, first derived in 1909, is shown to correspond to the equations studied independently by Ramanujan in 1916.

Multiplicative noise: A mechanism leading to nonextensive statistical mechanics

Abstract: A large variety of microscopic or mesoscopic models lead to generic results that accommodate naturally within Boltzmann–Gibbs statistical mechanics [based on S1≡−k∫du p(u)ln p(u)]. Similarly, other classes of models point toward nonextensive statistical mechanics [based on Sq≡k[1−∫du[p(u)]q]/[q−1], where the value of the entropic index q∈R depends on the specific model]. We show here a family of models, with multiplicative noise, which belongs to the nonextensive class. More specifically, we consider Langevin equations of the type u=f(u)+g(u)ξ(t)+η(t), where ξ(t) and η(t) are independent zero-mean Gaussian white noises with respective amplitudes M and A. This leads to the Fokker–Planck equation ∂tP(u,t)=−∂u[f(u)P(u,t)]+M∂u{g(u)∂u[g(u)P(u,t)]}+A∂uuP(u,t). Whenever the deterministic drift is proportional to the noise induced one, i.e., f(u)=−τg(u)g′(u), the stationary solution is shown to be P(u,∞)∝{1−(1−q)β[g(u)]2}1/(1−q) [with q≡(τ+3M)/(τ+M) and β=(τ+M/2A)]. This distribution is precisely the one optimiz...

Foldy–Wouthuysen transformation for relativistic particles in external fields

Abstract: A method of Foldy–Wouthuysen transformation for relativistic spin-1/2 particles in external fields is proposed It permits the determination of the Hamilton operator in the Foldy–Wouthuysen representation with any accuracy Interactions between a particle having an anomalous magnetic moment and nonstationary electromagnetic and electroweak fields are investigated

Noncommutative geometry of angular momentum space U(su(2))

Abstract: We study the standard angular momentum algebra [xi,xj]=iλeijkxk as a noncommutative manifold Rλ3. We show that there is a natural 4D differential calculus and obtain its cohomology and Hodge * operator. We solve the spin 0 wave equation and some aspects of the Maxwell or electromagnetic theory including solutions for a uniform electric current density, and we find a natural Dirac operator ∂/. We embed Rλ3 inside a 4D noncommutative space–time which is the limit q→1 of q-Minkowski space and show that Rλ3 has a natural quantum isometry group given by the quantum double C(SU(2))⋊U(su(2)) which is a singular limit of the q-Lorentz group. We view Rλ3 as a collection of all fuzzy spheres taken together. We also analyze the semiclassical limit via minimum uncertainty states |j,θ,φ〉 approximating classical positions in polar coordinates.

General soliton matrices in the Riemann–Hilbert problem for integrable nonlinear equations

Abstract: We derive the soliton matrices corresponding to an arbitrary number of higher-order normal zeros for the matrix Riemann–Hilbert problem of arbitrary matrix dimension, thus giving the complete solution to the problem of higher-order solitons. Our soliton matrices explicitly give all higher-order multisoliton solutions to the nonlinear partial differential equations integrable through the matrix Riemann–Hilbert problem. We have applied these general results to the three-wave interaction system, and derived new classes of higher-order soliton and two-soliton solutions, in complement to those from our previous publication [Stud. Appl. Math. 110, 297 (2003)], where only the elementary higher-order zeros were considered. The higher-order solitons corresponding to nonelementary zeros generically describe the simultaneous breakup of a pumping wave (u3) into the other two components (u1 and u2) and merger of u1 and u2 waves into the pumping u3 wave. The two-soliton solutions corresponding to two simple zeros gener...

Scattering on compact manifolds with infinitely thin horns

Abstract: The quantum-mechanical scattering on a compact manifold with semi-axes attached to the manifold (“hedgehog-shaped manifold”) is considered. The complete description of the spectral structure of Schrodinger operators on such a manifold is done, the proof of existence and uniqueness of scattering states is presented, an explicit form for the scattering matrix is obtained and unitarity of this matrix is proven. It is shown that the positive part of the spectrum of the Schrodinger operator on the initial compact manifold as well as the spectrum of a point perturbation of such an operator may be recovered from the scattering amplitude for one attached half-line. Moreover, the positive part of the spectrum of the initial Schrodinger operator is fully determined by the conductance properties of an “electronic device” consisting of the initial manifold and two “wires” attached to it.

Bound states in one and two spatial dimensions

Abstract: In this article we study the number of bound states for potentials in one and two spatial dimensions. We first show that in addition to the well-known fact that an arbitrarily weak attractive potential has a bound state, it is easy to construct examples where weak potentials have an infinite number of bound states. These examples have potentials which decrease at infinity faster than expected. Using somewhat stronger conditions, we derive explicit bounds on the number of bound states in one dimension, using known results for the three-dimensional zero angular momentum. A change of variables which allows us to go from the one-dimensional case to that of two dimensions results in a bound for the zero angular momentum case. Finally, we obtain a bound on the total number of bound states in two dimensions, first for the radial case and then, under stronger conditions, for the noncentral case.

The fourth Painlevé equation and associated special polynomials

Abstract: In this article rational solutions and associated polynomials for the fourth Painleve equation are studied. These rational solutions of the fourth Painleve equation are expressible as the logarithmic derivative of special polynomials, the Okamoto polynomials. The structure of the roots of these Okamoto polynomials is studied and it is shown that these have a highly regular structure. The properties of the Okamoto polynomials are compared and contrasted with those of classical orthogonal polynomials. Further representations are given of the associated rational solutions in the form of determinants through Schur functions.

Progress in relativistic gravitational theory using the inverse scattering method

Abstract: The increasing interest in compact astrophysical objects (neutron stars, binaries, galactic black holes) has stimulated the search for rigorous methods, which allow a systematic general relativistic description of such objects. This article is meant to demonstrate the use of the inverse scattering method, which allows, in particular cases, the treatment of rotating body problems. The idea is to replace the investigation of the matter region of a rotating body by the formulation of boundary values along the surface of the body. In this way we construct solutions describing rotating black holes and disks of dust (“galaxies”). Physical properties of the solutions and consequences of the approach are discussed. Among other things, the balance problem for two black holes can be tackled.

“Massive” spin-2 field in de Sitter space

Abstract: In this paper we present a covariant quantization of the “massive” spin-2 field on de Sitter (dS) space. By “massive” we mean a field which carries a specific principal series representation of the dS group. The work is in the direct continuation of previous ones concerning the scalar, the spinor, and the vector cases. The quantization procedure, independent of the choice of the coordinate system, is based on the Wightman-Garding axiomatic and on analyticity requirements for the two-point function in the complexified pseudo-Riemanian manifold. Such a construction is necessary in view of preparing and comparing with the dS conformal spin-2 massless case (dS linear quantum gravity) which will be considered in a forthcoming paper and for which specific quantization methods are needed.

Tiling spaces are inverse limits

Abstract: Let M be an arbitrary Riemannian homogeneous space, and let Ω be a space of tilings of M, with finite local complexity (relative to some symmetry group Γ) and closed in the natural topology. Then Ω is the inverse limit of a sequence of compact finite-dimensional branched manifolds. The branched manifolds are (finite) unions of cells, constructed from the tiles themselves and the group Γ. This result extends previous results of Anderson and Putnam, of Ormes, Radin, and Sadun, of Bellissard, Benedetti, and Gambaudo, and of Gahler. In particular, the construction in this paper is a natural generalization of Gahler’s.

Dynamical structure of irregular constrained systems

Abstract: Hamiltonian systems with functionally dependent constraints (irregular systems), for which the standard Dirac procedure is not directly applicable, are discussed. They are classified according to their behavior in the vicinity of the constraint surface into two fundamental types. If the irregular constraints are multilinear (type I), then it is possible to regularize the system so that the Hamiltonian and Lagrangian descriptions are equivalent. When the constraints are power of a linear function (type II), regularization is not always possible and the Hamiltonian and Lagrangian descriptions may be dynamically inequivalent. It is shown that the inequivalence between the two formalisms can occur if the kinetic energy is an indefinite quadratic form in the velocities. It is also shown that a system of type I can evolve in time from a regular configuration into an irregular one, without any catastrophic changes. Irregularities have important consequences in the linearized approximation to nonlinear theories, as well as for the quantization of such systems. The relevance of these problems to Chern–Simons theories in higher dimensions is discussed.

R-matrix presentation for super-Yangians Y(osp(m|2n))

Abstract: We give a RTT presentation of super-Yangians Y(g) for g=osp(m|2n), thereby unifying the formalism with the cases of g=so(n) and g=sp(2n).

An invariant action for noncommutative gravity in four dimensions

Abstract: Two main problems face the construction of noncommutative actions for gravity with star products: the complex metric and finding an invariant measure. The only gauge groups that could be used with star products are the unitary groups. I propose an invariant gravitational action in D=4 dimensions based on the constrained gauge group U(2,2) broken to U(1,1)×U(1,1). No metric is used, thus giving a naturally invariant measure. This action is generalized to the noncommutative case by replacing ordinary products with star products. The four-dimensional noncommutative action is studied and the deformed action to first order in deformation parameter is computed.

Twistor theory of hyper-Kähler metrics with hidden symmetries

Abstract: We review the hierarchy for the hyper-Kahler equations and define a notion of symmetry for solutions of this hierarchy. A four-dimensional hyper-Kahler metric admits a hidden symmetry if it embeds into a hierarchy with a symmetry. It is shown that a hyper-Kahler metric admits a hidden symmetry if it admits a certain Killing spinor. We show that if the hidden symmetry is tri-holomorphic, then this is equivalent to requiring symmetry along a higher time and the hidden symmetry determines a “twistor group” action as introduced by Bielawski [Twistor Quotients of Hyper-Kahler Manifolds (World Scientific, River Edge, NJ, 2001)]. This leads to a construction for the solution to the hierarchy in terms of linear equations and variants of the generalized Legendre transform for the hyper-Kahler metric itself given by Ivanov and Rocek [Commun. Math. Phys. 182, 291 (1996)]. We show that the ALE spaces are examples of hyper-Kahler metrics admitting three tri-holomorphic Killing spinors. These metrics are in this sense analogous to the “finite gap” solutions in soliton theory. Finally we extend the concept of a hierarchy from that of our earlier work [Commun. Math. Phys. 213, 641 (2000)] for the four-dimensional hyper-Kahler equations to a generalization of the conformal anti-self-duality equations and briefly discuss hidden symmetries for these equations.

Magnetoencephalography in ellipsoidal geometry

Abstract: An exact analytic solution for the forward problem in the theory of biomagnetics of the human brain is known only for the (1D) case of a sphere and the (2D) case of a spheroid, where the excitation field is due to an electric dipole within the corresponding homogeneous conductor. In the present work the corresponding problem for the more realistic ellipsoidal brain model is solved and the leading quadrupole approximation for the exterior magnetic field is obtained in a form that exhibits the anisotropic character of the ellipsoidal geometry. The results are obtained in a straightforward manner through the evaluation of the interior electric potential and a subsequent calculation of the surface integral over the ellipsoid, using Lame functions and ellipsoidal harmonics. The basic formulas are expressed in terms of the standard elliptic integrals that enter the expressions for the exterior Lame functions. The laborious task of reducing the results to the spherical geometry is also included.