A Geometric Model of Arbitrary Spin Massive Particle

A new model of relativistic massive particle with arbitrary spin (($m,s$)-particle) is suggested. Configuration space of the model is a product of Minkowski space and two-dimensional sphere, ${\cal M}^6 = {\Bbb R}^{3,1} \times S^2$. The system describes Zitterbewegung at the classical level. Together with explicitly realized Poincar\'e symmetry, the action functional turns out to be invariant under two types of gauge transformations having their origin in the presence of two Abelian first-class constraints in the Hamilton formalism. These constraints correspond to strong conservation for the phase-space counterparts of the Casimir operators of the Poincar\'e group. Canonical quantization of the model leads to equations on the wave functions which prove to be equivalent to the relativistic wave equations for the massive spin-$s$ field.Comment: 25 pages; v2: eq. (45.b) correcte

Massive spinning particle on anti-de Sitter space

To describe a massive particle with fixed, but arbitrary, spin on $d=4$ anti-de Sitter space $M^4$, we propose the point-particle model with configuration space ${\cal M}^6 = M^{4}\times S^{2}$, where the sphere $S^2$ corresponds to the spin degrees of freedom. The model possesses two gauge symmetries expressing strong conservation of the phase-space counterparts of the second- and fourth-order Casimir operators for $so(3,2)$. We prove that the requirement of energy to have a global positive minimum $E_o$ over the configuration space is equivalent to the relation $E_o > s$, $s$ being the particle's spin, what presents the classical counterpart of the quantum massive condition. States with the minimal energy are studied in detail. The model is shown to be exactly solvable. It can be straightforwardly generalized to describe a spinning particle on $d$-dimensional anti-de Sitter space $M^d$, with ${\cal M}^{2(d-1)} = M^d \times S^{(d-2)}$ the corresponding configuration space.Comment: 23 pages, LaTe

Why the general Zakharov-Shabat equations form a hierarchy?

The totality of all Zakharov-Shabat equations (ZS), i.e., zero-curvature equations with rational dependence on a spectral parameter, if properly defined, can be considered as a hierarchy. The latter means a collection of commuting vector fields in the same phase space. Further properties of the hierarchy are discussed, such as additional symmetries, an analogue to the string equation, a Grassmannian related to the ZS hierarchy, and a Grassmannian definition of soliton solutions.Comment: 13p

Spectrum Generating Algebras for the free motion in S3S^3

We construct the spectrum generating algebra (SGA) for a free particle in the three dimensional sphere $S^3$ for both, classical and quantum descriptions. In the classical approach, the SGA supplies time-dependent constants of motion that allow to solve algebraically the motion. In the quantum case, the SGA include the ladder operators that give the eigenstates of the free Hamiltonian. We study this quantum case from two equivalent points of view.Comment: 29 pages, 1 figur

Spectral Difference Equations Satisfied by KP Soliton Wavefunctions

The Baker-Akhiezer (wave) functions corresponding to soliton solutions of the KP hierarchy are shown to satisfy eigenvalue equations for a commutative ring of translational operators in the spectral parameter. In the rational limit, these translational operators converge to the differential operators in the spectral parameter previously discussed as part of the theory of "bispectrality". Consequently, these translational operators can be seen as demonstrating a form of bispectrality for the non-rational solitons as well.Comment: to appear in "Inverse Problems

Coherent (spin-)tensor fields on D=4 anti-de Sitter space

The coherent states associated to the discrete serie representations $D(E_o,s)$ of $SO(3,2)$ are constructed in terms of (spin-)tensor fields on $D=4$ anti-de Sitter space. For $E_o>s+5$ the linear space ${\cal H}_{E_o,s}$ spanned by these states is proved to carry the unitary irreducible representation $D(E_o,s)$. The $SO(3,2)$-covariant generalized Fourier transform in this space is exhibited. The quasiclassical properties of the coherent states are analyzed. In particular, these states are shown to be localized on the time-like geodesics of anti-de Sitter space.Comment: 15 pages, LaTe

Intertwining technique for a system of difference Schroedinger equations and new exactly solvable multichannel potentials

The intertwining operator technique is applied to difference Schroedinger equations with operator-valued coefficients. It is shown that these equations appear naturally when a discrete basis is used for solving a multichannel Schroedinger equation. New families of exactly solvable multichannel Hamiltonians are found

Geometry of W-algebras from the affine Lie algebra point of view

To classify the classical field theories with W-symmetry one has to classify the symplectic leaves of the corresponding W-algebra, which are the intersection of the defining constraint and the coadjoint orbit of the affine Lie algebra if the W-algebra in question is obtained by reducing a WZNW model. The fields that survive the reduction will obey non-linear Poisson bracket (or commutator) relations in general. For example the Toda models are well-known theories which possess such a non-linear W-symmetry and many features of these models can only be understood if one investigates the reduction procedure. In this paper we analyze the SL(n,R) case from which the so-called W_n-algebras can be obtained. One advantage of the reduction viewpoint is that it gives a constructive way to classify the symplectic leaves of the W-algebra which we had done in the n=2 case which will correspond to the coadjoint orbits of the Virasoro algebra and for n=3 which case gives rise to the Zamolodchikov algebra. Our method in principle is capable of constructing explicit representatives on each leaf. Another attractive feature of this approach is the fact that the global nature of the W-transformations can be explicitly described. The reduction method also enables one to determine the ``classical highest weight (h. w.) states'' which are the stable minima of the energy on a W-leaf. These are important as only to those leaves can a highest weight representation space of the W-algebra be associated which contains a ``classical h. w. state''.Comment: 17 pages, LaTeX, revised 1. and 7. chapter