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Jun 2005

Volume 46, Issue 6, Articles (06xxxx)

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Scattering kernel for polyatomic molecules

J. Gilbert Méolans and S. Kokou Dadzie

J. Math. Phys. 46, 062101 (2005); http://dx.doi.org/10.1063/1.1904703 (10 pages) | Cited 2 times

Online Publication Date: 12 May 2005

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A polyatomic scattering kernel phenomenologically presented in a previous paper is derived from an integral operator formulation. The five parameters involved in the scattering kernel expression are shown to be equal to the accommodation coefficients of various fluxes at the wall, namely, the fluxes of the three components of the momentum and the fluxes of the rotational and vibrational energies of molecules. Under its present form the model is especially convenient for the diatomic molecules.
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34.50.-s Scattering of atoms and molecules
33.15.Mt Rotation, vibration, and vibration-rotation constants
02.30.Rz Integral equations

On the infimum of quantum effects

Aurelian Gheondea, Stanley Gudder, and Peter Jonas

J. Math. Phys. 46, 062102 (2005); http://dx.doi.org/10.1063/1.1904704 (11 pages) | Cited 7 times

Online Publication Date: 12 May 2005

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The quantum effects for a physical system can be described by the set E(H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator. While a general effect may be unsharp, the collection of sharp effects is described by the set of orthogonal projections P(H)⊆E(H). Under the natural order, E(H) becomes a partially ordered set that is not a lattice if dim H ≥ 2. A physically significant and useful characterization of the pairs A,B ∊ E(H) such that the infimum A∧B exists is called the infimum problem. We show that A∧P exists for all A ∊ E(H), P ∊ P(H) and give an explicit expression for A∧P. We also give a characterization of when A∧(I−A) exists in terms of the location of the spectrum of A. We present a counterexample which shows that a recent conjecture concerning the infimum problem is false. Finally, we compare our results with the work of Ando on the infimum problem.
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03.65.Fd Algebraic methods

Distinguishing bipartitite orthogonal states using LOCC: Best and worst cases

Michael Nathanson

J. Math. Phys. 46, 062103 (2005); http://dx.doi.org/10.1063/1.1914731 (15 pages) | Cited 22 times

Online Publication Date: 16 May 2005

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Two types of results are presented for distinguishing pure bipartite quantum states using local operations and classical communications. We examine sets of states that can be perfectly distinguished, in particular showing that any three orthogonal maximally entangled states in C3⊗C3 form such a set. In cases where orthogonal states cannot be distinguished, we obtain upper bounds for the probability of error using LOCC taken over all sets of k orthogonal states in Cn⊗Cm. In the process of proving these bounds, we identify some sets of orthogonal states for which perfect distinguishability is not possible.
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03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)

Time reversal and n-qubit canonical decompositions

Stephen S. Bullock, Gavin K. Brennen, and Dianne P. O’Leary

J. Math. Phys. 46, 062104 (2005); http://dx.doi.org/10.1063/1.1900293 (19 pages) | Cited 5 times

Online Publication Date: 17 May 2005

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On pure states of n quantum bits, the concurrence entanglement monotone returns the norm of the inner product of a pure state with its spin-flip. The monotone vanishes for n odd, but for n even there is an explicit formula for its value on mixed states, i.e., a closed-form expression computes the minimum over all ensemble decompositions of a given density. For n even a matrix decomposition ν = k1ak2 of the unitary group is explicitly computable and allows for study of the monotone’s dynamics. The side factors k1 and k2 of this concurrence canonical decomposition (CCD) are concurrence symmetries, so the dynamics reduce to consideration of the a factor. This unitary a phases a basis of entangled states, and the concurrence dynamics of u are determined by these relative phases. In this work, we provide an explicit numerical algorithm computing ν = k1ak2 for n odd. Further, in the odd case we lift the monotone to a two-argument function. The concurrence capacity of ν according to the double argument lift may be nontrivial for n odd and reduces to the usual concurrence capacity in the literature for n even. The generalization may also be studied using the CCD, leading again to maximal capacity for most unitaries. The capacity of ν⊗I2 is at least that of ν, so odd-qubit capacities have implications for even-qubit entanglement. The generalizations require considering the spin-flip as a time reversal symmetry operator in Wigner’s axiomatization, and the original Lie algebra homomorphism defining the CCD may be restated entirely in terms of this time reversal. The polar decomposition related to the CCD then writes any unitary evolution as the product of a time-symmetric and time-antisymmetric evolution with respect to the spin-flip. En route we observe a Kramers’ nondegeneracy: the existence of a nondegenerate eigenstate of any time reversal symmetric n-qubit Hamiltonian demands (i) n even and (ii) maximal concurrence of said eigenstate. We provide examples of how to apply this work to study the kinematics and dynamics of entanglement in spin chain Hamiltonians.
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03.67.Mn Entanglement measures, witnesses, and other characterizations
03.67.Lx Quantum computation architectures and implementations
02.10.Yn Matrix theory
02.20.Sv Lie algebras of Lie groups

An isoperimetric problem for leaky loops and related mean-chord inequalities

Pavel Exner

J. Math. Phys. 46, 062105 (2005); http://dx.doi.org/10.1063/1.1914728 (10 pages) | Cited 3 times

Online Publication Date: 17 May 2005

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We consider a class of Hamiltonians in L2(math2) with attractive interaction supported by piecewise C2 smooth loops Γ of a fixed length L, formally given by −Δ−αδ(x−Γ) with α>0. It is shown that the ground state of this operator is locally maximized by a circular Γ. We also conjecture that this property holds globally and show that the problem is related to an interesting family of geometric inequalities concerning mean values of chords of Γ.
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02.40.-k Geometry, differential geometry, and topology

Operational distance and fidelity for quantum channels

Viacheslav P. Belavkin, Giacomo Mauro D’Ariano, and Maxim Raginsky

J. Math. Phys. 46, 062106 (2005); http://dx.doi.org/10.1063/1.1904510 (23 pages) | Cited 12 times

Online Publication Date: 23 May 2005

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We define and study a fidelity criterion for quantum channels, which we term the minimax fidelity, through a noncommutative generalization of maximal Hellinger distance between two positive kernels in classical probability theory. Like other known fidelities for quantum channels, the minimax fidelity is well defined for channels between finite-dimensional algebras, but it also applies to a certain class of channels between infinite-dimensional algebras (explicitly, those channels that possess an operator-valued Radon-Nikodym density with respect to the trace in the sense of Belavkin-Staszewski) and induces a metric on the set of quantum channels that is topologically equivalent to the CB-norm distance between channels, precisely in the same way as the Bures metric on the density operators associated with statistical states of quantum-mechanical systems, derived from the well-known fidelity (“generalized transition probability”) of Uhlmann, is topologically equivalent to the trace-norm distance.
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03.65.Fd Algebraic methods
02.10.Yn Matrix theory
02.40.Pc General topology

Generalized density functional theories using the k-electron densities: Development of kinetic energy functionals

Paul W. Ayers

J. Math. Phys. 46, 062107 (2005); http://dx.doi.org/10.1063/1.1922071 (22 pages) | Cited 39 times

Online Publication Date: 26 May 2005

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Several explicit formulas for the kinetic energy of a many-electron system as a functional of the k-electron density are derived, with emphasis on the electron pair density. The emphasis is on general techniques for deriving approximate kinetic energy functionals and features generalized Weisacker bounds and methods using density-matrix reconstruction. Adapting results from statistical mechanics, a hierarchy of equations is derived that links electron pairs, triplets, quadruplets, etc.; this may be used to derive more accurate approximations. Several methods for defining the exact kinetic energy functional are presented, including the generalizations of the Levy and Lieb formulations of density-functional theory. Together with N-representability constraints on the k-density, this paper provides the basis for “generalized density functional theories” based on the electron pair density. There are also implications for conventional density-functional theory, notably regarding the development of more accurate density functionals for the kinetic energy.
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05.20.Dd Kinetic theory
02.10.Yn Matrix theory

Analytic solution of the Schrödinger equation for the Coulomb-plus-linear potential. I. The wave functions

Guillaume Plante and Adel F. Antippa

J. Math. Phys. 46, 062108 (2005); http://dx.doi.org/10.1063/1.1931041 (20 pages) | Cited 4 times

Online Publication Date: 27 May 2005

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We solve the Schrödinger equation for a quark–antiquark system interacting via a Coulomb-plus-linear potential, and obtain the wave functions as power series, with their coefficients given in terms of the combinatorics functions.
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03.65.Ge Solutions of wave equations: bound states
14.65.-q Quarks
12.38.Gc Lattice QCD calculations
13.40.Gp Electromagnetic form factors

Solvable PT-symmetric model with a tunable interspersion of nonmerging levels

Miloslav Znojil

J. Math. Phys. 46, 062109 (2005); http://dx.doi.org/10.1063/1.1925249 (18 pages) | Cited 21 times

Online Publication Date: 27 May 2005

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We study the spectrum in such a PT-symmetric square well (of a diameter L ⩽ ∞) where the “strength of the non-Hermiticity” is controlled by the two parameters, viz., by an imaginary coupling ig and by the distance ℓ<L of its onset from the origin. We solve this problem and confirm that the spectrum is discrete and real in a nonempty interval of g ⩽ g0(ℓ,L). Surprisingly, a specific distinction between the bound states is found in their asymptotic stability∕instability with respect to an unlimited growth of g beyond g0(ℓ,L). In our model, all of the low-lying levels remain asymptotically unstable at the small ℓ⪡L and finite L while only the stable levels survive near ℓ ≈ L<∞ or in the purely imaginary force limit with 0<ℓ<L = ∞. In between these two extremes, an unusual and tunable, variable pattern of the interspersed “robust” and “fragile” subspectra of the real levels is obtained.
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11.30.Er Charge conjugation, parity, time reversal, and other discrete symmetries
11.10.Jj Asymptotic problems and properties
11.10.St Bound and unstable states; Bethe-Salpeter equations

Superconductivity by means of the subquantum medium coherence

M. Agop, P. D. Ioannou, and P. Nica

J. Math. Phys. 46, 062110 (2005); http://dx.doi.org/10.1063/1.1904163 (24 pages) | Cited 5 times

Online Publication Date: 31 May 2005

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In the hydrodynamic formulation of the scale relativity theory one shows that a stable vortices distribution of bipolaron type induces superconducting pairs by means of the quantum potential. Then, usual mechanisms (as, for example, the exchange interaction used in the bipolaron theory) are reduced to the coherence on the subquantum medium, the superconducting pairs resulting as a one-dimensional projection of a fractal. The temperature dependences of the superconducting parameters (coherence length, critical speed, pair breaking time, carriers concentration, penetration depth, critical field, critical current) and the concordance with the experimental data and other theories are analyzed.
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03.75.Lm Tunneling, Josephson effect, Bose-Einstein condensates in periodic potentials, solitons, vortices, and topological excitations
71.38.Mx Bipolarons
74.25.Sv Critical currents
74.20.Fg BCS theory and its development
74.25.Ha Magnetic properties including vortex structures and related phenomena
74.25.Op Mixed states, critical fields, and surface sheaths

Representation of the contextual statistical model by hyperbolic amplitudes

Andrei Khrennikov

J. Math. Phys. 46, 062111 (2005); http://dx.doi.org/10.1063/1.1931042 (12 pages) | Cited 2 times

Online Publication Date: 6 June 2005

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We continue the development of a so-called contextual statistical model (here context has the meaning of a complex of physical conditions). It is shown that, besides contexts producing the conventional trigonometric cos-interference, there exist contexts producing the hyperbolic cos-interference. Starting with the corresponding interference formula of total probability we represent such contexts by hyperbolic probabilistic amplitudes or in the abstract formalism by normalized vectors of a hyperbolic analogue of the Hilbert space. There is obtained a hyperbolic Born’s rule. Incompatible observables are represented by noncommutative operators. This paper can be considered as the first step towards hyperbolic quantum probability. We also discuss possibilities of experimental verification of hyperbolic quantum mechanics: in physics of elementary particles, string theory as well as in experiments with nonphysical systems, e.g., in psychology, cognitive sciences, and economy.
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03.65.Db Functional analytical methods
02.50.Cw Probability theory
11.25.-w Strings and branes

Lifting Bell inequalities

Stefano Pironio

J. Math. Phys. 46, 062112 (2005); http://dx.doi.org/10.1063/1.1928727 (11 pages) | Cited 9 times

Online Publication Date: 10 June 2005

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A Bell inequality defined for a specific experimental configuration can always be extended to a situation involving more observers, measurement settings, or measurement outcomes. In this article, such “liftings” of Bell inequalities are studied. It is shown that if the original inequality defines a facet of the polytope of local joint outcome probabilities then the lifted one also defines a facet of the more complex polytope.
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03.65.Ta Foundations of quantum mechanics; measurement theory
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Neveu–Schwarz fivebrane and tachyon condensation

Debashis Ghoshal, Dileep P. Jatkar, and Maximilian Kreuzer

J. Math. Phys. 46, 062301 (2005); http://dx.doi.org/10.1063/1.1922069 (12 pages)

Online Publication Date: 13 May 2005

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We argue that a semi-infinite D6-brane ending on an NS5-brane can be obtained from the condensation of the tachyon on the unstable D9-brane of type IIA theory. The construction uses a combination of the descriptions of these branes as solitons of the worldvolume theory of the D9-brane. The NS5-brane, in particular, involves a gauge bundle which is operator valued, and hence is better thought of as a gerbe.
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11.27.+d Extended classical solutions; cosmic strings, domain walls, texture
14.80.-j Other particles (including hypothetical)
11.25.Uv D branes

Normal ordering and boundary conditions in open bosonic strings

Nelson R. F. Braga, Hector L. Carrion, and Cresus F. L. Godinho

J. Math. Phys. 46, 062302 (2005); http://dx.doi.org/10.1063/1.1914727 (5 pages) | Cited 6 times

Online Publication Date: 13 May 2005

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Boundary conditions play a nontrivial role in string theory. For instance, the rich structure of D-branes is generated by choosing appropriate combinations of Dirichlet and Neumann boundary conditions. Furthermore, when an antisymmetric background is present at the string end points (corresponding to mixed boundary conditions) space time becomes noncommutative there. We show here how to build up normal ordered products for bosonic string position operators that satisfy both equations of motion and open string boundary conditions at the quantum level. We also calculate the equal time commutator of these normal ordered products in the presence of an antisymmetric tensor background.
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11.25.Uv D branes
11.25.Wx String and brane phenomenology
11.27.+d Extended classical solutions; cosmic strings, domain walls, texture

Bose–Einstein condensate and spontaneous breaking of conformal symmetry on Killing horizons

Valter Moretti and Nicola Pinamonti

J. Math. Phys. 46, 062303 (2005); http://dx.doi.org/10.1063/1.1917310 (29 pages) | Cited 1 time

Online Publication Date: 18 May 2005

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Local scalar quantum field theory (in Weyl algebraic approach) is constructed on degenerate semi-Riemannian manifolds corresponding to Killing horizons in spacetime. Covariance properties of the C*-algebra of observables with respect to the conformal group PSL(2,math) are studied. It is shown that, in addition to the state studied by Guido, Longo, Roberts, and Verch for bifurcated Killing horizons, which is conformally invariant and KMS at Hawking temperature with respect to the Killing flow and defines a conformal net of von Neumann algebras, there is a further wide class of algebraic (coherent) states representing spontaneous breaking of PSL(2,math) symmetry. This class is labeled by functions in a suitable Hilbert space and their GNS representations enjoy remarkable properties. The states are nonequivalent extremal KMS states at Hawking temperature with respect to the residual one-parameter subgroup of PSL(2,math) associated with the Killing flow. The KMS property is valid for the two local subalgebras of observables uniquely determined by covariance and invariance under the residual symmetry unitarily represented. These algebras rely on the physical region of the manifold corresponding to a Killing horizon cleaned up by removing the unphysical points at infinity [necessary to describe the whole PSL(2,math) action]. Each of the found states can be interpreted as a different thermodynamic phase, containing Bose–Einstein condensate, for the considered quantum field. It is finally suggested that the found states could describe different black holes.
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11.10.-z Field theory
11.30.Qc Spontaneous and radiative symmetry breaking
04.70.Dy Quantum aspects of black holes, evaporation, thermodynamics
97.60.Lf Black holes
11.30.Ly Other internal and higher symmetries
02.30.Oz Bifurcation theory
02.20.-a Group theory
02.10.-v Logic, set theory, and algebra

The Epstein–Glaser approach to perturbative quantum field theory: graphs and Hopf algebras

Alexander Lange

J. Math. Phys. 46, 062304 (2005); http://dx.doi.org/10.1063/1.1893215 (33 pages)

Online Publication Date: 20 May 2005

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The paper aims at investigating perturbative quantum field theory in the approach of Epstein and Glaser (EG) and, in particular, its formulation in the language of graphs and Hopf algebras (HAs). Various HAs are encountered, each one associated with a special combination of physical concepts such as normalization, localization, pseudounitarity, causal regularization, and renormalization. The algebraic structures, representing the perturbative expansion of the S-matrix, are imposed on operator-valued distributions equipped with appropriate graph indices. Translation invariance ensures the algebras to be analytically well defined and graded total symmetry allows to formulate bialgebras. The algebraic results are given embedded in the corresponding physical framework, covering the two EG versions by Fredenhagen and Scharf that differ with respect to the concrete recursive implementation of causality. Besides, the ultraviolet divergences occurring in Feynman’s representation are mathematically reasoned. As a final result, the change of the renormalization scheme in the context of EG is modeled via a HA and interpreted as the EG analog of Kreimer’s HA.
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11.10.Gh Renormalization
11.55.Bq Analytic properties of S matrix
02.10.Ox Combinatorics; graph theory

Derivation of particle, string, and membrane motions from the Born–Infeld electromagnetism

Yann Brenier and Wen-An Yong

J. Math. Phys. 46, 062305 (2005); http://dx.doi.org/10.1063/1.1925248 (17 pages) | Cited 3 times

Online Publication Date: 25 May 2005

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We derive classical particle, string, and membrane motion equations from a rigorous asymptotic analysis of the Born–Infeld nonlinear electromagnetic theory. We first add to the Born–Infeld equations the corresponding energy-momentum conservation laws and write the resulting system as a nonconservative symmetric 10×10 system of first-order PDEs. Then we show that four rescaled versions of the system have smooth solutions existing in the (finite) time interval where the corresponding limit problems have smooth solutions. Our analysis is based on a continuation principle previously formulated by Yong for (singular) limit problems.
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03.50.De Classical electromagnetism, Maxwell equations
02.30.Jr Partial differential equations

Asymptotics of regulated field commutators for Einstein-Rosen waves

J. Fernando Barbero G., Guillermo A. Mena Marugán, and Eduardo J. S. Villaseñor

J. Math. Phys. 46, 062306 (2005); http://dx.doi.org/10.1063/1.1864251 (21 pages) | Cited 2 times

Online Publication Date: 26 May 2005

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We discuss the asymptotic behavior of regulated field commutators for linearly polarized, cylindrically symmetric gravitational waves and the mathematical techniques needed for this analysis. We concentrate our attention on the effects brought about by the introduction of a physical cutoff in the study of the microcausality of the model and describe how the different physically relevant regimes are affected by its presence. Specifically we discuss how genuine quantum gravity effects can be disentangled from those originating in the introduction of a regulator.
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03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
04.30.Db Wave generation and sources
04.60.-m Quantum gravity

Covariant q-differential operators and unitary highest weight representations for Uqmathn,n

Dmitry Shklyarov and Genkai Zhang

J. Math. Phys. 46, 062307 (2005); http://dx.doi.org/10.1063/1.1927077 (24 pages)

Online Publication Date: 13 June 2005

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We investigate a one-parameter family of quantum Harish–Chandra modules of Uqmath2n. This family is an analog of the holomorphic discrete series of representations of the group SU(n,n) for the quantum group Uqmathn,n. We introduce a q-analog of “the wave” operator (a determinant-type differential operator) and prove certain covariance property of its powers. This result is applied to the study of some quotients of the above-mentioned quantum Harish–Chandra modules. We also prove an analog of a known result by J. Faraut and A. Koranyi on the expansion of reproducing kernels which determines the analytic continuation of the holomorphic discrete series.
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03.65.Fd Algebraic methods
02.20.Uw Quantum groups
02.30.Hq Ordinary differential equations
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Energy transport in the Vaidya system

J. P. Krisch and E. N. Glass

J. Math. Phys. 46, 062501 (2005); http://dx.doi.org/10.1063/1.1915290 (12 pages) | Cited 2 times

Online Publication Date: 16 May 2005

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Energy transport mechanisms can be generated by imposing relations between null tetrad Ricci components. Several kinds of mass and density transport generated by these relations are studied for the generalized Vaidya system.
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04.40.Nr Einstein-Maxwell spacetimes, spacetimes with fluids, radiation or classical fields
05.60.-k Transport processes

Random partial orders, posts, and the causal set approach to discrete quantum gravity

Avner Ash and Patrick McDonald

J. Math. Phys. 46, 062502 (2005); http://dx.doi.org/10.1063/1.1922070 (13 pages) | Cited 3 times

Online Publication Date: 19 May 2005

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We study a collection of Markov chains with values in the collection of partial orderings of the natural numbers. These systems arise naturally in the causal set approach to discrete quantum gravity and include the well-studied random partial orders of Alon, Bollobas, Brightwell, and Janson [ Ann. Appl. Probab. 4, 108–123 (1994) ]. We prove that under the dynamics associated to Markov chains in our collection, posts occur infinitely often, almost surely.
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04.60.-m Quantum gravity
05.40.Fb Random walks and Levy flights

Wyman’s solution, self-similarity, and critical behavior

G. Oliveira-Neto and F. I. Takakura

J. Math. Phys. 46, 062503 (2005); http://dx.doi.org/10.1063/1.1920308 (6 pages)

Online Publication Date: 24 May 2005

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The Wyman solution depends on two parameters, the mass M and the scalar charge Σ. If one fixes M to a positive value, say M0, and lets Σ2 take values along the real line, we show that this solution exhibits a type of critical behavior, in analogy with the nonstatic massless scalar field solutions. For Σ2>0 the space-times have naked singularities, for Σ2 = 0 one has a Schwarzschild black hole of mass M0 and finally for −M02 ⩽ Σ2<0 one has “wormhole-like” solutions. We also show that the Wyman solution is not self-similar, i.e., it does not admit a homothetic Killing vector.
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04.70.-s Physics of black holes
97.60.Lf Black holes

An exact solution of the five-dimensional Einstein equations with four-dimensional de Sitter-like expansion

Tomáš Liko and Paul S. Wesson

J. Math. Phys. 46, 062504 (2005); http://dx.doi.org/10.1063/1.1926168 (4 pages) | Cited 7 times

Online Publication Date: 1 June 2005

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We present an exact solution to the Einstein field equations which is Ricci and Riemann flat in five dimensions, but in four dimensions is a good model for the early vacuum-dominated universe.
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04.20.Jb Exact solutions
98.80.Cq Particle-theory and field-theory models of the early Universe (including cosmic pancakes, cosmic strings, chaotic phenomena, inflationary universe, etc.)

Multipolar expansions for closed and open systems of relativistic particles

David Alba, Luca Lusanna, and Massimo Pauri

J. Math. Phys. 46, 062505 (2005); http://dx.doi.org/10.1063/1.1897841 (36 pages) | Cited 4 times

Online Publication Date: 13 June 2005

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Dixon’s multipoles for a system of N relativistic positive-energy scalar particles are evaluated in the rest-frame instant form of dynamics. The Wigner hyperplanes (intrinsic rest frame of the isolated system) turn out to be the natural framework for describing multipole kinematics. Classical concepts like the barycentric tensor of inertia turn out to be extensible to special relativity only by means of the quadrupole moments of the isolated system. Two new applications of the multipole technique are worked out for systems of interacting particles and fields. In the rest frame of the isolated system of either free or interacting positive energy particles it is possible to define a unique world line which embodies the properties of the most relevant centroids introduced in the literature as candidates for the collective motion of the system. This is no longer true, however, in the case of open subsystems of the isolated system. While effective mass, 3-momentum and angular momentum in the rest frame can be calculated from the definition of the subsystem energy-momentum tensor, the definitions of effective center of motion and effective intrinsic spin of the subsystem are not unique. Actually, each of the previously considered centroids corresponds to a different world line in the case of open systems. The pole–dipole description of open subsystems is compared to their description as effective extended objects. Hopefully, the technique developed here could be instrumental for the relativistic treatment of binary star systems in metric gravity.
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03.30.+p Special relativity
97.80.-d Binary and multiple stars
95.30.Sf Relativity and gravitation
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On asymptotic Hopf invariant for Hamiltonian systems

Mikhail V. Deryabin

J. Math. Phys. 46, 062701 (2005); http://dx.doi.org/10.1063/1.1904511 (8 pages)

Online Publication Date: 13 May 2005

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We give a definition and discuss main properties of the asymptotic Hopf invariant, or helicity, for Hamiltonian systems. The definition relies on a technical construction which is a definition of a relative Hopf invariant for divergence-free vector fields in multiconnected domains of the following type: (flat domain in math2)×(circle). We prove correctness of the definitions, and discuss ergodic interpretation of the Hamiltonian Hopf invariant, together with its relation with the Calabi invariant.
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02.10.Ud Linear algebra
02.40.Pc General topology
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