Showing papers in "Journal of Mathematical Physics in 2011"
TL;DR: In this article, the authors derived the cyclotomic harmonic polylogarithms and harmonic sums and studied their algebraic and structural relations, and derived the basis representations for weight w = 1,2 sums up to cyclotomy l = 20.
Abstract: The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin transforms of Poincare- iterated integrals including denominators of higher cyclotomic polynomials. We derive the cyclotomic harmonic polylogarithms and harmonic sums and study their algebraic and structural relations. The analytic continuation of cyclotomic harmonic sums to complex values of N is performed using analytic representations. We also consider special values of the cyclotomic harmonic polylogarithms at argument x = 1, resp., for the cyclotomic harmonic sums at N → ∞, which are related to colored multiple zeta values, deriving various of their relations, based on the stuffle and shuffle alge bras and three multiple ar- gument relations. We also consider infinite generalized nested harmonic sums at roots of unity which are related to the infinite cyclotomic harmonic sums. Basis representations are derived for weight w = 1,2 sums up to cyclotomy l = 20.
265 citations
TL;DR: In this article, the authors studied various methods to generate ensembles of random density matrices of a fixed size N, obtained by partial trace of pure states on composite systems.
Abstract: We study various methods to generate ensembles of random density matrices of a fixed size N, obtained by partial trace of pure states on composite systems. Structured ensembles of random pure states, invariant with respect to local unitary transformations are introduced. To analyze statistical properties of quantum entanglement in bi-partite systems we analyze the distribution of Schmidt coefficients of random pure states. Such a distribution is derived in the case of a superposition of k random maximally entangled states. For another ensemble, obtained by performing selective measurements in a maximally entangled basis on a multi-partite system, we show that this distribution is given by the Fuss-Catalan law and find the average entanglement entropy. A more general class of structured ensembles proposed, containing also the case of Bures, forms an extension of the standard ensemble of structureless random pure states, described asymptotically, as N → ∞, by the Marchenko-Pastur distribution.
152 citations
TL;DR: In this article, Tsirelson's problem concerning the set of quantum correlations and Connes' embedding problem on finite approximations in von Neumann algebras are essentially equivalent.
Abstract: We show that Tsirelson's problem concerning the set of quantum correlations and Connes' embedding problem on finite approximations in von Neumann algebras (known to be equivalent to Kirchberg's QWEP conjecture) are essentially equivalent. Specifically, Tsirelson's problem asks whether the set of bipartite quantum correlations generated between tensor product separated systems is the same as the set of correlations between commuting C*-algebras. Connes' embedding problem asks whether any separable II1 factor is a subfactor of the ultrapower of the hyperfinite II1 factor. We show that an affirmative answer to Connes' question implies a positive answer to Tsirelson's. Conversely, a positive answer to a matrix valued version of Tsirelson's problem implies a positive one to Connes' problem.
139 citations
TL;DR: In this paper, the authors investigated the geometry of the space of N-valent SU(2) intertwiners and proposed a set of holomorphic operators acting on this space and a new set of coherent states which are covariant under U(N) transformations.
Abstract: We investigate the geometry of the space of N-valent SU(2) intertwiners. We propose a new set of holomorphic operators acting on this space and a new set of coherent states which are covariant under U(N) transformations. These states are labeled by elements of the Grassmannian GrN, 2, they possess a direct geometrical interpretation in terms of framed polyhedra and are shown to be related to the well-known coherent intertwiners.
130 citations
TL;DR: In this article, the authors studied the asymptotic position distribution of general quantum walks on a lattice, including walks with a random coin, which is chosen from step to step by a general Markov chain.
Abstract: We study the asymptotic position distribution of general quantum walks on a lattice, including walks with a random coin, which is chosen from step to step by a general Markov chain. In the unitary (i.e., nonrandom) case, we allow any unitary operator which commutes with translations and couples only sites at a finite distance from each other. For example, a single step of the walk could be composed of any finite succession of different shift and coin operations in the usual sense, with any lattice dimension and coin dimension. We find ballistic scaling and establish a direct method for computing the asymptotic distribution of position divided by time, namely as the distribution of the discrete time analog of the group velocity. In the random case, we let a Markov chain (control process) pick in each step one of finitely many unitary walks, in the sense described above. In ballistic order, we find a nonrandom drift which depends only on the mean of the control process and not on the initial state. In diffusive scaling, the limiting distribution is asymptotically Gaussian, with a covariance matrix (diffusion matrix) depending on momentum. The diffusion matrix depends not only on the mean but also on the transition rates of the control process. In the nonrandom limit, i.e., when the coins chosen are all very close or the transition rates of the control process are small, leading to long intervals of ballistic evolution, the diffusion matrix diverges. Our method is based on spatial Fourier transforms, and the first and second order perturbation theory of the eigenvalue 1 of the transition operator for each value of the momentum.
123 citations
TL;DR: In this paper, a spin-12-particle moving on a one-dimensional lattice subject to disorder induced by a random, space-dependent quantum coin is studied, and sufficient conditions on the probability distribution of the coins such that the system exhibits dynamical localization are derived.
Abstract: We study a spin-12-particle moving on a one-dimensional lattice subject to disorder induced by a random, space-dependent quantum coin. The discrete time evolution is given by a family of random unitary quantum walk operators, where the shift operation is assumed to be deterministic. Each coin is an independent identically distributed random variable with values in the group of two-dimensional unitary matrices. We derive sufficient conditions on the probability distribution of the coins such that the system exhibits dynamical localization. Put differently, the tunneling probability between two lattice sites decays rapidly for almost all choices of random coins and after arbitrary many time steps with increasing distance. Our findings imply that this effect takes place if the coin is chosen at random from the Haar measure, or some measure continuous with respect to it, but also for a class of discrete probability measures which support consists of two coins, one of them being the Hadamard coin.
110 citations
TL;DR: In this article, the quantum group spin-foam model is defined and a finite partition function on a fixed triangulation is given, which is a spinfoam quantization of discrete gravity with a cosmological constant.
Abstract: We study the quantum group deformation of the Lorentzian EPRL spin-foam model. The construction uses the harmonic analysis on the quantum Lorentz group. We show that the quantum group spin-foam model so defined is free of the infra-red divergence, thus gives a finite partition function on a fixed triangulation. We expect this quantum group spin-foam model is a spin-foam quantization of discrete gravity with a cosmological constant.
100 citations
TL;DR: In this article, the Dirac equation for the generalized Morse potential with arbitrary spin-orbit quantum number κ was solved by using an improved approximation scheme to deal with the centrifugal (pseudo-centrifugal) term.
Abstract: By using an improved approximation scheme to deal with the centrifugal (pseudo-centrifugal) term, we solve the Dirac equation for the generalized Morse potential with arbitrary spin-orbit quantum number κ. In the presence of spin and pseudospin symmetry, the analytic bound state energy eigenvalues and the associated upper- and lower-spinor components of two Dirac particles are found by using the basic concepts of the Nikiforov-Uvarov method. We study the special cases when κ = ±1 (l=l=0, s-wave), the non-relativistic limit and the limit when α becomes zero (Kratzer potential model). The present solutions are compared with those obtained by other methods.
90 citations
TL;DR: In this article, a multisoliton system of nonlinear equations that generalizes the short pulse (SP) equation describing the propagation of ultra-short pulses in optical fibers is proposed.
Abstract: We propose a novel multi-component system of nonlinear equations that generalizes the short pulse (SP) equation describing the propagation of ultra-short pulses in optical fibers. By means of the bilinear formalism combined with a hodograph transformation, we obtain its multisoliton solutions in the form of a parametric representation. Notably, unlike the determinantal solutions of the SP equation, the proposed system is found to exhibit solutions expressed in terms of pfaffians. The proof of the solutions is performed within the framework of an elementary theory of determinants. The reduced 2-component system deserves a special consideration. In particular, we show by establishing a Lax pair that the system is completely integrable. The properties of solutions such as loop solitons and breathers are investigated in detail, confirming their solitonic behavior. A variant of the 2-component system is also discussed with its multisoliton solutions.
84 citations
TL;DR: In this paper, the moments of the eigenvalue densities of matrices in the Gaussian, Laguerre, and Jacobi ensembles for all the symmetry classes β ∈ {1, 2, 4} and finite matrix dimension n were derived.
Abstract: We develop a method to compute the moments of the eigenvalue densities of matrices in the Gaussian, Laguerre, and Jacobi ensembles for all the symmetry classes β ∈ {1, 2, 4} and finite matrix dimension n. The moments of the Jacobi ensembles have a physical interpretation as the moments of the transmission eigenvalues of an electron through a quantum dot with chaotic dynamics. For the Laguerre ensemble we also evaluate the finite n negative moments. Physically, they correspond to the moments of the proper delay times, which are the eigenvalues of the Wigner-Smith matrix. Our formulae are well suited to an asymptotic analysis as n → ∞.
73 citations
TL;DR: In this article, the first β-deformed corrections in the one-cut and the two-cut cases were presented, as well as two applications to supersymmetric gauge theories: the calculation of superpotentials in N=1 gauge theories, and the calculation vevs of surface operators in superconformal N=2 theories and their Liouville duals.
Abstract: We study matrix models in the β-ensemble by building on the refined recursion relation proposed by Chekhov and Eynard. We present explicit results for the first β-deformed corrections in the one-cut and the two-cut cases, as well as two applications to supersymmetric gauge theories: the calculation of superpotentials in N=1 gauge theories, and the calculation of vevs of surface operators in superconformal N=2 theories and their Liouville duals. Finally, we study the β-deformation of the Chern–Simons matrix model. Our results indicate that this model does not provide an appropriate description of the Ω-deformed topological string on the resolved conifold, and therefore that the β-deformation might provide a different generalization of topological string theory in toric Calabi–Yau backgrounds.
TL;DR: In this paper, the analogue of the relativistic Landau quantization in the Aharonov-Casher setup can be achieved in the Lorentz-symmetry violation background.
Abstract: Based on the discussions about the Aharonov-Casher effect in the Lorentz symmetry violation background, we show that the analogue of the relativistic Landau quantization in the Aharonov-Casher setup can be achieved in the Lorentz-symmetry violation background.
TL;DR: In this article, the authors explicitly determine all magnetic curves corresponding to the Killing magnetic fields on the 3D Euclidean space and show that these curves correspond to the magnetic fields.
Abstract: We explicitly determine all magnetic curves corresponding to the Killing magnetic fields on the 3-dimensional Euclidean space.
TL;DR: In this article, the graviton propagator is constructed on de Sitter background in exact de Donder gauge, and it is shown that the propagator must break De Sitter invariance, just like the massless, minimally coupled scalar.
Abstract: We construct the graviton propagator on de Sitter background in exact de Donder gauge. We prove that it must break de Sitter invariance, just like the propagator of the massless, minimally coupled scalar. Our explicit solutions for its two scalar structure functions preserve spatial homogeneity and isotropy so that the propagator can be used within the larger context of inflationary cosmology; however, it is simple to alter the residual symmetry. Because our gauge condition is de Sitter invariant (although no solution for the propagator can be) renormalization should be simpler using this propagator than one based on a noncovariant gauge. It remains to be seen how other computational steps compare.
TL;DR: In this paper, Dirac spinors in Bianchi type-I cosmological models with torsional f(R)-gravity were studied, and it was shown that the resulting dynamic behavior of the universe depends on the particular choice of function f (R), and that the singularity problem can be avoided due to the presence of torsion.
Abstract: We study Dirac spinors in Bianchi type-I cosmological models, within the framework of torsional f(R)-gravity We find four types of results: the resulting dynamic behavior of the universe depends on the particular choice of function f(R); some f(R) models do not isotropize and have no Einstein limit, so that they have no physical significance, whereas for other f(R) models isotropization and Einsteinization occur, and so they are physically acceptable, suggesting that phenomenological arguments may select f(R) models that are physically meaningful; the singularity problem can be avoided, due to the presence of torsion; the general conservation laws holding for f(R)-gravity with torsion ensure the preservation of the Hamiltonian constraint, so proving that the initial value problem is well-formulated for these models
TL;DR: In this article, the authors develop the off-shell nilpotent finite field dependent Becchi-Rouet-Stora-Tyutin (BRST) transformations and show that for different choices of the BRST parameter these transformations connect the generating functionals corresponding to different effective theories.
Abstract: We develop the off-shell nilpotent finite field dependent Becchi–Rouet–Stora–Tyutin (BRST) transformations and show that for different choices of the finite field dependent parameter these transformations connect the generating functionals corresponding to different effective theories. We also construct both on-shell and off-shell finite field dependent anti-BRST transformations for Yang–Mills theories and show that these transformations play the similar role in connecting different generating functionals of different effective theories. Analogous to the finite field dependent BRST transformations, the nontrivial Jacobians of the path integral measure which arise due to the finite field dependent anti-BRST transformations are responsible for the new results. We consider several explicit examples in each case to demonstrate the results.
TL;DR: In this paper, the authors apply Hilbert's projective metric in the context of quantum information theory and obtain bounds on measures for statistical distinguishability of quantum states and on the decrease of entanglement under protocols involving local quantum operations and classical communication or under other conepreserving operations.
Abstract: We introduce and apply Hilbert's projective metric in the context of quantum information theory. The metric is induced by convex cones such as the sets of positive, separable or positive partial transpose operators. It provides bounds on measures for statistical distinguishability of quantum states and on the decrease of entanglement under protocols involving local quantum operations and classical communication or under other cone-preserving operations. The results are formulated in terms of general cones and base norms and lead to contractivity bounds for quantum channels, for instance, improving Ruskai's trace-norm contraction inequality. A new duality between distinguishability measures and base norms is provided. For two given pairs of quantum states we show that the contraction of Hilbert's projective metric is necessary and sufficient for the existence of a probabilistic quantum operation that maps one pair onto the other. Inequalities between Hilbert's projective metric and the Chernoff bound, the ...
TL;DR: In this paper, a generalized integrable two-component system mt=[m(uxvx−uv+uvx−uxv)]x,nt=[n(u−uxx and n=v−vxx, where m =u−u2 and n = v−v xx, is proposed.
Abstract: In this paper, a new integrable two-component system, mt=[m(uxvx−uv+uvx−uxv)]x,nt=[n(uxvx−uv+uvx−uxv)]x, where m=u−uxx and n=v−vxx, is proposed. Our system is a generalized version of the integrable system mt=[m(ux2−u2)]x, which was shown having cusped solution (cuspon) and W/M-shape soliton solutions by Qiao [J. Math. Phys. 47, 112701 (2006). The new system is proven integrable not only in the sense of Lax-pair but also in the sense of geometry, namely, it describes pseudospherical surfaces. Accordingly, infinitely many conservation laws are derived through recursion relations. Furthermore, exact solutions such as cuspons and W/M-shape solitons are also obtained.
TL;DR: In this article, it is shown that it is possible to generate an infinite set of rational extensions from every exceptional first category translationally shape invariant potential by using Darboux-Backlund transformations based on unphysical regular Riccati-Schrodinger functions which are obtained from specific symmetries associated with the considered family of potentials.
Abstract: We show that it is possible to generate an infinite set of solvable rational extensions from every exceptional first category translationally shape invariant potential. This is made by using Darboux-Backlund transformations based on unphysical regular Riccati-Schrodinger functions which are obtained from specific symmetries associated with the considered family of potentials.
TL;DR: In this paper, an algorithm for the explicit numerical calculation of SU(N) and SL(N,C) Clebsch-Gordan coefficients, based on the Gelfand-Tsetlin pattern calculus, is presented.
Abstract: We present an algorithm for the explicit numerical calculation of SU(N) and SL(N,C) Clebsch–Gordan coefficients, based on the Gelfand–Tsetlin pattern calculus. Our algorithm is well suited for numerical implementation; we include a computer code in an appendix. Our exposition presumes only familiarity with the representation theory of SU(2).
TL;DR: In this paper, a twisted version of the Yang-Baxter equation, called the Hom-Yang-baxter equation (HYBE), was introduced and several more classes of solutions of the HYBE were constructed.
Abstract: Motivated by recent work on Hom–Lie algebras, a twisted version of the Yang–Baxter equation, called the Hom–Yang–Baxter equation (HYBE), was introduced by Yau [J. Phys. A 42, 165202 (2009)]. In this paper, several more classes of solutions of the HYBE are constructed. Some of the solutions of the HYBE are closely related to the quantum enveloping algebra of sl(2), the Jones–Conway polynomial, and Yetter-Drinfel'd modules. Under some invertibility conditions, we construct a new infinite sequence of solutions of the HYBE from a given one.
TL;DR: In this article, the authors studied coordinate-invariance of some asymptotic invariants, such as the Arnowitt-Deser-Misner mass or the Chruściel-Herzlich momentum, given by an integral over a "boundary at infinity".
Abstract: We study coordinate-invariance of some asymptotic invariants, such as the Arnowitt-Deser-Misner mass or the Chruściel–Herzlich momentum, given by an integral over a “boundary at infinity.” When changing the coordinates at infinity, some terms in the change of integrand do not decay fast enough to have a vanishing integral at infinity; but they may be gathered in a divergence, thus having vanishing integral over any closed hypersurface. This fact could only be checked after direct calculation (and was called a “curious cancellation”). We give a conceptual explanation thereof.
TL;DR: In this article, the binary Bell polynomials are applied to succinctly construct bilinear formulism, Backlund transformations, Lax pairs, and Darboux covariant Lax pair for the (2+1)-dimensional breaking soliton equation.
Abstract: In this paper, the binary Bell polynomials are applied to succinctly construct bilinear formulism, bilinear Backlund transformations, Lax pairs, and Darboux covariant Lax pairs for the (2+1)-dimensional breaking soliton equation. An extra auxiliary variable is introduced to get the bilinear formulism. The infinitely local conservation laws of the equation are found by virtue of its Lax equation and a generalized Miura transformation. All conserved densities and fluxes are given with explicit recursion formulas.
TL;DR: In this article, the bilinear equations of the constrained BKP hierarchy were derived from the calculus of pseudodifferential operators, and the full hierarchy equations can be expressed in Hirota's bilinearly form characterized by the functions ρ, σ, and τ.
Abstract: We derive the bilinear equations of the constrained BKP hierarchy from the calculus of pseudodifferential operators. The full hierarchy equations can be expressed in Hirota's bilinear form characterized by the functions ρ, σ, and τ. Besides, we also give a modification of the original Orlov–Schulman additional symmetry to preserve the constrained form of the Lax operator for this hierarchy. The vector fields associated with the modified additional symmetry turn out to satisfy a truncated centerless Virasoro algebra.
TL;DR: In this article, the scaling of the average upward heat transport, the Nusselt number Nu, in terms of the non-dimensionalized temperature forcing, the Rayleigh number Ra, was investigated.
Abstract: We consider Rayleigh–Benard convection as modelled by the Boussinesq equations in the infinite-Prandtl-number limit. We are interested in the scaling of the average upward heat transport, the Nusselt number Nu, in terms of the non-dimensionalized temperature forcing, the Rayleigh number Ra. Experiments, asymptotics and heuristics suggest that Nu ∼ Ra1/3. This work is mostly inspired by two earlier rigorous work on upper bounds of Nu in terms of Ra. (1) The work of Constantin and Doering establishing Nu ≲ Ra1/3ln 2/3Ra with help of a (logarithmically failing) maximal regularity estimate in L∞ on the level of the Stokes equation. (2) The work of Doering, Reznikoff and the first author establishing Nu ≲ Ra1/3ln 1/3Ra with help of the background field method. The paper contains two results. (1) The background field method can be slightly modified to yield Nu ≲ Ra1/3ln 1/15Ra. (2) The estimates behind the background field method can be combined with the maximal regularity in L∞ to yield Nu ≲ Ra1/3ln 1/3ln Ra —...
TL;DR: In this article, it was shown that the existence of a SIC-POVM in dimension d is equivalent to a certain structure in the adjoint representation of gl (d,C).
Abstract: Examples of symmetric informationally complete positive operator-valued measures (SIC-POVMs) have been constructed in every dimension ⩽67. However, it remains an open question whether they exist in all finite dimensions. A SIC-POVM is usually thought of as a highly symmetric structure in quantum state space. However, its elements can equally well be regarded as a basis for the Lie algebra gl (d,C). In this paper we examine the resulting structure constants, which are calculated from the traces of the triple products of the SIC-POVM elements and which, it turns out, characterize the SIC-POVM up to unitary equivalence. We show that the structure constants have numerous remarkable properties. In particular we show that the existence of a SIC-POVM in dimension d is equivalent to the existence of a certain structure in the adjoint representation of gl (d,C). We hope that transforming the problem in this way, from a question about quantum state space to a question about Lie algebras, may help to make the existence problem tractable.
TL;DR: The authors showed that the linear group of automorphism of Hermitian matrices which preserves the set of separable states is generated by natural automorphisms: change of an orthonormal basis in each tensor factor.
Abstract: We show that the linear group of automorphism of Hermitian matrices which preserves the set of separable states is generated by natural automorphisms: change of an orthonormal basis in each tensor factor, partial transpose in each tensor factor, and interchanging two tensor factors of the same dimension. We apply our results to the preservers of the product numerical range.
TL;DR: This work generalizes existing work on exponential decay to an arbitrary decay not necessarily of an exponential or polynomial rate.
Abstract: It is by now well known that a necessary condition for the exponential (polynomial) decay of the energy of a problem arising in viscoelasticity is that the kernel (which appears in the memory term) itself be of exponential (polynomial) type. By a kernel of exponential (polynomial) type we mean that the product of this kernel with an exponential (polynomial) function is summable. Some researchers have started from this condition to seek other (sufficient) conditions ensuring exponential or polynomial decay of the energy. In this work we generalize these works to an arbitrary decay not necessarily of an exponential or polynomial rate.
TL;DR: In this paper, the authors proposed new Wightman functions as vacuum expectation values of products of field operators in the non-commutative space-time, and proved the spin-statistics theorem for the simplest case of a scalar field.
Abstract: We propose new Wightman functions as vacuum expectation values of products of field operators in the noncommutative space–time. These Wightman functions involve the ⋆-product among the fields, compatible with the twisted Poincare symmetry of the noncommutative quantum field theory (NC QFT). In the case of only space–space noncommutativity (θ0i = 0), we prove the CPT theorem using the noncommutative form of the Wightman functions. We also show that the spin-statistics theorem, demonstrated for the simplest case of a scalar field, holds in NC QFT within this formalism.
TL;DR: In this article, the authors present a procedure to construct (n + 1)-ary Hom-Nambu-Lie algebras with a generalized trace function, and the implications of the compatibility conditions, that are necessary for this construction, can be understood in terms of the kernel of the trace function and the range of the twisting maps.
Abstract: As n-ary operations, generalizing Lie and Poisson algebras, arise in many different physical contexts, it is interesting to study general ways of constructing explicit realizations of such multilinear structures. Generically, they describe the dynamics of a physical system, and there is a need of understanding their quantization. Hom-Nambu-Lie algebras provide a framework that might be an appropriate setting in which n-Lie algebras (n-ary Nambu-Lie algebras) can be deformed, and their quantization studied. We present a procedure to construct (n + 1)-ary Hom-Nambu-Lie algebras from n-ary Hom-Nambu-Lie algebras equipped with a generalized trace function. It turns out that the implications of the compatibility conditions, that are necessary for this construction, can be understood in terms of the kernel of the trace function and the range of the twisting maps. Furthermore, we investigate the possibility of defining (n + k)-Lie algebras from n-Lie algebras and a k-form satisfying certain conditions.