Showing papers in "Journal of Mathematical Physics in 2007"

Bounds for the adiabatic approximation with applications to quantum computation

Abstract: We present straightforward proofs of estimates used in the adiabatic approximation. The gap dependence is analyzed explicitly. We apply the result to interpolating Hamiltonians of interest in quantum computing.

Evenly distributed unitaries: On the structure of unitary designs

Abstract: We clarify the mathematical structure underlying unitary t-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any tth order polynomial over the design equals the average over the entire unitary group. We present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design. Lower bounds for the number of elements of 2-designs are derived. We show how to turn mutually unbiased bases into approximate 2-designs whose cardinality is optimal in leading order. Designs of higher order are discussed and an example of a unitary 5-design is presented. We comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups. Further, we sketch connections to problems in linear optics and questions regarding typical entanglement.

A (q,γ) analog of the W1+∞ algebra

Abstract: A (q,γ) analog of the W1+∞ algebra is introduced Irreducible quasifinite highest weight modules of this algebra and R matrices acting on a tensor product of these modules are investigated A connection with the q deformed Virasoro and WN algebras is also discussed

Point vortex dynamics: A classical mathematics playground

Abstract: The idealization of a two-dimensional, ideal flow as a collection of point vortices embedded in otherwise irrotational flow yields a surprisingly large number of mathematical insights and connects to a large number of areas of classical mathematics. Several examples are given including the integrability of the three-vortex problem, the interplay of relative equilibria of identical vortices and the roots of certain polynomials, addition formulas for the cotangent and the Weierstras ζ function, projective geometry, and other topics. The hope and intent of the article is to garner further participation in the exploration of this intriguing dynamical system from the mathematical physics community.

Lagrangian coherent structures in n-dimensional systems

Abstract: Numerical simulations and experimental observations reveal that unsteady fluid systems can be divided into regions of qualitatively different dynamics. The key to understanding transport and stirring is to identify the dynamic boundaries between these almost-invariant regions. Recently, ridges in finite-time Lyapunov exponent fields have been used to define such hyperbolic, almost material, Lagrangian coherent structures in two-dimensional systems. The objective of this paper is to develop and apply a similar theory in higher dimensional spaces. While the separatrix nature of these structures is their most important property, a necessary condition is their almost material nature. This property is addressed in this paper. These results are applied to a model of Rayleigh-Benard convection based on a three-dimensional extension of the model of Solomon and Gollub.

Fractional embedding of differential operators and Lagrangian systems

Abstract: This paper is a contribution to the general program of embedding theories of dynamical systems. Following our previous work on the stochastic embedding theory developed with Darses [C. R. Acad. Sci. Ser. I: Math 342, 333 (2006); (preprint IHES 06/27, p. 87, 2006)], we define the fractional embedding of differential operators and ordinary differential equations. We construct an operator combining in a symmetric way the left and right (Riemann-Liouville) fractional derivatives. For Lagrangian systems, our method provides a fractional Euler-Lagrange equation. We prove, developing the corresponding fractional calculus of variations, that such equation can be derived via a fractional least-action principle. We then obtain naturally a fractional Noether theorem and a fractional Hamiltonian formulation of fractional Lagrangian systems. All these constructions are coherents, i.e., the embedding procedure is compatible with the fractional calculus of variations. We then extend our results to cover the Ostrogradski...

Navier-Stokes equations with regularity in one direction

Abstract: We consider sufficient conditions for the regularity of Leray-Hopf solutions of the Navier-Stokes equations. We prove that if the third derivative of the velocity ∂u∕∂x3 belongs to the space Lts0Lxr0, where 2∕s0+3∕r0⩽2 and 9∕4⩽r0⩽3, then the solution is regular. This extends a result of Beirao da Veiga [Chin. Ann. Math., Ser. B 16, 407–412 (1995); C. R. Acad. Sci, Ser. I: Math. 321, 405–408 (1995)] by making a requirement only on one direction of the velocity instead of on the full gradient. The derivative ∂u∕∂x3 can be substituted with any directional derivative of u.

Lorentzian version of the noncommutative geometry of the standard model of particle physics

Abstract: A formulation of the noncommutative geometry for the standard model of particle physics with a Lorentzian signature metric is presented. The elimination of the fermion doubling in the Lorentzian case is achieved by a modification of Connes’ internal space geometry [“Gravity coupled with matter and the foundation of non-commutative geometry,” Commun. Math. Phys. 182, 155–176 (1996)] so that it has signature 6 (mod 8) rather than 0. The fermionic part of the Connes-Chamseddine spectral action can be formulated, and it is shown that it allows an extension with right-handed neutrinos and the correct mass terms for the seesaw mechanism of neutrino mass generation.

Generalized fractional Schrödinger equation with space-time fractional derivatives

Abstract: In this paper the generalized fractional Schrodinger equation with space and time fractional derivatives is constructed. The equation is solved for free particle and for a square potential well by the method of integral transforms, Fourier transform and Laplace transform, and the solution can be expressed in terms of Mittag-Leffler function. The Green function for free particle is also presented in this paper. Finally, we discuss the relationship between the cases of the generalized fractional Schrodinger equation and the ones in standard quantum.

Some solutions to the space fractional Schrödinger equation using momentum representation method

Abstract: The space fractional Schrodinger equation with linear potential, delta-function potential, and Coulomb potential is studied under momentum representation using Fourier transformation. By use of Mellin transform and its inverse transform, we obtain the energy levels and wave functions expressed in H function for a particle in linear potential field. The wave function expressed also by the H function and the unique energy level of the bound state for the particle of even parity state in delta-function potential well, which is proved to have no action on the particle of odd parity state, is also obtained. The integral form of the wave functions for a particle in Coulomb potential field is shown and the corresponding energy levels which have been discussed in Laskin’s paper [Phys. Rev. E 66, 056108 (2002)] are proved to satisfy an equality of infinite limit of the H function. All of these results contain the ones of the standard quantum mechanics as their special cases.

New integrable hierarchy, its parametric solutions, cuspons, one-peak solitons, and M/W-shape peak solitons

Abstract: In this paper, we propose a new completely integrable hierarchy. Particularly in the hierarchy we draw two new soliton equations: (1) mt=12(1∕m2)xxx−12(1∕m2)x; (2) mt+mx(u2−ux2)+2m2ux=0, m=u−uxx. The first one is the second positive member in the hierarchy while the second one is the second negative member in the hierarchy. Both equations can be derived from the two-dimensional Euler equation by using the approximation procedure. All equations in the hierarchy are proven to have bi-Hamiltonian operators and Lax pairs through solving a crucial matrix equation. Moreover, we develop parametric solutions of the entire hierarchy through constructing two kinds of constraints; one is for all negative members of the hierarchy on a symplectic submanifold, and the other is for all positive members in the standard symplectic space. The most interesting things are both equations possess new type of peaked solitons—continuous and piecewise smooth “W-/M-shape peak” soliton solutions. In addition, we find new cusp solit...

Mutually unbiased bases and Hadamard matrices of order six

Abstract: We report on a search for mutually unbiased bases (MUBs) in six dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organize our results. Finally, we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.

Weighted complex projective 2-designs from bases : Optimal state determination by orthogonal measurements

Abstract: We introduce the problem of constructing weighted complex projective 2-designs from the union of a family of orthonormal bases. If the weight remains constant across elements of the same basis, then such designs can be interpreted as generalizations of complete sets of mutually unbiased bases, being equivalent whenever the design is composed of d+1 bases in dimension d. We show that, for the purpose of quantum state determination, these designs specify an optimal collection of orthogonal measurements. Using highly nonlinear functions on Abelian groups, we construct explicit examples from d+2 orthonormal bases whenever d+1 is a prime power, covering dimensions d=6, 10, and 12, for example, where no complete sets of mutually unbiased bases have thus far been found.

Kazhdan-Lusztig-dual quantum group for logarithimic extensions of Virasoro minimal models

Abstract: We derive and study a quantum group gp,q that is Kazhdan-Lusztig dual to the W-algebra Wp,q of the logarithmic (p,q) conformal field theory model. The algebra Wp,q is generated by two currents W+(z) and W−(z) of dimension (2p−1)(2q−1) and the energy-momentum tensor T(z). The two currents generate a vertex-operator ideal R with the property that the quotient Wp,q∕R is the vertex-operator algebra of the (p,q) Virasoro minimal model. The number (2pq) of irreducible gp,q representations is the same as the number of irreducible Wp,q representations on which R acts nontrivially. We find the center of gp,q and show that the modular group representation on it is equivalent to the modular group representation on the Wp,q characters and “pseudocharacters.” The factorization of the gp,q ribbon element leads to a factorization of the modular group representation on the center. We also find the gp,q Grothendieck ring, which is presumably the “logarithmic” fusion of the (p,q) model.

Magnetic vortex filament flows

Abstract: We exhibit a variational approach to study the magnetic flow associated with a Killing magnetic field in dimension 3. In this context, the solutions of the Lorentz force equation are viewed as Kirchhoff elastic rods and conversely. This provides an amazing connection between two apparently unrelated physical models and, in particular, it ties the classical elastic theory with the Hall effect. Then, these magnetic flows can be regarded as vortex filament flows within the localized induction approximation. The Hasimoto transformation can be used to see the magnetic trajectories as solutions of the cubic nonlinear Schrodinger equation showing the solitonic nature of those.

Homological error correction: Classical and quantum codes

Abstract: We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming bound. In the quantum case, we show that for nonorientable surfaces it is impossible to construct homological codes based on qudits of dimension D>2, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension D. We give a method to obtain planar homological codes based on the construction of quantum codes on compact surfaces without boundaries. We show how the original Shor’s 9-qubit code can be visualized as a homological quantum code. We study the problem of constructing quantum codes with optimal encoding rate. In the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality. For homological quantum codes on surfaces of arbitrary genus we also construct a family of codes asymptotically attaining the maximum possible encoding rate. We provide the tools of homology group theory for graphs embedded on surfaces in a self-contained manner.

Logarithmic intertwining operators and W(2,2p−1) algebras

Abstract: For every p⩾2, we obtained an explicit construction of a family of W(2,2p−1) modules, which decompose as direct sum of simple Virasoro algebra modules. Furthermore, we classified all irreducible self-dual W(2,2p−1) modules, we described their internal structure, and computed their graded dimensions. In addition, we constructed certain hidden logarithmic intertwining operators among two ordinary and one logarithmic W(2,2p−1) modules. This work, in particular, gives a mathematically precise formulation and interpretation of what physicists have been referring to as “logarithmic conformal field theory” of central charge cp,1=1−[6(p−1)2∕p], p⩾2. Our explicit construction can be easily applied for computations of correlation functions. Techniques from this paper can be used to study the triplet vertex operator algebra W(2,(2p−1)3) and other logarithmic models.

Dual monogamy inequality for entanglement

Abstract: We establish duality for monogamy of entanglement: whereas monogamy of entanglement inequalities provide an upper bound for bipartite sharability of entanglement in a multipartite system, as quantified by linear entropy, we prove that the same quantity (namely, linear entropy) provides a lower bound for distribution of bipartite entanglement in a multipartite system Our theorem for monogamy of entanglement is used to establish relations between bipartite entanglement that separate one qubit from the rest versus separating two qubits from the rest

Transport properties of quasi-free fermions

Abstract: Using the scattering approach to the construction of Non-Equilibrium Steady States proposed by Ruelle we study the transport properties of systems of independent electrons. We show that Landauer-Buttiker and Green-Kubo formulas hold under very general conditions.

Duality and exact correlations for a model of heat conduction

Abstract: We study a model of heat conduction with stochastic diffusion of energy. We obtain a dual particle process which describes the evolution of all the correlation functions. An exact expression for the covariance of the energy exhibits long-range correlations in the presence of a current. We discuss the formal connection of this model with the simple symmetric exclusion process.

Many-body wave scattering by small bodies and applications

Abstract: A rigorous reduction of the many-body wave scattering problem to solving a linear algebraic system is given bypassing solving the usual system of integral equation. The limiting case of infinitely many small particles embedded into a medium is considered and the limiting equation for the field in the medium is derived. The impedance boundary conditions are imposed on the boundaries of small bodies. The case of Neumann boundary conditions (acoustically hard particles) is also considered. Applications to creating materials with a desired refraction coefficient are given. It is proved that by embedding a suitable number of small particles per unit volume of the original material with suitable boundary impedances, one can create a new material with any desired refraction coefficient. The governing equation is a scalar Helmholtz equation, which one obtains by Fourier transforming the wave equation.

Tikekar superdense stars in electric fields

Abstract: We present exact solutions to the Einstein-Maxwell system of equations with a specified form of the electric field intensity by assuming that the hypersurface {t=constant} are spheroidal. The solution of the Einstein-Maxwell system is reduced to a recurrence relation with variable rational coefficients which can be solved in general using mathematical induction. New classes of solutions of linearly independent functions are obtained by restricting the spheroidal parameter K and the electric field intensity parameter α. Consequently, it is possible to find exact solutions in terms of elementary functions, namely, polynomials and algebraic functions. Our result contains models found previously including the superdense Tikekar neutron star model [J. Math. Phys. 31, 2454 (1990)] when K=−7 and α=0. Our class of charged spheroidal models generalize the uncharged isotropic Maharaj and Leach solutions [J. Math. Phys. 37, 430 (1996)]. In particular, we find an explicit relationship directly relating the spheroidal...

From Dirac spinor fields to eigenspinoren des ladungskonjugationsoperators

Abstract: Dual-helicity eigenspinors of the charge conjugation operator [eigenspinoren des ladungskonjugationsoperators (ELKO) spinor fields] belong—together with Majorana spinor fields—to a wider class of spinor fields, the so-called flagpole spinor fields, corresponding to the class (5), according to Lounesto spinor field classification based on the relations and values taken by their associated bilinear covariants. There exists only six such disjoint classes: the first three corresponding to Dirac spinor fields, and the other three, respectively, corresponding to flagpole, flag-dipole, and Weyl spinor fields. This paper is devoted to investigate and provide the necessary and sufficient conditions to map Dirac spinor fields to ELKO, in order to naturally extend the standard model to spinor fields possessing mass dimension 1. As ELKO is a prime candidate to describe dark matter, an adequate and necessary formalism is introduced and developed here, to better understand the algebraic, geometric, and physical propert...

Biorthogonal quantum systems

Abstract: Models of PT symmetric quantum mechanics provide examples of biorthogonal quantum systems. The latter incorporate all the structure of PT symmetric models, and allow for generalizations, especially in situations where the PT construction of the dual space fails. The formalism is illustrated by a few exact results for models of the form H=(p+ν)2+∑k>0μkexp(ikx). In some nontrivial cases, equivalent Hermitian theories are obtained and shown to be very simple: They are just free (chiral) particles. Field theory extensions are briefly considered.

An approximate solution of Dirac-Hulthén problem with pseudospin and spin symmetry for any κ state

Abstract: For any spin-orbit quantum number κ, the analytical solutions of the Dirac equation are presented for the Hulthen potential by applying an approximation to spin-orbit coupling potential for the case of spin symmetry, Δ(r)=C=const, and pseudospin symmetry Σ(r)=C=const. The bound state energy eigenvalues and the corresponding spinors are obtained in the closed forms.

Analytical study of certain magnetohydrodynamic-α models

Abstract: In this paper we present an analytical study of a subgrid scale turbulence model of the three-dimensional magnetohydrodynamic (MHD) equations, inspired by the Navier-Stokes-α (also known as the viscous Camassa-Holm equations or the Lagrangian-averaged Navier-Stokes-α model). Specifically, we show the global well-posedness and regularity of solutions of a certain MHD-α model (which is a particular case of the Lagrangian averaged magnetohydrodynamic-α model without enhancing the viscosity for the magnetic field). We also introduce other subgrid scale turbulence models, inspired by the Leray-α and the modified Leray-α models of turbulence. Finally, we discuss the relation of the MHD-α model to the MHD equations by proving a convergence theorem, that is, as the length scale α tends to zero, a subsequence of solutions of the MHD-α equations converges to a certain solution (a Leray-Hopf solution) of the three-dimensional MHD equations.

Uncertainty principle and quantum Fisher information. II.

Abstract: Heisenberg and Schrodinger uncertainty principles give lower bounds for the product of variances Varρ(A)Varρ(B) if the observables A,B are not compatible, namely, if the commutator [A,B] is not zero. In this paper, we prove an uncertainty principle in Schrodinger form where the bound for the product of variances Varρ(A)Varρ(B) depends on the area spanned by the commutators i[ρ,A] and i[ρ,B] with respect to an arbitrary quantum version of the Fisher information.

Reduced models for fluid flows with strong constraints

Abstract: The presence of a dominant balance in the equations for fluid flow can be exploited to derive an asymptotically exact but simpler set of governing equations. These permit semianalytical and/or numerical explorations of parameter regimes that would otherwise be inaccessible to direct numerical simulation. The derivation of the resulting reduced models is illustrated here for (i) rapidly rotating convection in a plane layer, (ii) convection in a strong magnetic field, and (iii) the magnetorotational instability in accretion disks and the results used to extend our understanding of these systems in the strongly nonlinear regime.

Simple finite non-Abelian flavor groups

Abstract: The recently measured unexpected neutrino mixing patterns have caused a resurgence of interest in the study of finite flavor groups with two- and three-dimensional irreducible representations. This paper details the mathematics of the two finite simple groups with such representations, the icosahedral group A5, a subgroup of SO(3), and PSL2(7), a subgroup of SU(3).

Maximally symmetric vector propagator

Abstract: We derive the propagator for a massive vector field on a de Sitter background of arbitrary dimension. This propagator is de Sitter invariant and possesses the proper flat space-time and massless limits. Moreover, the retarded Green’s function inferred from it produces the correct classical response to a test source. Our result is expressed in a tensor basis which is convenient for performing quantum-field-theory computations using dimensional regularization.