arXiv.org: Mathematical Physics

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• Codimension one symplectic foliations and regular Poisson structures. (arXiv:1009.1175v1 [math.SG])
  

  In this short note we give a complete characterization of a certain class of compact corank one Poisson manifolds, those equipped with a closed one-form defining the symplectic foliation and a closed two-form extending the symplectic form on each leaf. If such a manifold has a compact leaf, then all the leaves are compact, and furthermore the manifold is a mapping torus of a compact leaf. These manifolds and their regular Poisson structures admit an extension as the critical hypersurface of a b-Poisson manifold as we consider in another paper.

  
  
  
• Hopf Maps, Lowest Landau Level, and Fuzzy Spheres. (arXiv:1009.1192v1 [hep-th])
  

  This paper is a review of monopoles, lowest Landau level, fuzzy spheres, and their mutual relations. The Hopf maps of division algebras provide a prototype relation between monopoles and fuzzy spheres. Generalization of complex numbers to Clifford algebra is exactly analogous to generalization of fuzzy two-spheres to higher dimensional fuzzy spheres. Higher dimensional fuzzy spheres have an interesting hierarchical structure made of ''compounds'' of lower dimensional spheres. We give a physical interpretation for such particular structure of fuzzy spheres by utilizing Landau models in generic even dimensions. With Grassmann algebra, we also introduce a graded version of the Hopf map, and discuss its relation to fuzzy supersphere in context of supersymmetric Landau model.

  
  
  
• Entanglement can increase asymptotic rates of zero-error classical communication over classical channels. (arXiv:1009.1195v1 [quant-ph])
  

  It is known that the number of different classical messages which can be communicated with a single use of a classical channel with zero probability of decoding error can sometimes be increased by using entanglement shared between sender and receiver. It has been an open question to determine whether entanglement can ever offer an advantage in terms of the zero-error communication rates achievable in the limit of many channel uses. In this paper we show, by explicit examples, that entanglement can indeed increase asymptotic zero-error capacity. Interestingly, in our examples the quantum protocols are based on the root systems of the exceptional Lie groups E7 and E8.

  
  
  
• Symmetries in Fluctuations Far from Equilibrium. (arXiv:1009.1243v1 [cond-mat.stat-mech])
  

  Fluctuations arise universally in Nature as a reflection of the discrete microscopic world at the macroscopic level. Despite their apparent noisy origin, fluctuations encode fundamental aspects of the physics of the system at hand, crucial to understand irreversibility and nonequilibrium behavior. In order to sustain a given fluctuation, a system traverses a precise optimal path in phase space. Here we show that by demanding invariance of optimal paths under symmetry transformations, new and general fluctuation relations valid arbitrarily far from equilibrium are unveiled. This opens an unexplored route toward a deeper understanding of nonequilibrium physics by bringing symmetry principles to the realm of fluctuations. We illustrate this concept studying symmetries of the current distribution out of equilibrium. In particular we derive an isometric fluctuation relation which links in a strikingly simple manner the probabilities of any pair of isometric current fluctuations. This relation, which results from the time-reversibility of the dynamics, includes as a particular instance the Gallavotti-Cohen fluctuation theorem in this context but adds a completely new perspective on the high level of symmetry imposed by time-reversibility on the statistics of nonequilibrium fluctuations. We confirm the validity of the new symmetry relation in extensive numerical simulations, and suggest that the idea of symmetry in fluctuations as invariance of optimal paths has far-reaching consequences in diverse fields.

  
  
  
• A general and solvable random matrix model for spin decoherence. (arXiv:1009.1282v1 [math-ph])
  

  We propose and solve a simple but very general quantum model of an SU(2) spin interacting with a large external system with N states. The coupling is described by a random hamiltonian in a new general gaussian SU(2)xU(N) random matrix ensemble, that we introduce in this paper. We solve the model in the large N limit, for any value of the spin j and for any choice of the coupling matrix element distributions in the different possible angular momentum channels l (and provided that the internal dynamics of the spin is slow). Besides its mathematical interest as a non-trivial random matrix model, it allows to study and illustrate in a simple framework various phenomena: the decoherence dynamics, the conditions of emergence of the classical phase space for the spin, the properties quantum diffusion in phase space. The large time evolution for the spin is shown to be non-Markovian in general, the Markov property emerging in some specific case for the dynamics and the initial conditions.

  
  
  
• Deterministic spectral properties of Anderson-type Hamiltonians. (arXiv:1009.1353v1 [math.FA])
  

  This paper concerns the deterministic spectral properties of Anderson-type Hamiltonians. The main result states that under mild cyclicity conditions the essential parts of two realizations are almost surely unitary equivalent modulo a rank one perturbation. Its proof is based on the techniques and results developed by A.G.~Poltoratski in previous work.

  
  
  
• Oceanic Internal Wave Field: Theory of Scale-invariant Spectra. (arXiv:math-ph/0505050v3 UPDATED)
  

  Steady scale-invariant solutions of a kinetic equation describing the statistics of oceanic internal gravity waves based on wave turbulence theory are investigated. It is shown in the non-rotating scale-invariant limit that the collision integral in the kinetic equation diverges for almost all spectral power-law exponents. These divergences come from resonant interactions with the smallest horizontal wavenumbers and/or the largest horizontal wavenumbers with extreme scale-separations. We identify a small domain in which the scale-invariant collision integral converges and numerically find a convergent power-law solution. This numerical solution is close to the Garrett--Munk spectrum. Power-law exponents which potentially permit a balance between the infra-red and ultra-violet divergences are investigated. The balanced exponents are generalizations of an exact solution of the scale-invariant kinetic equation, the Pelinovsky--Raevsky spectrum. A balance between oppositely signed divergences states that infinity minus infinity may be approximately equal to zero. A small but finite Coriolis parameter representing the effects of rotation is introduced into the kinetic equation to determine solutions over the divergent part of the domain using rigorous asymptotic arguments. This gives rise to the induced diffusion regime. The derivation of the kinetic equation is based on an assumption of weak nonlinearity. Dominance of the nonlocal interactions puts the self-consistency of the kinetic equation at risk. Yet these weakly nonlinear stationary states are consistent with much of the observational evidence.

  
  
  
• hbar-expansion of KP hierarchy: Recursive construction of solutions. (arXiv:0912.4867v2 [math-ph] UPDATED)
  

  The \hbar-dependent KP hierarchy is a formulation of the KP hierarchy that depends on the Planck constant \hbar and reduces to the dispersionless KP hierarchy as \hbar -> 0. A recursive construction of its solutions on the basis of a Riemann-Hilbert problem for the pair (L,M) of Lax and Orlov-Schulman operators is presented. The Riemann-Hilbert problem is converted to a set of recursion relations for the coefficients X_n of an \hbar-expansion of the operator X = X_0 + \hbar X_1 + \hbar^2 X_2 +... for which the dressing operator W is expressed in the exponential form W = \exp(X/\hbar). Given the lowest order term X_0, one can solve the recursion relations to obtain the higher order terms. The wave function \Psi associated with W turns out to have the WKB form \Psi = \exp(S/\hbar), and the coefficients S_n of the \hbar-expansion S = S_0 + \hbar S_1 + \hbar^2 S_2 +..., too, are determined by a set of recursion relations. This WKB form is used to show that the associated tau function has an \hbar-expansion of the form \log\tau = \hbar^{-2}F_0 + \hbar^{-1}F_1 + F_2 + ...

  
  
  
• Group analysis and exact solutions for equations of axion electrodynamics. (arXiv:1002.0064v2 [math-ph] UPDATED)
  

  The group classification of models of axion electrodynamics with arbitrary self interaction of axionic field is carried out. Using the In\"on\"u-Wigner contraction the non-relativistic limit of equations of axion electrodynamics is found. With using the three-dimensional subalgebras of the Lie algebra of Poincar\'e group an extended class of exact solutions for the electromagnetic and axionic fields is obtained.

  
  
  
• Quantum Homogeneous Spaces. (arXiv:1007.2438v2 [math.OA] UPDATED)
  

  The aim of this paper is to present and analyze a new definition of a quantum homogeneous space of a locally compact quantum group G. It is shown to be an appropriate quantum counterpart of the classical notion of homogeneity, providing an operator algebraic characterization of the transitive group actions. Furthermore our framework covers different classes of examples such as the quotient of a locally compact quantum group by its closed quantum subgroup due to S. Vaes and (generically non-quotient) quantum homogeneous spaces of a compact quantum group studied by P. Podles as well as the Rieffel deformation of G-homogeneous spaces. On the other hand, the paradoxical examples of non-compact quantum homogeneous spaces of a compact quantum group due to S.L. Woronowicz are ruled out. We also prove that the homogeneity of a quantum G-space implies its G-simplicity.

  
  
  
• Translational-invariant noncommutative gauge theory. (arXiv:1008.5064v2 [hep-th] UPDATED)
  

  A generalized translational invariant noncommutative field theory is analyzed in detail, and a complete description of translational invariant noncommutative structures is worked out. The relevant gauge theory is described, and the planar and nonplanar axial anomalies are obtained.

  
  
  
• Analytic and algebraic conditions for bifurcations of homoclinic orbits I: Saddle equilibria. (arXiv:1009.0977v2 [math.DS] UPDATED)
  

  We study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four-dimensional systems which may be Hamiltonian or not. Only one parameter is enough to treat these types of bifurcations in Hamiltonian systems but two parameters are needed in general systems. We apply a version of Melnikov's method due to Gruendler to obtain saddle-node and pitchfork types of bifurcation results for homoclinic orbits. Furthermore we prove that if these bifurcations occur, then the variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory under the assumption that the homoclinic orbits lie on analytic invariant manifolds. We illustrate our theories with an example which arises as stationary states of coupled real Ginzburg-Landau partial differential equations, and demonstrate the theoretical results by numerical ones.

  
  
  
• Galoisian approach for a Sturm-Liouville problem on the infinite interval. (arXiv:1009.0979v2 [math.DS] UPDATED)
  

  We study a Sturm-Liouville type eigenvalue problem for second-order differential equations on the infinite interval. Here the eigenfunctions are nonzero solutions exponentially decaying at infinity. We prove that at any discrete eigenvalue the differential equations are integrable in the setting of differential Galois theory under general assumptions. Our result is illustrated with two examples for a stationary Schroedinger equation having a generalized Hulthen potential and an eigenvalue problem for a traveling front in the Allen-Cahn equation.

  
  
  
• Higher order statistics in the annulus square billiard: transport and polyspectra. (arXiv:1009.1019v2 [nlin.CD] UPDATED)
  

  Classical transport in a doubly connected polygonal billiard, i.e. the annulus square billiard, is considered. Dynamical properties of the billiard flow with a fixed initial direction are analyzed by means of the moments of arbitrary order of the number of revolutions around the inner square, accumulated by the particles during the evolution. An "anomalous" diffusion is found: the moment of order q exhibits an algebraic growth in time with an exponent different from q/2, like in the normal case. Transport features are related to spectral properties of the system, which are reconstructed by Fourier transforming time correlation functions. An analytic estimate for the growth exponent of integer order moments is derived as a function of the scaling index at zero frequency of the spectral measure, associated to the angle spanned by the particles. The n-th order moment is expressed in terms of a multiple-time correlation function, depending on n-1 time intervals, which is shown to be linked to higher order density spectra (polyspectra), by a generalization of the Wiener-Khincin Theorem. Analytic results are confirmed by numerical simulations.

  
  
  
• The Heisenberg-Lorentz quantum group. (arXiv:0807.3401v4 [math.OA] CROSS LISTED)
  

  In this article we present a new C*-algebraic deformation of the Lorentz group. It is obtained by means of the Rieffel deformation applied to SL(2,C). We give a detailed description of the resulting quantum group in terms of generators - the quantum counterparts of the matrix coefficients of the fundamental representation of SL(2,C). In order to construct the most involved of four generators, we first define it on the quantum Borel subgroup, then on the quantum complement of the Borel subgroup and finally we perform the gluing procedure. In order to classify representations of the C*-algebra of the Heisenberg-Lorentz quantum group and to analyze the action of the comultiplication on the generators we employ the duality in the theory of locally compact quantum groups.

  
  
  
• The Nakayama permutation of the nimreps associated to SU(3) modular invariants. (arXiv:1008.1003v1 [math.OA] CROSS LISTED)
  

  We determine the Nakayama permutation of the almost Calabi-Yau algebra associated to the braided subfactors or nimrep graphs associated to SU(3) modular invariants and the associated Hilbert series of dimensions.