Showing papers in "Journal of Mathematical Physics in 1999"
TL;DR: In this paper, the authors proposed a generalization of Hermiticity for complex deformation H =p2+x2(ix)e of the harmonic oscillator Hamiltonian, where e is a real parameter.
Abstract: This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition H†=H on the Hamiltonian, where † represents the mathematical operation of complex conjugation and matrix transposition. This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian H has a real spectrum. However, replacing this mathematical condition by the weaker and more physical requirement H‡=H, where ‡ represents combined parity reflection and time reversal PT, one obtains new classes of complex Hamiltonians whose spectra are still real and positive. This generalization of Hermiticity is investigated using a complex deformation H=p2+x2(ix)e of the harmonic oscillator Hamiltonian, where e is a real parameter. The system exhibits two phases: When e⩾0, the energy spectrum of H is real and positive as a consequence of PT symmetry. However, when −1
1,268 citations
TL;DR: In this article, the special quasiperiodic solution of the (2+1)-dimensional Kadometsev-Petviashvili equation is separated into three systems of ordinary differential equations, which are the second, third, and fourth members in the confocal involutive hierarchy associated with the nonlinearized Zakharov-Shabat eigenvalue problem.
Abstract: The special quasiperiodic solution of the (2+1)-dimensional Kadometsev–Petviashvili equation is separated into three systems of ordinary differential equations, which are the second, third, and fourth members in the well-known confocal involutive hierarchy associated with the nonlinearized Zakharov–Shabat eigenvalue problem. The explicit theta function solution of the Kadometsev–Petviashvili equation is obtained with the help of this separation technique. A generating function approach is introduced to prove the involutivity and the functional independence of the conserved integrals which are essential for the Liouville integrability.
267 citations
TL;DR: In this paper, simple general expressions for the explicit Killing spinors on the n-sphere, for arbitrary n, were derived and extended to the hyperbolic spaces Hn and Hn.
Abstract: We derive simple general expressions for the explicit Killing spinors on the n-sphere, for arbitrary n. Using these results we also construct the Killing spinors on various AdS×Sphere supergravity backgrounds, including AdS5×S5, AdS4×S7, and AdS7×S4. In addition, we extend previous results to obtain the Killing spinors on the hyperbolic spaces Hn.
253 citations
TL;DR: In this article, a space-time view of variational integrators is employed and time step adaptation is used to impose the constraint of conservation of energy and momentum, and some numerical examples for the solvability of the time steps are given.
Abstract: The purpose of this paper is to develop variational integrators for conservative mechanical systems that are symplectic and energy and momentum conserving. To do this, a space–time view of variational integrators is employed and time step adaptation is used to impose the constraint of conservation of energy. Criteria for the solvability of the time steps and some numerical examples are given.
251 citations
TL;DR: In this article, it was shown that the Camassa-holm equation is a geodesic flow on the Bott-Virasoro group, which is the case κ = 0.
Abstract: Misiolek [J. Geom. Phys. 24, 203–208 (1998)] has shown that the Camassa–Holm equation is a geodesic flow on the Bott–Virasoro group. In this paper it is shown that the Camassa–Holm equation for the case κ=0 is the geodesic spray of the weak Riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating the H1 inner product over the entire group. This paper uses the right-trivialization technique to rigorously verify that the Euler–Poincare theory for Lie groups can be applied to diffeomorphism groups. The observation made in this paper has led to physically meaningful generalizations of the CH-equation to higher dimensional manifolds.
229 citations
TL;DR: In this article, it is shown that differential forms and their discrete counterparts (cochains) provide a natural bridge between the continuum and the lattice versions of the theory, allowing for a natural factorization of the field equations into topological field equations (i.e., invariant under homeomorphisms) and metric field equations.
Abstract: The language of differential forms and topological concepts are applied to study classical electromagnetic theory on a lattice. It is shown that differential forms and their discrete counterparts (cochains) provide a natural bridge between the continuum and the lattice versions of the theory, allowing for a natural factorization of the field equations into topological field equations (i.e., invariant under homeomorphisms) and metric field equations. The various potential sources of inconsistency in the discretization process are identified, distinguished, and discussed. A rationale for a consistent extension of the lattice theory to more general situations, such as to irregular lattices, is considered.
203 citations
TL;DR: In this article, the relative modular operator is used to define a generalized relative entropy for any convex operator function g on (0,∞) satisfying g(1)=0.
Abstract: We use the relative modular operator to define a generalized relative entropy for any convex operator function g on (0,∞) satisfying g(1)=0. We show that these convex operator functions can be partitioned into convex subsets, each of which defines a unique symmetrized relative entropy, a unique family (parametrized by density matrices) of continuous monotone Riemannian metrics, a unique geodesic distance on the space of density matrices, and a unique monotone operator function satisfying certain symmetry and normalization conditions. We describe these objects explicitly in several important special cases, including g(w)=−log w, which yields the familiar logarithmic relative entropy. The relative entropies, Riemannian metrics, and geodesic distances obtained by our procedure all contract under completely positive, trace-preserving maps. We then define and study the maximal contraction associated with these quantities.
188 citations
TL;DR: In this paper, an algebraic structure related to discrete zero curvature equations is established, which is used to give an approach for generating master symmetries of the first degree for systems of discrete evolution equations.
Abstract: An algebraic structure related to discrete zero curvature equations is established. It is used to give an approach for generating master symmetries of the first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries. The key of the theory is to generate nonisospectral flows (λt=λl, l⩾0) from the discrete spectral problem associated with a given system of discrete evolution equations. Three examples are given.
171 citations
TL;DR: In this paper, a method for deriving the nonrelativistic quantum Hamiltonian of a free massive fermion from the relativistic Lagrangian of the Lorentz-violating standard model extension is presented.
Abstract: A method is presented for deriving the nonrelativistic quantum Hamiltonian of a free massive fermion from the relativistic Lagrangian of the Lorentz-violating standard-model extension. It permits the extraction of terms at arbitrary order in a Foldy–Wouthuysen expansion in inverse powers of the mass. The quantum particle Hamiltonian is obtained and its nonrelativistic limit is given explicitly to third order.
168 citations
TL;DR: In this article, it was shown that if the quality judgment is based solely on measurements of single output clones, there is again a unique optimal cloning device, which coincides with the one found previously.
Abstract: We consider quantum devices for turning a finite number N of d-level quantum systems in the same unknown pure state σ into M>N systems of the same kind, in an approximation of the M-fold tensor product of the state σ. In a previous paper it was shown that this problem has a unique optimal solution, when the quality of the output is judged by arbitrary measurements, involving also the correlations between the clones. We show in this paper, that if the quality judgment is based solely on measurements of single output clones, there is again a unique optimal cloning device, which coincides with the one found previously.
164 citations
TL;DR: In this article, the authors proposed a method to solve a set of problems in the context of artificial neural networks, and showed that the proposed method can be applied in the real world.
Abstract: Acad Sinica, Inst Theoret Phys, Beijing 100080, Peoples R China; Hunan Univ, Dept Phys, Changsha 410082, Peoples R China
TL;DR: In this paper, the authors studied the Jordanian quantization of Lie algebras using the factorizable twist element F, universal R-matrix and the corresponding canonical element T. The construction of the twist is generalized to a certain type of inhomogenious Lie algesbras.
Abstract: Jordanian quantizations of Lie algebras are studied using the factorizable twists. For a restricted Borel subalgebra B∨ of sl(N) the explicit expressions are obtained for the twist element F, universal R-matrix and the corresponding canonical element T. It is shown that the twisted Hopf algebra UF(B∨) is self-dual. The cohomological properties of the involved Lie bialgebras are studied to justify the existence of a contraction from the Dinfeld–Jimbo quantization to the Jordanian one. The construction of the twist is generalized to a certain type of inhomogenious Lie algebras.
TL;DR: In this work, a general procedure for constructing the recursion operators for nonlinear integrable equations admitting Lax representation is developed and the recursions operators for some KdV-type systems of integrability equations are found.
Abstract: In this work we develop a general procedure for constructing the recursion operators for nonlinear integrable equations admitting Lax representation. Several new examples are given. In particular, we find the recursion operators for some KdV-type systems of integrable equations.
TL;DR: In this paper, the existence of superintegrable systems with n = 2 degrees of freedom possessing three independent globally defined constants of motion which are quadratic in the velocities is studied on the two-dimensional sphere S2 and on the hyperbolic plane H2.
Abstract: The existence of superintegrable systems with n=2 degrees of freedom possessing three independent globally defined constants of motion which are quadratic in the velocities is studied on the two-dimensional sphere S2 and on the hyperbolic plane H2. The approach used is based on enforcing the conditions for the existence of two independent integrals (further than the energy). This is done in a way which allows us to discuss at once the cases of the sphere S2 and the hyperbolical plane H2, by considering the curvature κ as a parameter. Different superintegrable potentials are obtained as the solutions of certain systems of two κ-dependent second order partial differential equations. The Euclidean results are directly recovered for κ=0, and the superintegrable potentials on either the standard unit sphere (radius R=1) or the unit Lobachewski plane (“radius” R=1) appear as the particular values of the κ-dependent superintegrable potentials for the values κ=1 and κ=−1. Some new superintegrable potentials are f...
TL;DR: In this paper, the Euler-Kirchhoffer equations for elastic rods with circular cross sections are used to provide a classification of the different shapes a filament can assume.
Abstract: Euler–Kirchhoff filaments are solutions of the static Kirchhoff equations for elastic rods with circular cross sections. These equations are known to be formally equivalent to the Euler equations for spinning tops. This equivalence is used to provide a classification of the different shapes a filament can assume. Explicit formulas for the different possible configurations and specific results for interesting particular cases are given. In particular, conditions for which the filament has points of self-intersection, self-tangency, vanishing curvature or when it is closed or localized in space are provided. The average properties of generic filaments are also studied. They are shown to be equivalent to helical filaments on long length scales.
TL;DR: In this article, a precise calculation of the ground-state energy of the complex PT-symmetric Hamiltonian H =p2+14x2+iλx3, is performed using high-order Rayleigh-Schrodinger perturbation theory.
Abstract: A precise calculation of the ground-state energy of the complex PT-symmetric Hamiltonian H=p2+14x2+iλx3, is performed using high-order Rayleigh–Schrodinger perturbation theory. The energy spectrum of this Hamiltonian has recently been shown to be real using numerical methods. Here we present convincing numerical evidence that the Rayleigh–Schrodinger perturbation series is Borel summable, and show that Pade summation provides excellent agreement with the real energy spectrum. Pade analysis provides strong numerical evidence that the once-subtracted ground-state energy considered as a function of λ2 is a Stieltjes function. The analyticity properties of this Stieltjes function lead to a dispersion relation that can be used to compute the imaginary part of the energy for the related real but unstable Hamiltonian H=p2+14 x2−ex3.
TL;DR: In this paper, a solution concept for the geodesic deviation equation based on embedding the distributional metric into the Colombeau algebra of generalized functions is presented, using a universal regularization procedure.
Abstract: The geodesic as well as the geodesic deviation equation for impulsive gravitational waves involve highly singular products of distributions (θδ,θ2δ,δ2). A solution concept for these equations based on embedding the distributional metric into the Colombeau algebra of generalized functions is presented. Using a universal regularization procedure we prove existence and uniqueness results and calculate the distributional limits of these solutions explicitly. The obtained limits are regularization independent and display the physically expected behavior.
TL;DR: In this article, it was shown that certain partial differential equations associated to nonisospectral scattering problems in 2+1 dimensions provide a key to associated integrable hierarchies of both ordinary and partial differential equation.
Abstract: We show that certain partial differential equations associated to nonisospectral scattering problems in 2+1 dimensions provide a key to associated integrable hierarchies of both ordinary and partial differential equations. This is illustrated using (an extension of) a known second-order and two new third-order nonisospectral scattering problems. These scattering problems allow us to derive new hierarchies of integrable partial differential equations, in both 1+1 and 2+1 dimensions, together with their underlying linear problems (isospectral and nonisospectral); and also new hierarchies of integrable ordinary differential equations, again with their underlying linear problems.
TL;DR: In this paper, the eigenvalue problems for the Ginzburg-Landau operator with a large parameter in bounded domains in R2 under gauge invariant boundary conditions are studied.
Abstract: In this paper we study the eigenvalue problems for the Ginzburg–Landau operator with a large parameter in bounded domains in R2 under gauge invariant boundary conditions. The estimates for the eigenvalues are obtained and the asymptotic behavior of the associated eigenfunctions is discussed. These results play a key role in estimating the critical magnetic field in the mathematical theory of superconductivity.
TL;DR: In this paper, a new family of boson coherent states using a specially designed function which is a solution of a functional equation de(q,x)/dx=e( q,qx) with 0⩽q⩾1 and e(q 0)=1.
Abstract: We construct a new family of boson coherent states using a specially designed function which is a solution of a functional equation de(q,x)/dx=e(q,qx) with 0⩽q⩽1 and e(q,0)=1. We use this function in place of the usual exponential to generate new coherent states |q,z〉 from the vacuum, which are normalized and continuous in their label z. These states allow the resolution of unity, and a corresponding weight function is furnished by the exact solution of the associated Stieltjes moment problem. They also permit exact evaluation of matrix elements of an arbitrary polynomial given as a normally-ordered function of boson operators. We exemplify this by showing that the photon number statistics for these states is sub-Poissonian. For any q<1 the states |q,z〉 are squeezed; we obtain and discuss their signal to quantum noise ratio. The function e(q,x) allows a natural generation of multiboson coherent states of arbitrary multiplicity, which is impossible for the usual coherent states. For q=1 all the above results reduce to those for conventional coherent states. Finally, we establish a link with q-deformed bosons.
TL;DR: In this article, it was shown that a real symmetric positive definite matrix V is congruent to a diagonal matrix modulo a pseudo-orthogonal [pseudo-unitary] matrix in SO(m,n) for any choice of partition N =m+n.
Abstract: It is shown that a N×N real symmetric [complex Hermitian] positive definite matrix V is congruent to a diagonal matrix modulo a pseudo-orthogonal [pseudo-unitary] matrix in SO(m,n)[SU(m,n)], for any choice of partition N=m+n. It is further shown that the method of proof in this context can easily be adapted to obtain a rather simple proof of Williamson’s theorem which states that if N is even then V is congruent also to a diagonal matrix modulo a symplectic matrix in Sp(N,R)[Sp(N,C)]. Applications of these results considered include a generalization of the Schweinler–Wigner method of “orthogonalization based on an extremum principle” to construct pseudo-orthogonal and symplectic bases from a given set of linearly independent vectors.
TL;DR: In this article, the singular manifold method is used to perform a complete study of an equation in 2+1 dimensions and the Lax pair, Darboux transformation and τ functions in such a way that a plethora of different solutions with solitonic behavior can be constructed iteratively.
Abstract: Painleve analysis and the singular manifold method are the tools used in this paper to perform a complete study of an equation in 2+1 dimensions. This procedure has allowed us to obtain the Lax pair, Darboux transformation and τ functions in such a way that a plethora of different solutions with solitonic behavior can be constructed iteratively.
TL;DR: In this paper, all nonwisting Petrov-type N solutions of vacuum Einstein field equations with cosmological constant Λ are summarized and shown to belong either to the nonexpanding Kundt class or to the expanding Robinson-Trautman class.
Abstract: All nontwisting Petrov-type N solutions of vacuum Einstein field equations with cosmological constant Λ are summarized. They are shown to belong either to the nonexpanding Kundt class or to the expanding Robinson–Trautman class. Invariant subclasses of each class are defined and the corresponding metrics are given explicitly in suitable canonical coordinates. Relations between the subclasses and their geometrical properties are analyzed. In the subsequent paper these solutions are interpreted as exact gravitational waves propagating in de Sitter or anti-de Sitter spacetimes.
TL;DR: In this paper, a nonlinear partial differential equation can be solved by solving ordinary different equations or even algebraic equations using a formal variable separation approach, and some explicit solitary wave solutions which are induced by background source and nonlinearity or dispersion are obtained.
Abstract: Using a formal variable separation approach, a nonlinear partial differential equation can be solved by solving ordinary different equations or even algebraic equations. Taking the KdV–Burgers and modified KdV–Burgers equations with background interaction as simple examples, some explicit solitary wave solutions which are induced by background source and nonlinearity or dispersion are obtained.
TL;DR: In this article, a proof is presented that demonstrates that the intertwiner of a vertex of such a spin network is uniquely determined, up to normalization, by the representations on the incident edges and the constraints.
Abstract: Barrett and Crane have proposed a model of simplicial Euclidean quantum gravity in which a central role is played by a class of Spin(4) spin networks called “relativistic spin networks” which satisfy a series of physically motivated constraints. Here a proof is presented that demonstrates that the intertwiner of a vertex of such a spin network is uniquely determined, up to normalization, by the representations on the incident edges and the constraints. Moreover, the constraints, which were formulated for four valent spin networks only, are extended to networks of arbitrary valence, and the generalized relativistic spin networks proposed by Yetter are shown to form the entire solution set (mod normalization) of the extended constraints. Finally, using the extended constraints, the Barrett–Crane model is generalized to arbitrary polyhedral complexes (instead of just simplicial complexes) representing space-time. It is explained how this model, like the Barret–Crane model can be derived from BF theory, a simple topological field theory [G. Horowitz, Commun. Math. Phys. 125, 417 (1989)], by restricting the sum over histories to ones in which the left-handed and right-handed areas of any 2-surface are equal. It is known that the solutions of classical Euclidean general relativity form a branch of the stationary points of the BF action with respect to variations preserving this condition.
TL;DR: In this article, a nonlinear evolution equation is suggested to describe the propagation of waves in a relaxing medium, and for low-frequency approach this equation is reduced to the KdVB equation.
Abstract: A nonlinear evolution equation is suggested to describe the propagation of waves in a relaxing medium. It is shown that for low-frequency approach this equation is reduced to the KdVB equation. The high-frequency perturbations are described by a new nonlinear equation. This equation has ambiguous looplike solutions. It is established that a dissipative term, with a dissipation parameter less than some limit value, does not destroy these looplike solutions.
TL;DR: In this paper, the ground state energy of a spineless particle minimally coupled to a massless quantized radiation field with an ultraviolet cutoff is derived and the Hamiltonian of the system is defined in terms of functional integrals.
Abstract: The system of N-nonrelativistic spineless particles minimally coupled to a massless quantized radiation field with an ultraviolet cutoff is considered. The Hamiltonian of the system is defined for arbitrary coupling constants in terms of functional integrals. It is proved that the ground state of the system with a class of external potentials, if they exist, is unique. Moreover an expression of the ground state energy is obtained and it is shown that the ground state energy is a monotonously increasing, concave, and continuous function with respect to the square of a coupling constant.
TL;DR: In this article, the supersymmetric partners of the Lame potentials are computed for integer values a = 1, 2, 3, 4, 5, 6, 7, 8, 9,
Abstract: Using the formalism of supersymmetric quantum mechanics, we obtain a large number of new analytically solvable one-dimensional periodic potentials and study their properties. More specifically, the supersymmetric partners of the Lame potentials ma(a+1)sn2(x,m) are computed for integer values a=1,2,3,… . For all cases (except a=1), we show that the partner potential is distinctly different from the original Lame potential, even though they both have the same energy band structure. We also derive and discuss the energy band edges of the associated Lame potentials pm sn2(x,m)+qm cn2(x,m)/dn2(x,m), which constitute a much richer class of periodic problems. Computation of their supersymmetric partners yields many additional new solvable and quasiexactly solvable periodic potentials.
TL;DR: The classical n-dimensional Calogero-Moser system is a maximally superintegrable system endowed with a rich variety of symmetries and constants of motion as discussed by the authors.
Abstract: The classical n-dimensional Calogero–Moser system is a maximally superintegrable system endowed with a rich variety of symmetries and constants of motion. In the first part of the article some properties related with the existence of several families of constants of motion are analyzed. In the second part, the master symmetries and the time-dependent symmetries of this system are studied.
TL;DR: In this paper, the authors propose to modify Einstein's equations by embedding them in a larger symmetric hyperbolic system, where the additional dynamical variables of the modified system are essentially first integrals of the original constraints.
Abstract: We introduce a proposal to modify Einstein’s equations by embedding them in a larger symmetric hyperbolic system. The additional dynamical variables of the modified system are essentially first integrals of the original constraints. The extended system of equations reproduces the usual dynamics on the constraint surface of general relativity, and therefore naturally includes the solutions to Einstein gravity. The main feature of this extended system is that, at least for a linearized version of it, the constraint surface is an attractor of the time evolution. This feature suggests that this system may be a useful alternative to Einstein’s equations when obtaining numerical solutions to full, nonlinear gravity.