The chiral WZNW phase space and its Poisson-Lie groupoid

The precise relationship between the arbitrary monodromy dependent 2-form appearing in the chiral WZNW symplectic form and the ‘exchange r-matrix' that governs the corresponding Poisson brackets is established. Generalizing earlier results related to diagonal monodromy, the exchange r-matrices are shown to satisfy a new dynamical generalization of the classical modified Yang-Baxter equation, which is found to admit an interpretation in terms of (new) Poisson-Lie groupoids. Dynamical exchange r-matrices for which right multiplication yields a classical or a Poisson-Lie symmetry on the chiral WZNW phase space are presented explicitly.     

The nearest-neighbor resonating-valence bond state in a Grassmannian form

The nearest-neighbor resonating-valence bond (NNRVB) state is studied using classical anticommuting (Grassmann) variables. The classical partition function corresponding to the self-overlap of the NNRVB wavefunction is generated from a local (bond) Hamiltonian expressed in terms of four anticommuting variables. It is shown that the one-particle-per-site constraint introduces an interaction term which is a local product of all four variables. Two approaches are applied to study this Hamiltonian: (i) a self-consistent field decoupling scheme and (ii) a systematic perturbation expansion around the unconstrained soluble point. Bounds on the norm of the wavefunction are derived. Extensions to the presence of holes, long-range valence bonds, and the introduction of phase fluctuations [which violate the Marshall sign rule and yield aU(1) gauge theory] are discussed.     

New formulation for squeezed fermion-pair states as a counterpart of Grassmannian evolution

A new formulation for deriving the squeezing fermion-pair operators, which manifestly shows the correspondence between pseudo-classical transformation in Grassmann number space and quantum mechanical Fermi operators, is presented. In this formulation new fermionic operator formulas regarding Bogoliubov-Valatin transformation are obtained. Received 17 February 2000     

Invariant star-product on a Poisson-Lie group and h-deformation of the corresponding Lie algebra

In this paper we prove that a left-invariant star-product on a Poisson-Lie group leads to the quantum Lie algebra structure on the corresponding Lie algebra of the Lie group.     

Electromagnetism as a second order effect

An historical survey of the theories of the forces between current elements is given and Ampère's formula is compared with the Biot-Savart law. It is shown that the Biot-Savart law, in differential form, is consistent with both Classical Electromagnetism and the Special Theory of Relativity.     

Classification of all Poisson-Lie structures on an infinite-dimensional jet group

A local classification of all Poisson-Lie structures on an infinite-dimensional group G of formal power series is given. All Lie bialgebra structures on the Lie algebra G of G are also classified.     

The Schouten-Nijenhuis bracket and interior products

The main properties of the Schouten-Nijenhuis bracket are reviewed and a new formula is proven, which relates that bracket with the right interior product of multivectors by one-forms.     

Differential operators and immersions of a Riemann surface into a Grassmannian

We consider equivariant holomorphic immersions of a universal cover of a compact Riemann surface X into a Grassmannian satisfying a nondegeneracy condition. The equivariance condition says that there is a homomorphism ρ of the Galois group to that takes the natural action of the Galois group on to the action of the Galois group on defined using ρ. We prove that the space of such embeddings are in bijective correspondence with the space of all holomorphic differential operators of order two on a rank n vector bundle over X with the property that the symbol of the operator is an isomorphism.     

The Parthasarathy formula and a spectral triple for the quantum Lagrangian Grassmannian of rank two

Letters in Mathematical Physics - We show that the Dolbeault–Dirac operator on the quantum Lagrangian Grassmannian of rank two, an example of a quantum irreducible flag manifold, satisfies an...     

Dimer as a challenge to the second law

A simple system consisting of an asymmetric dimer cooperating with two baths is presented as a challenge to the second law of thermodynamics. This requires specific coupling of the dimer to the baths and a specific thermodynamic regime.Received: 22 March 2003, Published online: 4 August 2003PACS: 05.30.-d Quantum statistical mechanics - 05.70.-a Thermodynamics This articles has been submitted by V. Capek a few weeks before he passed away and it is presumably his last paper. It has been edited by D.P. Sheehan (University of San Diego, USA).     

活动标架系下O（n）非线性σ模型Laxpair矩阵的Poisson—Lie结构

在活动标架系下给出了Ｏ（ｎ）／Ｏ（ｎ－１）对称空间上非线性σ模型的Ｐｏｉｓｓｏｎ－Ｌｉｅ括号，并用协变分解的方法讨论了活动标架与固定标架系的协交关系，得到的ｒ、ｓ矩阵可以明显地看出其依赖于场量的部分是由于Ｓｎ－１流形上的联络引起的．     

Invariant fourth root Finsler metrics on the Grassmannian manifolds

Fourth root metrics are a special and important class of Finsler metrics, which have been applied to physics. In this paper, we study invariant fourth root Finsler metrics on the Grassmannian manifolds SO(p+q)/SO(p)×SO(q)$SO(p+q)/SO\left(p\right)\times SO\left(q\right)$. By using the results from the theory of invariant polynomials of Lie groups, we obtain a complete classification of such metrics. Further, some invariant 2m$2m$-th root Finsler metrics are also given.     

The electron field and the Dirac bracket

A recent quantization rule of Fermi systems starts from the new symmetric brackets of classical mechanics. As a consequence, Fermi and Bose quantization can be put on an equal footing, instead of the standardad hoc procedure. We prove that the rule gives the right anticommutation relations when applied to the case of the relativistic electron. We show that this is a crucial test of the rule.For completeness, Dirac's Hamiltonian mechanics and the plus and minus Dirac bracket formalisms are developed for the electron's field.     

Infinite sequence of nonlocal conserved currents in a supersymmetric Grassmannian model

An infinite sequence of nonlocal conserved currents is constructed for the supersymmetric generalization of the U(N)/U(N-p) U(p) chiral model. The generating function for nonlocal charges is explicitly obtained in terms of the component fields.     

The Euler-Poincaré equations and double bracket dissipation

This paper studies the perturbation of a Lie-Poisson (or, equivalently an Euler-Poincaré) system by a special dissipation term that has Brockett's double bracket form. We show that a formally unstable equilibrium of the unperturbed system becomes a spectrally and hence nonlinearly unstable equilibrium after the perturbation is added. We also investigate the geometry of this dissipation mechanism and its relation to Rayleigh dissipation functions. This work complements our earlier work (Bloch, Krishnaprasad, Marsden and Ratiu [1991, 1994]) in which we studied the corresponding problem for systems with symmetry with the dissipation added to the internal variables; here it is added directly to the group or Lie algebra variables. The mechanisms discussed here include a number of interesting examples of physical interest such as the Landau-Lifschitz equations for ferromagnetism, certain models for dissipative rigid body dynamics and geophysical fluids, and certain relative equilibria in plasma physics and stellar dynamics.Research partially supported by the National Science Foundation PYI grant DMS-91-57556, and AFOSR grant F49620-93-1-0037.Research partially supported by the AFOSR University Research Initiative Program under grants AFOSR-87-0073 and AFOSR-90-0105 and by the National Science Foundation's Engineering Research Centers Program NSFD CDR 8803012.Research partially supported by, DOE contract DE-FG03-92ER-25129, a Fairchild Fellowship at Caltech, and the Fields Institute for Research in the Mathematical Sciences.Research partially supported by NSF Grant DMS 91-42613, DOE contract DE-FG03-92ER-25129, the Fields Institute, the Erwin Schrödinger Institute, and the Miller Institute of the University of California.