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Dec 2012

Volume 53, Issue 12, Articles (12xxxx)

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J. Math. Phys. 53, 122207 (2012); http://dx.doi.org/10.1063/1.4769382 (32 pages)

Wan-Jung Kuo, Gregory Quiroz, Gerardo Andres Paz-Silva, and Daniel A. Lidar
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Editorial: Wedge Causal Manifolds - An Unfinished Work of Hans-Jürgen Borchers

Stefan Hollands and Karl-Henning Rehren

J. Math. Phys. 53, 120401 (2012); http://dx.doi.org/10.1063/1.4772961 (3 pages)

Online Publication Date: 19 December 2012

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Abstract Unavailable
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01.30.Ww Editorials
02.40.Pc General topology
04.20.Gz Spacetime topology, causal structure, spinor structure
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back to top Quantum Mechanics (General and Nonrelativistic)

Partial order and a T0-topology in a set of finite quantum systems

A. Vourdas

J. Math. Phys. 53, 122101 (2012); http://dx.doi.org/10.1063/1.4764858 (22 pages) | Cited 1 time

Online Publication Date: 6 November 2012

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A “whole-part” theory is developed for a set of finite quantum systems Σ(n) with variables in math(n). The partial order “subsystem” is defined, by embedding various attributes of the system Σ(m) (quantum states, density matrices, etc.) into their counterparts in the supersystem Σ(n) (for m|n). The compatibility of these embeddings is studied. The concept of ubiquity is introduced for quantities which fit with this structure. It is shown that various entropic quantities are ubiquitous. The sets of various quantities become T0-topological spaces with the divisor topology, which encapsulates fundamental physical properties. These sets can be converted into directed-complete partial orders, by adding “top elements.” The continuity of various maps among these sets is studied.
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03.65.Ta Foundations of quantum mechanics; measurement theory
05.70.Ce Thermodynamic functions and equations of state
02.40.Pc General topology
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Group-theoretical derivation of angular momentum eigenvalues in spaces of arbitrary dimensions

Tamar Friedmann and C. R. Hagen

J. Math. Phys. 53, 122102 (2012); http://dx.doi.org/10.1063/1.4758928 (4 pages) | Cited 1 time

Online Publication Date: 13 November 2012

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The spectrum of the square of the angular momentum in arbitrary dimensions is derived using only group theoretical techniques. This is accomplished by application of the Lie algebra of the noncompact group O(2, 1).
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03.65.Fd Algebraic methods
02.20.Uw Quantum groups
02.20.Sv Lie algebras of Lie groups
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Integrability and supersymmetry of Schrödinger-Pauli equations for neutral particles

A. G. Nikitin

J. Math. Phys. 53, 122103 (2012); http://dx.doi.org/10.1063/1.4768464 (14 pages)

Online Publication Date: 28 November 2012

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Integrable quantum mechanical systems for neutral particles with spin ½ and nontrivial dipole momentum are classified. It is demonstrated that such systems give rise to new exactly solvable problems of quantum mechanics with clear physical content. Solutions for three of them are given in explicit form. The related symmetry algebras and superalgebras are discussed. The presented classification is restricted to two-dimensional systems, which admit matrix integrals of motion linear in momenta.
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03.65.Ge Solutions of wave equations: bound states
02.10.-v Logic, set theory, and algebra
03.65.Db Functional analytical methods
03.65.Fd Algebraic methods

The quantum Lane-Emden-type Kanai-Caldirola oscillators

V. H. L. Bessa and I. Guedes

J. Math. Phys. 53, 122104 (2012); http://dx.doi.org/10.1063/1.4768702 (8 pages)

Online Publication Date: 4 December 2012

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We obtain the exact wave functions of two Lane-Emden-type Kanai-Caldirola (LE-KC) oscillators by using the dynamical invariant method. To do so, we analytically solve the respective Milne-Pinney equation for each oscillator. We also calculate the uncertainty product, the transition probability and the quantum-mechanical energy expectation value for each oscillator. We compare the results with those of the well-known KC oscillator. The quantum-mechanical energy expectation value of the KC oscillator goes to zero faster than that of the LE-KC oscillators, indicating that the KC system is more damped than the LE-KC ones.
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03.65.Ge Solutions of wave equations: bound states
03.65.Ta Foundations of quantum mechanics; measurement theory

Supercharges in the hyper-Kähler with torsion supersymmetric sigma models

A. V. Smilga

J. Math. Phys. 53, 122105 (2012); http://dx.doi.org/10.1063/1.4769452 (9 pages)

Online Publication Date: 12 December 2012

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We construct explicitly classical and quantum supercharges satisfying the standard N = 4 supersymmetry algebra in the supersymmetric sigma models describing the motion over hyper-Kähler with torsion manifolds. One member of the family of superalgebras thus obtained is equivalent to the superalgebra derived and formulated earlier in purely mathematical framework.
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11.30.Pb Supersymmetry
02.10.De Algebraic structures and number theory
03.65.Fd Algebraic methods
11.10.Cd Axiomatic approach
11.10.Lm Nonlinear or nonlocal theories and models

Nonrelativistic quantum dynamics on a cone with and without a constraining potential

C. Filgueiras, E. O. Silva, and F. M. Andrade

J. Math. Phys. 53, 122106 (2012); http://dx.doi.org/10.1063/1.4770048 (9 pages) | Cited 2 times

Online Publication Date: 12 December 2012

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In this paper we investigate the bound state problem of nonrelativistic quantum particles on a conical surface. This kind of surface appears as a topological defect in ordinary semiconductors as well as in graphene sheets. Specifically, we compare and discuss the results stemming from two different approaches. In the first one, it is assumed that the charge carriers are bound to the surface by a constraining potential, while the second one is based on the Klein-Gordon type equation on surfaces, without the constraining potential. The main difference between both theories is the presence/absence of a potential which contains the mean curvature of a given surface. This fact changes the dependence of the bound states on the angular momentum l. Moreover, there are bound states that are absent in the Klein-Gordon theory, which instead appear in the Schrödinger one.
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03.65.Ge Solutions of wave equations: bound states
03.65.Ta Foundations of quantum mechanics; measurement theory
02.30.Hq Ordinary differential equations
02.40.Pc General topology

Characteristic particle trajectories for an eigenfunction

A. Elçi

J. Math. Phys. 53, 122107 (2012); http://dx.doi.org/10.1063/1.4770045 (38 pages)

Online Publication Date: 18 December 2012

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The jet space of the Schrödinger equation and Noether's theorem are used to determine a set of particle trajectories that belong exclusively to an eigenfunction. These trajectories depend on a vector field math which satisfies two partial differential equations. Characteristic trajectories exist for all eigenfunctions, including those for which probability current densities vanish. This paper mathematically demonstrates Einstein's assertion that a wave function is not a complete description of a particle's physical state.
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03.65.Ge Solutions of wave equations: bound states
03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
02.10.Ud Linear algebra
02.50.Cw Probability theory

Relation between primes and nontrivial zeros in the Riemann hypothesis; Legendre polynomials, modified zeta function and Schrödinger equation

Seongsoo Choi, J. W. Chung, and Kwang S. Kim

J. Math. Phys. 53, 122108 (2012); http://dx.doi.org/10.1063/1.4770050 (16 pages)

Online Publication Date: 18 December 2012

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We study the dependence between prime numbers and the real and imaginary parts of the nontrivial zeros of the Riemann zeta function. The Legendre polynomials and the partial derivatives of the Riemann zeta function are used to investigate the above dependence along with the Riemann hypothesis with physical interpretations. A modified zeta function with finite terms is defined as a new implement for the study of the zeta function and its zeros.
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02.10.De Algebraic structures and number theory
02.60.Lj Ordinary and partial differential equations; boundary value problems

Transport in the random Kronig-Penney model

Maxim Drabkin, Werner Kirsch, and Hermann Schulz-Baldes

J. Math. Phys. 53, 122109 (2012); http://dx.doi.org/10.1063/1.4769219 (15 pages)

Online Publication Date: 19 December 2012

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The Kronig-Penney model with random Dirac potentials on the lattice math has critical energies at which the Lyapunov exponent vanishes and the density of states has a van Hove singularity. This leads to a non-trivial quantum diffusion even though the spectrum is known to be pure-point.
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05.60.Gg Quantum transport
02.50.-r Probability theory, stochastic processes, and statistics
03.65.Yz Decoherence; open systems; quantum statistical methods
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
back to top Quantum Information and Computation

Entanglement transformation between sets of bipartite pure quantum states using local operations

H. F. Chau, Chi-Hang Fred Fung, Chi-Kwong Li, Edward Poon, and Nung-Sing Sze

J. Math. Phys. 53, 122201 (2012); http://dx.doi.org/10.1063/1.4765298 (11 pages)

Online Publication Date: 8 November 2012

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Alice and Bob are given an unknown initial state chosen from a set of pure quantum states. Their task is to transform the initial state to a corresponding final pure state using local operations only. We prove necessary and sufficient conditions on the existence of such a transformation. We also provide efficient algorithms that can quickly rule out the possibility of transforming a set of initial states to a set of final states.
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03.67.Mn Entanglement measures, witnesses, and other characterizations

Isolated Hadamard matrices from mutually unbiased product bases

Daniel McNulty and Stefan Weigert

J. Math. Phys. 53, 122202 (2012); http://dx.doi.org/10.1063/1.4764884 (16 pages) | Cited 1 time

Online Publication Date: 28 November 2012

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A new construction of complex Hadamard matrices of composite order d = pq, with primes p, q, is presented which is based on pairs of mutually unbiased bases containing only product states. For product dimensions d < 100, we illustrate the method by deriving many previously unknown complex Hadamard matrices. We obtain at least 12 new isolated matrices of Butson type, with orders ranging from 9 to 91.
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02.10.Yn Matrix theory

Extremality conditions for generalized channels

Anna Jenčová

J. Math. Phys. 53, 122203 (2012); http://dx.doi.org/10.1063/1.4764885 (14 pages)

Online Publication Date: 28 November 2012

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A generalized channel is a completely positive map that preserves trace on a given subspace. We find conditions under which a generalized channel with respect to a positively generated subspace J is an extreme point in the set of all such generalized channels. As a special case, this yields extremality conditions for quantum protocols. In particular, we obtain new extremality conditions for quantum 1-testers with 2 outcomes, which correspond to yes/no measurements on the set of quantum channels.
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03.67.Hk Quantum communication
03.67.Lx Quantum computation architectures and implementations
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Operator extension of strong subadditivity of entropy

Isaac H. Kim

J. Math. Phys. 53, 122204 (2012); http://dx.doi.org/10.1063/1.4769176 (3 pages)

Online Publication Date: 30 November 2012

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We prove an operator inequality that extends strong subadditivity of entropy: after taking a trace, the operator inequality becomes the strong subadditivity of entropy.
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05.70.Ce Thermodynamic functions and equations of state
02.30.Tb Operator theory

Quantum state discrimination bounds for finite sample size

Koenraad M. R. Audenaert, Milán Mosonyi, and Frank Verstraete

J. Math. Phys. 53, 122205 (2012); http://dx.doi.org/10.1063/1.4768252 (23 pages)

Online Publication Date: 7 December 2012

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In the problem of quantum state discrimination, one has to determine by measurements the state of a quantum system, based on the a priori side information that the true state is one of the two given and completely known states, ρ or σ. In general, it is not possible to decide the identity of the true state with certainty, and the optimal measurement strategy depends on whether the two possible errors (mistaking ρ for σ, or the other way around) are treated as of equal importance or not. Results on the quantum Chernoff and Hoeffding bounds and the quantum Stein's lemma show that, if several copies of the system are available then the optimal error probabilities decay exponentially in the number of copies, and the decay rate is given by a certain statistical distance between ρ and σ (the Chernoff distance, the Hoeffding distances, and the relative entropy, respectively). While these results provide a complete solution to the asymptotic problem, they are not completely satisfying from a practical point of view. Indeed, in realistic scenarios one has access only to finitely many copies of a system, and therefore it is desirable to have bounds on the error probabilities for finite sample size. In this paper we provide finite-size bounds on the so-called Stein errors, the Chernoff errors, the Hoeffding errors, and the mixed error probabilities related to the Chernoff and the Hoeffding errors.
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03.65.Aa Quantum systems with finite Hilbert space
03.65.Fd Algebraic methods
02.50.Cw Probability theory
05.70.Ce Thermodynamic functions and equations of state

Some applications of hypercontractive inequalities in quantum information theory

Ashley Montanaro

J. Math. Phys. 53, 122206 (2012); http://dx.doi.org/10.1063/1.4769269 (15 pages) | Cited 3 times

Online Publication Date: 10 December 2012

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Hypercontractive inequalities have become important tools in theoretical computer science and have recently found applications in quantum computation. In this note we discuss how hypercontractive inequalities, in various settings, can be used to obtain (fairly) concise proofs of several results in quantum information theory: a recent lower bound of Lancien and Winter on the bias achievable by local measurements which are 4-designs; spectral concentration bounds for k-local Hamiltonians; and a recent result of Pellegrino and Seoane-Sepúlveda giving general lower bounds on the classical bias obtainable in multiplayer XOR games.
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03.67.Lx Quantum computation architectures and implementations
03.65.Ta Foundations of quantum mechanics; measurement theory

Universality proof and analysis of generalized nested Uhrig dynamical decoupling

Wan-Jung Kuo, Gregory Quiroz, Gerardo Andres Paz-Silva, and Daniel A. Lidar

J. Math. Phys. 53, 122207 (2012); http://dx.doi.org/10.1063/1.4769382 (32 pages)

Online Publication Date: 18 December 2012

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Nested Uhrig dynamical decoupling (NUDD) is a highly efficient quantum error suppression scheme that builds on optimized single axis UDD sequences. We prove the universality of NUDD and analyze its suppression of different error types in the setting of generalized control pulses. We present an explicit lower bound for the decoupling order of each error type, which we relate to the sequence orders of the nested UDD layers. We find that the error suppression capabilities of NUDD are strongly dependent on the parities and relative magnitudes of all nested UDD sequence orders. This allows us to predict the optimal arrangement of sequence orders. We test and confirm our analysis using numerical simulations.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.67.Lx Quantum computation architectures and implementations

Linear maps preserving maximal deviation and the Jordan structure of quantum systems

Jan Hamhalter

J. Math. Phys. 53, 122208 (2012); http://dx.doi.org/10.1063/1.4771671 (10 pages)

Online Publication Date: 19 December 2012

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In the algebraic approach to quantum theory, a quantum observable is given by an element of a Jordan algebra and a state of the system is modelled by a normalized positive functional on the underlying algebra. Maximal deviation of a quantum observable is the largest statistical deviation one can obtain in a particular state of the system. The main result of the paper shows that each linear bijective transformation between JBW algebras preserving maximal deviations is formed by a Jordan isomorphism or a minus Jordan isomorphism perturbed by a linear functional multiple of an identity. It shows that only one numerical statistical characteristic has the power to determine the Jordan algebraic structure completely. As a consequence, we obtain that only very special maps can preserve the diameter of the spectra of elements. Nonlinear maps preserving the pseudometric given by maximal deviation are also described. The results generalize hitherto known theorems on preservers of maximal deviation in the case of self-adjoint parts of von Neumann algebras proved by Molnár.
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03.65.Fd Algebraic methods
03.65.Ta Foundations of quantum mechanics; measurement theory
02.50.-r Probability theory, stochastic processes, and statistics
02.60.-x Numerical approximation and analysis

Invariant measures on multimode quantum Gaussian states

C. Lupo, S. Mancini, A. De Pasquale, P. Facchi, G. Florio, and S. Pascazio

J. Math. Phys. 53, 122209 (2012); http://dx.doi.org/10.1063/1.4768712 (19 pages)

Online Publication Date: 20 December 2012

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We derive the invariant measure on the manifold of multimode quantum Gaussian states, induced by the Haar measure on the group of Gaussian unitary transformations. To this end, by introducing a bipartition of the system in two disjoint subsystems, we use a parameterization highlighting the role of nonlocal degrees of freedom—the symplectic eigenvalues—which characterize quantum entanglement across the given bipartition. A finite measure is then obtained by imposing a physically motivated energy constraint. By averaging over the local degrees of freedom we finally derive the invariant distribution of the symplectic eigenvalues in some cases of particular interest for applications in quantum optics and quantum information.
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03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
02.30.Uu Integral transforms
03.65.Fd Algebraic methods
02.10.Ud Linear algebra
back to top Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

Noncommuting local common causes for correlations violating the Clauser–Horne inequality

Gábor Hofer-Szabó and Péter Vecsernyés

J. Math. Phys. 53, 122301 (2012); http://dx.doi.org/10.1063/1.4763468 (12 pages) | Cited 1 time

Online Publication Date: 6 November 2012

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In the paper, the EPR-Bohm scenario will be reproduced in an algebraic quantum field theoretical setting with locally finite degrees of freedom. It will be shown that for a set of spatially separated correlating events (projections) maximally violating the Clauser–Horne inequality there can be given a common causal explanation if commutativity is abandoned between the common cause and the correlating events. Moreover, the noncommuting common cause will be local and supported in the common past of the correlating events.
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11.10.Nx Noncommutative field theory
03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)

Full counting statistics in the resonant-level model

Denis Bernard and Benjamin Doyon

J. Math. Phys. 53, 122302 (2012); http://dx.doi.org/10.1063/1.4763471 (25 pages)

Online Publication Date: 6 November 2012

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We derive the large deviation function, which provides the large-time full counting statistics for the charge transfer, in the non-equilibrium steady state of the resonant-level model. The general form of this function in free fermion models, in terms of transmission coefficients, was proposed by Levitov and Lesovik in 1993 using a particular measurement set-up involving an interacting spin. It was later suggested to hold as well for a proper quantum mechanical measurement of the transferred charge. We give a precise proof of both statements in the resonant-level model. We first give a full description of the model and its steady state. That is, we explain how the decoupled system prepared with a charge differential evolves, with the impurity coupling, towards the Hershfield non-equilibrium density matrix, in the sense of averages of finitely supported operators. We describe how this holds both for the usual resonant-level model with a point-like impurity, and for a regularized model with an impurity spread on a finite region, shedding light on subtleties associated to the point-like impurity. We then prove Levitov-Lesovik formula by recasting the problem into calculating averages of finitely supported operators.
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05.30.Fk Fermion systems and electron gas
03.65.Ta Foundations of quantum mechanics; measurement theory

Non-Grassmann mechanical model of the Dirac equation

A. A. Deriglazov, B. F. Rizzuti, G. P. Zamudio, and P. S. Castro

J. Math. Phys. 53, 122303 (2012); http://dx.doi.org/10.1063/1.4759500 (31 pages)

Online Publication Date: 8 November 2012

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We construct a new example of the spinning-particle model without Grassmann variables. The spin degrees of freedom are described on the base of an inner anti-de Sitter space. This produces both Γμ and Γμν-matrices in the course of quantization. Canonical quantization of the model implies the Dirac equation. We present the detailed analysis of both the Lagrangian and the Hamiltonian formulations of the model and obtain the general solution to the classical equations of motion. Comparing Zitterbewegung of the spatial coordinate with the evolution of spin, we ask on the possibility of space-time interpretation for the inner space of spin. We enumerate similarities between our analogous model of the Dirac equation and the two-body system subject to confining potential which admits only the elliptic orbits of the order of de Broglie wavelength. The Dirac equation dictates the perpendicularity of the elliptic orbits to the direction of center-of-mass motion.
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03.65.Pm Relativistic wave equations
02.10.Yn Matrix theory

Construction of Lax connections by exponentiation

Niklas Beisert and Florian Lücker

J. Math. Phys. 53, 122304 (2012); http://dx.doi.org/10.1063/1.4769824 (14 pages)

Online Publication Date: 14 December 2012

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We propose a method for constructing the Lax connection of two-dimensional relativistic integrable sigma models on coset spaces by means of exponentiation of a suitable operator. We derive a simple quadratic relation that this operator must satisfy for an entire one-parameter family of connections to be flat.
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11.10.Lm Nonlinear or nonlocal theories and models
11.10.Cd Axiomatic approach
back to top General Relativity and Gravitation

Some necessary and sufficient conditions for admitting a continuous sphere order representation of two-dimensional space-times

M. Vatandoost and Y. Bahrampour

J. Math. Phys. 53, 122501 (2012); http://dx.doi.org/10.1063/1.4761822 (8 pages)

Online Publication Date: 8 November 2012

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A space-time may be regarded as a partially ordered set by causal relations. Recently, as a poset, a sphere order representation of a space-time is considered and some interesting results are obtained. In this paper, we define a weakly causally convex open set and characterize connected open subsets of the n-dimensional Minkowski space-time which admit a continuous n-sphere order representation. Then, some necessary and sufficient conditions for admitting a continuous two-sphere order representation of two-dimensional space-times, are given. As a result, it is shown that any simply connected and causally simple space-time of dimension two can be causally isomorphically imbedded into the two-dimensional Minkowski space-time.
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04.20.Gz Spacetime topology, causal structure, spinor structure

Graviton propagator in a general invariant gauge on de Sitter

P. J. Mora, N. C. Tsamis, and R. P. Woodard

J. Math. Phys. 53, 122502 (2012); http://dx.doi.org/10.1063/1.4764882 (13 pages) | Cited 2 times

Online Publication Date: 26 November 2012

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We construct the graviton propagator on de Sitter background in the one parameter family of exact, de Sitter invariant gauges. Our result takes the form of a universal spin two part and a gauge dependent spin zero part. Scalar equations are derived for the structure functions of each part. There is no de Sitter invariant solution for either structure function, although the de Sitter breaking contribution to the spin zero part may drop out for certain choices of the gauge parameter. Our results imply that de Sitter breaking is universal for the graviton propagator, and hence that there is an error in the contrary results derived by analytic continuation of average gauge fixing techniques.
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04.60.Bc Phenomenology of quantum gravity
14.70.Kv Gravitons
11.15.-q Gauge field theories
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