N = 2 Supercomplexification of the Korteweg-de Vries,Sawada-Kotera and Kaup-Kupershmidt Equations

The supercomplexification is a special method of N = 2 supersymmetrization of the integrable equations in which the bosonic sector can be reduced to the complex version of these equations. The N = 2 supercomplex Korteweg-de Vries, Sawada-Kotera and Kaup-Kupershmidt equations are defined and investigated. The common attribute of the supercomplex equations is appearance of the odd Hamiltonian structures and superfermionic conservation laws. The odd bi-Hamiltonian structure, Lax representation and superfermionic conserved currents for new N = 2 supersymmetric Korteweg-de Vries equation and for Sawada-Kotera one, are given.     

Integrable extensions of N = 2 supersymmetric KdV hierarchy associated with the nonuniqueness of the roots of the Lax operator

We present new supersymmetric integrable extensions of the a = 4, N = 2 KdV hierarchy. The root of the supersymmetric Lax operator of the KdV equation is generalized, by including additional fields. This generalized root generates a new hierarchy of integrable equations, for which we investigate the Hamiltonian structure. In a special case our system describes the interaction of the KdV equation with the two MKdV equations.     

Hidden symmetries of integrable conformal mechanical systems

We split the generic conformal mechanical system into a “radial” and an “angular” part, where the latter is defined as the Hamiltonian system on the orbit of the conformal group, with the Casimir function in the role of the Hamiltonian. We reduce the analysis of the constants of motion of the full system to the study of certain differential equations on this orbit. For integrable mechanical systems, the conformal invariance renders them superintegrable, yielding an additional series of conserved quantities originally found by Wojciechowski in the rational Calogero model. Finally, we show that, starting from any N=4 supersymmetric “angular” Hamiltonian system one may construct a new system with full N=4 superconformal D(1,2;α) symmetry.     

The Hamiltonian structure of the N = 2 supersymmetric GNLS hierarchy

The first two Hamiltonian structures and the recursion operator connecting all evolution systems and Hamiltonian structures of the N = 2 supersymmetric (n, m)-GNLS hierarchy are constructed in terms of N = 2 superfields in two different superfield bases with local evolution equations. Their bosonic limits are studied in detail. New local and nonlocal bosonic and fermionic integrals both for the N = 2 supersymmetric (n, m)-GNLS hierarchy and its bosonic counterparts are derived. As an example, in the n = 1, m = 1 case, the algebra and the symmetry transformations for some of them are worked out and a rich N = 4 supersymmetry structure is uncovered.     

Entanglement,superselection rules and supersymmetric quantum mechanics

In this paper we show that the energy eigenstates of supersymmetric quantum mechanics (SUSYQM) with non-definite “fermion” number are entangled states. They are “physical states” of the model provided that observables with odd number of spin variables are allowed in the theory like it happens in the Jaynes–Cummings model. Those states generalize the so-called “spin-spring” states of the Jaynes–Cummings model which have played an important role in the study of entanglement.     

On the integrable hierarchies associated with the N = 2 super Wn algebra

A new Lax operator is proposed from the viewpoint of constructing the integrable hierarchies related with the N = 2 super W_n algebra. It is shown that the Poisson algebra associated to the second Hamiltonian structure for the resulting hierarchy contains the N = 2 super Virasoro algebra as a proper subalgebra. The simplest cases are discussed in detail. In particular, it is proved that the supersymmetric two-boson hierarchy is one of the N = 2 supersymmetric KdV hierarchies. Also, a Lax operator is supplied for one of the N = 2 supersymmetric Boussinesq hierarchies.     

Two new supersymmetric equations of Harry Dym type and their supersymmetric reciprocal transformations

A new N=1$N=1$ supersymmetric Harry Dym equation is constructed by applying supersymmetric reciprocal transformation to a trivial supersymmetric Harry Dym equation, and its recursion operator and Lax formulation are also obtained. Within the framework of symmetry approach, a class of 3rd order supersymmetric equations of Harry Dym type are considered. In addition to five known integrable equations, a new supersymmetric equation, admitting 5th order generalized symmetry, is shown to be linearizable through supersymmetric reciprocal transformation. Furthermore, its Lax representation and recursion operator are given so that the integrability of this new equation is confirmed.     

Relativistic interactions for the meson-two-nucleon system in the clothed-particle unitary representation

The method of unitary clothing transformations put forward in relativistic quantum field theory (QFT) by Greenberg and Schweber and developed by Shirokov is applied to construct a new family of interactions in the meson-two-nucleon system. Along with a brief exposition of its basic elements we show a specific transition from the initial “bare” one-meson and one-nucleon operators and states to their physical “clothed” counterparts. We emphasize that the clothing transformations in question do not alter the original total Hamiltonian, but provides a conceptually more transparent representation of the same Hamiltonian in terms of a new set of operators for particles with physical properties and their relativistic interactions. The Hermitian and energy-independent interaction operators for the processes πN → πN, NN → NN and NN ↔ πNN are derived starting from the Yukawa-type couplings between fermions (nucleons and antinucleons) and bosons (π−, η−, ρ−, ω− mesons, etc.). These types of interaction have a distinctive off-energy-shell structure which is naturally generated by the unitary transformation that removes from the Hamiltonian the (three-leg) πNN vertex coupling.     

Towards the construction of N = 2 supersymmetric integrable hierarchies

We formulate a conjecture for the three different Lax operators that describe the bosonic sectors of the three possible N = 2 supersymmetric integrable hierarchies with N = 2 super-W_n second Hamiltonian structure. We check this conjecture in the simplest cases, then we verify it in general in one of the three possible supersymmetric extensions. To this end we construct the N = 2 supersymmetric extensions of the Generalized Non-Linear Schrödinger hierarchy by exhibiting the corresponding super Lax operator. To find the correct Hamiltonians we are led to a new definition of super-residues for degenerate N = 2 supersymmetric pseudo-differential operators. We have found a new non-polynomials Miura-like realization for N = 2 superconformal algebra in terms of two bosonic chiral-anti-chiral free superfields.     

The N = 2 supersymmetric Toda lattice hierarchy

We propose a new integrable N = 2 supersymmetric Toda lattice hierarchy which may be relevant for constructing a supersymmetric one-matrix model. We define its first two Hamiltonian structures, the recursion operator and Lax-pair representation. We provide partial evidence for the existence of an infinite-dimensional N = 2 superalgebra of its flows. We study its bosonic limit and introduce new Lax-pair representations for the bosonic Toda lattice hierarchy. Finally we discuss the relevance this approach for constructing N = 2 supersymmetric generalized Toda lattice hierarchies.     

A supersymmetric Sawada-Kotera equation

A new supersymmetric equation is proposed for the Sawada-Kotera equation. The integrability of this equation is shown by the existence of Lax representation and infinite conserved quantities and a recursion operator.     

A comment on the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations” by T.D. Frank

The purpose of this comment is to correct mistaken assumptions and claims made in the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations” by T. D. Frank [T.D. Frank, Stochastic feedback, non-linear families of Markov processes, and nonlinear Fokker-Planck equations, Physica A 331 (2004) 391]. Our comment centers on the claims of a “non-linear Markov process” and a “non-linear Fokker-Planck equation.” First, memory in transition densities is misidentified as a Markov process. Second, the paper assumes that one can derive a Fokker-Planck equation from a Chapman-Kolmogorov equation, but no proof was offered that a Chapman-Kolmogorov equation exists for the memory-dependent processes considered. A “non-linear Markov process” is claimed on the basis of a non-linear diffusion pde for a 1-point probability density. We show that, regardless of which initial value problem one may solve for the 1-point density, the resulting stochastic process, defined necessarily by the conditional probabilities (the transition probabilities), is either an ordinary linearly generated Markovian one, or else is a linearly generated non-Markovian process with memory. We provide explicit examples of diffusion coefficients that reflect both the Markovian and the memory-dependent cases. So there is neither a “non-linear Markov process”, nor a “non-linear Fokker-Planck equation” for a conditional probability density. The confusion rampant in the literature arises in part from labeling a non-linear diffusion equation for a 1-point probability density as “non-linear Fokker-Planck,” whereas neither a 1-point density nor an equation of motion for a 1-point density can define a stochastic process. In a closely related context, we point out that Borland misidentified a translation invariant 1-point probability density derived from a non-linear diffusion equation as a conditional probability density. Finally, in the Appendix A we present the theory of Fokker-Planck pdes and Chapman-Kolmogorov equations for stochastic processes with finite memory.     

Graded Hopf maps and fuzzy superspheres

We argue supersymmetric generalizations of fuzzy two- and four-spheres based on the unitary-orthosymplectic algebras, uosp(N|2) and uosp(N|4), respectively. Supersymmetric version of Schwinger construction is applied to derive graded fully symmetric representation for fuzzy superspheres. As a classical counterpart of fuzzy superspheres, graded versions of 1st and 2nd Hopf maps are introduced, and their basic geometrical structures are studied. It is shown that fuzzy superspheres are represented as a “superposition” of fuzzy superspheres with lower supersymmetries. We also investigate algebraic structures of fuzzy two- and four-superspheres to identify su(2|N) and su(4|N) as their enhanced algebraic structures, respectively. Evaluation of correlation functions manifests such enhanced structure as quantum fluctuations of fuzzy supersphere.     

OPE for null Wilson loops and open spin chains

Maximal helicity-violating scattering amplitudes in N=4$N=4$ supersymmetric Yang–Mills theory are dual to Wilson loops on closed null polygons. We perform their operator product expansion analysis in two-dimensional kinematics in the soft-collinear approximation which corresponds to the case when some light-cone distances vanish. We construct the expansion in terms of multi-particle “heavy”–light operators, where the “heavy” fields are identified with the Wilson lines defining the OPE channel and the light fields emerge from the curvature of the contour. The correlation function of these define the remainder function. We study the dilatation operator for these operators at one-loop order and find that it corresponds to a non-compact open spin chain. This provides an alternative view on elementary excitations propagating on the GKP string at weak coupling, which now correspond to particles traveling along an open spin chain. The factorized structure of the Wilson loop in the soft limit allows one to represent the two-loop correction to the octagon Wilson loop as a convolution formula and find the corresponding remainder function.     

On higher derivatives in 3D gravity and higher-spin gauge theories

The general second-order massive field equations for arbitrary positive integer spin in three spacetime dimensions, and their “self-dual” limit to first-order equations, are shown to be equivalent to gauge-invariant higher-derivative field equations. We recover most known equivalences for spins 1 and 2, and find some new ones. In particular, we find a non-unitary massive 3D gravity theory with a 5th order term obtained by contraction of the Ricci and Cotton tensors; this term is part of an N=2 super-invariant that includes the “extended Chern-Simons” term of 3D electrodynamics. We also find a new unitary 6th order gauge theory for “self-dual” spin 3.     

Two-dimensional Brownian vortices

We introduce a stochastic model of 2D Brownian vortices associated with the canonical ensemble. The point vortices evolve through their usual mutual advection but they experience in addition a random velocity and a systematic drift generated by the system as a whole. The statistical equilibrium state of this stochastic model is the Gibbs canonical distribution. We consider a single species system and a system made of two types of vortices with positive and negative circulations. At positive temperatures, like-sign vortices repel each other (“plasma” case) and at negative temperatures, like-sign vortices attract each other (“gravity” case). We derive the stochastic equation satisfied by the exact vorticity field and the Fokker-Planck equation satisfied by the N-body distribution function. We present the BBGKY-like hierarchy of equations satisfied by the reduced distribution functions and close the hierarchy by considering an expansion of the solutions in powers of 1/N, where N is the number of vortices, in a proper thermodynamic limit. For spatially inhomogeneous systems, we derive the kinetic equations satisfied by the smooth vorticity field in a mean field approximation valid for N→+∞. For spatially homogeneous systems, we study the two-body correlation function, in a Debye-Hückel approximation valid at the order O(1/N). The results of this paper can also apply to other systems of random walkers with long-range interactions such as self-gravitating Brownian particles and bacterial populations experiencing chemotaxis. Furthermore, for positive temperatures, our study provides a kinetic derivation, from microscopic stochastic processes, of the Debye-Hückel model of electrolytes.     

Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems

We study the growth of correlations in systems with weak long-range interactions. Starting from the BBGKY hierarchy, we determine the evolution of the two-body correlation function by using an expansion of the solutions of the hierarchy in powers of 1/N in a proper thermodynamic limit N→+∞, where N is the number of particles. These correlations are responsible for the “collisional” evolution of the system beyond the Vlasov regime due to finite N effects. We obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. These peculiarities are specific to systems with unshielded long-range interactions. For spatially homogeneous systems with short memory time like plasmas, we recover the classical Landau (or Lenard-Balescu) equations. An interest of our approach is to develop a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems. This enlightens the basic physics and provides novel kinetic equations with a clear physical interpretation. However, unless we restrict ourselves to spatially homogeneous systems, closed kinetic equations can be obtained only if we ignore some collective effects between particles. General exact coupled equations taking into account collective effects are also given. We use this kinetic theory to discuss the processes of violent collisionless relaxation and slow collisional relaxation in systems with weak long-range interactions. In particular, we investigate the dependence of the relaxation time with the system size N and try to provide a coherent discussion of all the numerical results obtained for these systems.