G-Strands

A G-strand is a map g(t,s):?×?→G for a Lie group G that follows from Hamilton’s principle for a certain class of G-invariant Lagrangians. The SO(3)-strand is the G-strand version of the rigid body equation and it may be regarded physically as a continuous spin chain. Here, SO(3) _K -strand dynamics for ellipsoidal rotations is derived as an Euler–Poincaré system for a certain class of variations and recast as a Lie–Poisson system for coadjoint flow with the same Hamiltonian structure as for a perfect complex fluid. For a special Hamiltonian, the SO(3) _K -strand is mapped into a completely integrable generalization of the classical chiral model for the SO(3)-strand. Analogous results are obtained for the Sp(2)-strand. The Sp(2)-strand is the G-strand version of the Sp(2) Bloch–Iserles ordinary differential equation, whose solutions exhibit dynamical sorting. Numerical solutions show nonlinear interactions of coherent wave-like solutions in both cases. Diff(?)-strand equations on the diffeomorphism group G=Diff(?) are also introduced and shown to admit solutions with singular support (e.g., peakons).     

Chain of interacting SU(2)4 anyons and quantum SU(2)k×$\overline { SU(2)_k } $ doubles

We consider a chain of SU(2) ₄ anyons with transitions to a topologically ordered phase state. For half-integer and integer indices of the type of strongly correlated excitations, we find an effective low-energy Hamiltonian that is an analogue of the standard Heisenberg Hamiltonian for quantum magnets. We describe the properties of the Hilbert spaces of the system eigenstates. For the Drinfeld quantum SU(2)_k×[`(SU(2)_k )]\overline {SU(2)_k } doubles, we use numerical computations to show that the largest eigenvalues of the adjacency matrix for graphs that are extended Dynkin diagrams coincide with the total quantum dimensions for the levels k = 2, 3, 4, 5. We also formulate a hypothesis about the reason for the universal behavior of the system in the long-wave limit.     

On Dynamical Properties of a One-Parameter Family of Transformations Arising in Averaging of Semigroups

To describe the dynamics of quantum systems with degenerate symmetric but not self-adjoint Hamiltonian, we consider the Naimark extension of the Hamiltonian to a self-adjoint operator in an extended Hilbert space. We relate to the symmetric Hamiltonian a one-parameter family of averaged dynamical transformations of the set of quantum states obtained from a unitary group of transformations of the extended Hilbert space by using a conditional expected value to an algebra of bounded operators acting in the original space. We establish the absence of the semigroup property and injectivity of the family of averaged dynamical transformations. We obtain a representation of trajectories of the averaged family of dynamical transformations by maximum points of functionals on the space of mappings of the time interval into the set of quantum states.     

Pairings and actions for dynamical quantum groups

Dynamical quantum groups constructed from a FRST-construction using a solution of the quantum dynamical Yang-Baxter equation are equipped with a natural pairing. The interplay of the pairing with *-structures, corepresentations and dynamical representations is studied, and natural left and right actions are introduced. Explicit details for the elliptic U(2) dynamical quantum group are given, and the pairing is calculated explicitly in terms of elliptic hypergeometric functions. Dynamical analogues of spherical and singular vectors for corepresentations are introduced.     

The functional integral in the Hubbard model

In a new functional integral approach proposed for the model, we find the regime with a deformed integration measure in which the standard integral is replaced with the Jackson integral. We indicate the relation to a p-adic functional integral. For the magnetic and electronic subsystems in the effective functional that results from the operator formulation of the Hubbard model, we find the two-parametric quantum derivative resulting in the appearance of the quantum SU_rq (2) group. We establish the relation to the one-parametric quantum derivative and to the standard derivative.     

The spectrum of a commutator Hamiltonian is hydrogenlike

For any finite group, an element (commutator Hamiltonian) is defined in its group algebra so that in any representation of that group the image of this element is diagonalizable and has the spectrum contained in the set {1/n ²|n = 1,2,3,…}. The result is generalized onto an arbitrary compact group. In particular, it is pointed out that for the natural representation of the group SU(2, C) in the space of complex-valued functions with the square of absolute values integrable over the Haar measure the multiplicity of the eigenvalue 1/n ² of the commutator Hamiltonian is equal to n ².     

Mass Operator and Dynamical Implementation of Mass Superselection Rule

We start reviewing Giulini’s dynamical approach to Bargmann superselection rule proposing some improvements. First of all we discuss some general features of the central extensions of the Galilean group used in Giulini’s programme, in particular focussing on the interplay of classical and quantum picture, without making any particular choice for the multipliers. Preserving other features of Giulini’s approach, we modify the mass operator of a Galilei invariant quantum system to obtain a mass spectrum that is (i) positive and (ii) discrete, so giving rise to a standard (non-continuous) superselection rule. The model results to be invariant under time reversal but a further degree of freedom appears that can be interpreted as describing an internal conserved charge of the system (however, adopting a POVM approach, the unobservable degrees of freedom can be pictured as a generalized observable automatically gaining a positive mass operator without assuming the existence of such a charge). The effectiveness of Bargmann rule is shown to be equivalent to an averaging procedure over the unobservable degrees of freedom of the central extension of Galileian group. Moreover, viewing the Galileian invariant quantum mechanics as a non-relativistic limit, we prove that the above-mentioned averaging procedure giving rise to Bargmann superselection rule is nothing but an effective de-coherence phenomenon due to time evolution if assuming that real measurements includes a temporal averaging procedure. It happens when the added term Mc ² is taken in due account in the Hamiltonian operator since, in the dynamical approach, the mass M is an operator and cannot be trivially neglected as in classical mechanics. The presented results are quite general and rely upon the only hypothesis that the mass operator has point spectrum. These results explicitly show the interplay of the period of time of the averaging procedure, the energy content of the considered states, and the minimal difference of the mass operator eigenvalues.     

Semiclassics for Particles with Spin via a Wigner–Weyl-Type Calculus

We show how to relate the full quantum dynamics of a spin-½ particle on \({\mathbb{R}^d}\) to a classical Hamiltonian dynamics on the enlarged phase space \({\mathbb{R}^{2d} \times \mathbb{S}^{2}}\) up to errors of second order in the semiclassical parameter. This is done via an Egorov-type theorem for normal Wigner–Weyl calculus for \({\mathbb{R}^d}\) (Folland, Harmonic Analysis on Phase Space, 1989; Lein, Weyl Quantization and Semiclassics, 2010) combined with the Stratonovich–Weyl calculus for SU(2) (Varilly and Gracia-Bondia, Ann Phys 190:107–148, 1989). For a specific class of Hamiltonians, including the Rabi- and Jaynes–Cummings model, we prove an Egorov theorem for times much longer than the semiclassical time scale. We illustrate the approach for a simple model of the Stern–Gerlach experiment.     

Coarse-grained description ofq-deformation top and t-j model

From RTT relations the integrable Hamiltonian of the trigonometric Goryachev-Chaplygin gyrostat is established, which can be reduced to the Hamiltonian of t-j model by using multi-fermion realization ofSU _q(2) algebra and average-field approximation. Project supported in part by the National Natural Science Foundation of China (Grant No. 19377102).     

On projective representations for compact quantum groups

We study actions of compact quantum groups on type I-factors, which may be interpreted as projective representations of compact quantum groups. We generalize to this setting some of Woronowicz?s results concerning Peter-Weyl theory for compact quantum groups. The main new phenomenon is that for general compact quantum groups (more precisely, those which are not of Kac type), not all irreducible projective representations have to be finite-dimensional. As applications, we consider the theory of projective representations for the compact quantum groups associated with group von Neumann algebras of discrete groups, and consider a certain non-trivial projective representation for quantum SU(2).     

Continuous Limits of Classical Repeated Interaction Systems

We consider the physical model of a classical mechanical system (called “small system”) undergoing repeated interactions with a chain of identical small pieces (called “environment”). This physical setup constitutes an advantageous way of implementing dissipation for classical systems; it is at the same time Hamiltonian and Markovian. This kind of model has already been studied in the context of quantum mechanical systems, where it was shown to give rise to quantum Langevin equations in the limit of continuous time interactions (Attal and Pautrat in Ann Henri Poincaré 7:59–104, 2006), but it has never been considered for classical mechanical systems yet. The aim of this article is to compute the continuous limit of repeated interactions for classical systems and to prove that they give rise to particular stochastic differential equations (SDEs) in the limit. In particular, we recover the usual Langevin equations associated with the action of heat baths. In order to obtain these results, we consider the discrete-time dynamical system induced by Hamilton’s equations and the repeated interactions. We embed it into a continuous-time dynamical system and compute the limit when the time step goes to 0. This way, we obtain a discrete-time approximation of SDE, considered as a deterministic dynamical system on the Wiener space, which is not exactly of the usual Euler scheme type. We prove the L ^p and almost sure convergence of this scheme. We end up with applications to concrete physical examples such as a charged particle in a uniform electric field or a harmonic interaction. We obtain the usual Langevin equation for the action of a heat bath when considering a damped harmonic oscillator as the small system.     

Analysis on bifurcations of multiple limit cycles for a parametrically and externally excited mechanical system

Research on the bifurcations of the multiple limit cycles for a parametrically and externally excited mechanical system is presented in this paper. The original mechanical system is first transformed to the averaged equation in the Cartesian form, which is in the form of a Z₂-symmetric perturbed polynomial Hamiltonian system of degree 5. Then, using the bifurcation theory of planar dynamical system and the method of detection function, the bifurcations of the multiple limit cycles of the system are investigated and the configurations of compound eyes are also obtained.     

Chaos in M(atrix) theory

We consider the classical and quantum dynamics in M(atrix) theory. Using a simple ansatz we show that a classical trajectory exhibits a chaotic motion. We argue that the holographic feature of M(atrix) theory is related with the repulsive feature of energy eigenvalues in quantum chaotic system. Chaotic dynamics in N = 2 supersymmetric Yang—Mills theory is also discussed. We demonstrate that after the separation of “slow” and “fast” modes there is a singular contribution from the “slow” modes to the Hamiltonian of the “fast” modes.     

Superstrings,Cantorian-fractal Spacetime and Quantum-like Chaos in Atmospheric Flows

Atmospheric flows exhibit long-range spatiotemporal correlations manifested as the fractal geometry to the global cloud cover pattern concomitant with the inverse power law form for spectra of temporal fluctuations. Such non-local connections are ubiquitous to dynamical systems in nature and are identified as signatures of self-organized criticality. A recently developed cell dynamical system model for atmospheric flows predicts the observed self-organized criticality as a natural consequence of quantum-like mechanics governing flow dynamics. The model is based on the concept that spatial integration of enclosed small scale fluctuations results in the formation of large eddy circulations. The model predicts the following: (a) The flow structure consists of an overall logarithmic spiral trajectory with the quasiperiodic Penrose tiling pattern for the internal structure. (b) Conventional power spectrum analysis will resolve such spiral trajectories as a continuum of eddies with progressive increase in phase. (c) Increments in phase are concomitant with increases in period length and also represent the variance, a characteristic of quantum systems identified as Berrys phase. (d) The universal algorithm for self-organized criticality is expressed in terms of the universal Feigenbaum constants, a and d, as 2a²=πd, where the fractional volume intermittency of occurrence πd contributes to the total variance 2a² of fractal structures. (e) The Feigenbaum constants are expressed as functions of the golden mean. ( f) The quantum mechanical constants fine structure constant and ratio of proton mass to electron mass, which are pure numbers and are obtained by experimental observations only, are now derived in terms of the Feigenbaum constant, a. (g) Atmospheric flow structure follows Keplers third law of planetary motion. Therefore, Newtons inverse square law for gravitation also applies to eddy masses. The centripetal acceleration representing the inertial masses (of eddies) are equivalent to gravitational masses. The fractal-Cantorian structure of spacetime can also be visualized as a nested continuum of vortex (eddy) circulations, whose inertial masses obey Newtons inverse square law of gravitation. The model concept resembles a superstring model for subatomic dynamics which incorporates gravitational forces.     

“Classical” equations of motion in quantum mechanics with gauge fields

For the Schrödinger and Dirac equations in an external gauge field with symmetry groupSU(2), we construct to any degree of accuracy in powers ofh ^1/2,h0, approximate dynamical states in the form of wave packets—semiclassical trajectory-coherent states. For the quantum expectation values calculated with respect to these semiclassical states we obtain for the operators of the coordinates, momenta, and isospin of the particle Hamiltonian equations of motion that are equivalent (in the relativistic case after transition to the proper time) to Wong's well-known equations for a non-Abelian charge with isospin 1/2.Moscow Engineering Physics Institute. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 41–61, July, 1992.     

A theory for n-dimensional nonlinear dynamics on continuous vector fields

This paper systematically presents a theory for n-dimensional nonlinear dynamics on continuous vector fields. In this paper, a different view to look into the fundamental theory in dynamics is presented. The ideas presented herein are less formal and rigorous in an informal and lively manner. The ideas may give some inspirations in the field of nonlinear dynamics. The concepts of local and global flows are introduced to interpret the complexity of flows in nonlinear dynamic systems. Further, the global tangency and transversality of flows to the separatrix surface in nonlinear dynamical systems are discussed, and the corresponding necessary and sufficient conditions for such global tangency and transversality are presented. The ε-domains of flows in phase space are introduced from the first integral manifold surface. The domain of chaos in nonlinear dynamic systems is also defined, and such a domain is called a chaotic layer or band. The first integral quantity increment is introduced as an important quantity. Based on different reference surfaces, all possible expressions for the first integral quantity increment are given. The stability of equilibriums and periodic flows in nonlinear dynamical systems are discussed through the first integral quantity increment. Compared to the Lyapunov stability conditions, the weak stability conditions for equilibriums and periodic flows are developed. The criteria for resonances in the stochastic and resonant, chaotic layers are developed via the first integral quantity increment. To discuss the complexity of flows in nonlinear dynamical systems, the first integral manifold surface is used as a reference surface to develop the mapping structures of periodic and chaotic flows. The invariant set fragmentation caused by the grazing bifurcation is discussed. The global grazing bifurcation is a key to determine the global transversality to the separatrix. The local grazing bifurcation on the first integral manifold surface in a single domain without separatrix is a mechanism for the transition from one resonant periodic flow to another one. Such a transition may occur through chaos. The global grazing bifurcation on the separatrix surface may imply global chaos. The complexity of the global chaos is measured by invariant sets on the separatrix surface. The invariant set fragmentation of strange attractors on the separatrix surface is central to investigate the complexity of the global chaotic flows in nonlinear dynamical systems. Finally, the theory developed herein is applied to perturbed nonlinear Hamiltonian systems as an example. The global tangency and tranversality of the perturbed Hamiltonian are presented. The first integral quantity increment (or energy increment) for 2n-dimensional perturbed nonlinear Hamiltonian systems is developed. Such an energy increment is used to develop the iterative mapping relation for chaos and periodic motions in nonlinear Hamiltonian systems. Especially, the first integral quantity increment (or energy increment) for two-dimensional perturbed nonlinear Hamiltonian systems is derived, and from the energy increment, the Melnikov function is obtained under a certain perturbation approximation. Because of applying the perturbation approximation, the Melnikov function only can be used for a rough estimate of the energy increment. Such a function cannot be used to determine the global tangency and transversality to the separatrix surface. The global tangency and transversality to the separatrix surface only can be determined by the corresponding necessary and sufficient conditions rather than the first integral quantity increment. Using the first integral quantity increment, limit cycles in two-dimensional nonlinear systems is discussed briefly. The first integral quantity of any n-dimensional nonlinear dynamical system is very crucial to investigate the corresponding nonlinear dynamics. The theory presented in this paper needs to be further developed and to be treated more rigorously in mathematics.     

Some quantum optical states as realizations of Lie groups

We start with the Heisenberg–Weyl algebra and after the definitions of the Fock states we give the definition of the coherent state of this group. This is followed by the exposition of the SU(2) and SU(1, 1) algebras and their coherent states. From there we go on describing the binomial state and its extensions as realizations of the SU(2) group. This is followed by considering the negative binomial states, and some squeezed states as realizations of the SU(1, 1) group. Generation schemes based on physical systems are mentioned for some of these states.     

New Faddeev-Niemi-Type variables for the static Yang-Mills theory

Faddeev and Niemi introduced a nonlinear sigma model as a natural extension of the Faddeev $\mathbb{S}^2 $ chiral model. The field variables in the extended model are two chiral fields taking values in SU(3)/(U(1)×U(1)) and SU(3)/(SU(2)×U(1)). Shabanov showed that the energy functional of the extended model is bounded from below by a topological invariant and can therefore support knotlike excitations and a mass gap. We introduce new variables of the Faddeev-Niemi type for the static SU(3) Yang-Mills theory, which reveal a structure of a nonlinear sigma model in the Lagrangian.