Geometric algebra and star products on the phase space

Superanalysis can be deformed with a fermionic star product into a Clifford calculus that is equivalent to geometric algebra. With this multivector formalism it is then possible to formulate Riemannian geometry and an inhomogeneous generalization of exterior calculus. Moreover, it is shown here how symplectic and Poisson geometry fit in this context. The application of this formalism together with the bosonic star product formalism of deformation quantization leads then on space and space-time to a natural appearance of spin structures and on phase space to BRST structures that were found in the path integral formulation of classical mechanics. Furthermore it will be shown that Poincaré and Lie-Poisson reduction can be formulated in this formalism.     

On bi-hamiltonian structure of some integrable systems on so* (4)

Abstract

We classify quadratic Poisson structures on so*(4) and e*(3), which have the same foliations by symplectic leaves as canonical Lie-Poisson tensors. The separated variables for some of the corresponding bi-integrable systems are constructed.     

Reduction of Poisson manifolds

Reduction in the category of Poisson manifolds is defined and some basic properties are derived. The context is chosen to include the usual theorems on reduction of symplectic manifolds, as well as results such as the Dirac bracket and the reduction to the Lie-Poisson bracket.Research supported by DOE contract DE-AT03-85ER 12097.Supported by an A. P. Sloan Foundation fellowship.     

Lie-Poisson groups: Remarks and examples

The aim of this Letter is twofold. On the one hand, we discuss two possible definitions of complex structures on Poisson-Lie groups and we give a complete classification of the isomorphism classes of complex Lie-Poisson structures on the group SL(2, ). On the other hand, we give an algebraic characterization of a class of solutions of the Yang-Baxter equations which contains the well-known Drinfeld solutions [1]; in particular, we prove the existence of a nontrivial Lie-Poisson structure on any simply connected real semi-simple Lie Group G. Other low dimensional examples will appear elsewhere.Chercheur qualifié au FNRS.     

The Hamiltonian structure of general relativistic perfect fluids

We show that the evolution equations for a perfect fluid coupled to general relativity in a general lapse and shift, are Hamiltonian relative to a certain Poisson structure. For the fluid variables, a Lie-Poisson structure associated to the dual of a semi-direct product Lie algebra is used, while the bracket for the gravitational variables has the usual canonical symplectic structure. The evolution is governed by a Hamiltonian which is equivalent to that obtained from a canonical analysis. The relationship of our Hamiltonian structure with other approaches in the literature, such as Clebsch potentials, Lagrangian to Eulerian transformations, and its use in clarifying linearization stability, are discussed.Research supported in part by NSF grant MCS 81-08814(A02)Research supported in part by NSF grant MCS 81-07086     

On the symplectic structure of general relativity

The relation between the symplectic structures on the canonical and radiative phase spaces of general relativity is exhibited.Alfred P. Sloan Research Fellow. Supported in part by the NSF contract PHY80-08155 and by a grant from the Syracuse University Research and Equipment FundSupported in part by crédits ministériels, tranche spéciale     

Poincaré-Cartan Integral Variants and Invariants of Nonholonomic Constrained Systems

Usually there does not exist an integral invariant of Poincaré-Cartan's type for a nonholonomic system because a constraint submanifold does not admit symplectic structure in general. An integral variant of Poincaré-Cartan's type, depending on the nonholonomy of the constraints and nonconservative forces acting on the system, is derived from D'Alembert-Lagrange principle. For some nonholonomic constrained mechanical systems, there exists an alternative Lagrangian which determines the symplectic structure of a constraint submanifold. The integral invariants can then be constructed for such systems.     

An Area-Preserving Action of the Modular Group on Cubic Surfaces and the Painlevé VI Equation

We construct an area-preserving action of the modular group on a general 4-parameter family of affine cubic surfaces. We present a geometrical background behind this construction, that is, a natural symplectic structure on a moduli space of rank two linear monodromy representations over the 2-dimensional sphere with four punctures, and a natural symplectic action upon it of the braid group on three strings. Studying this action as a discrete dynamical system will be important in discussing the monodromy of the Painlevé VI equation.     

A Class of Integrable Flows on the Space of Symmetric Matrices

For a given skew symmetric real n × n matrix N, the bracket [X, Y] _N = XNY − YNX defines a Lie algebra structure on the space Sym(n, N) of symmetric n × n real matrices and hence a corresponding Lie-Poisson structure. The purpose of this paper is to investigate the geometry, integrability, and linearizability of the Hamiltonian system , or equivalently in Lax form, the equation on this space along with a detailed study of the Poisson geometry itself. If N has distinct eigenvalues, it is proved that this system is integrable on a generic symplectic leaf of the Lie-Poisson structure of Sym(n, N). This is established by finding another compatible Poisson structure. If N is invertible, several remarkable identifications can be implemented. First, (Sym(n, N), [·, ·]) is Lie algebra isomorphic with the symplectic Lie algebra associated to the symplectic form on given by N ⁻¹. In this case, the system is the reduction of the geodesic flow of the left invariant Frobenius metric on the underlying symplectic group Sp(n, N ⁻¹). Second, the trace of the product of matrices defines a non-invariant non-degenerate inner product on Sym(n, N) which identifies it with its dual. Therefore Sym(n, N) carries a natural Lie-Poisson structure as well as a compatible “frozen bracket” structure. The Poisson diffeomorphism from Sym(n, N) to maps our system to a Mischenko-Fomenko system, thereby providing another proof of its integrability if N is invertible with distinct eigenvalues. Third, there is a second ad-invariant inner product on Sym(n, N); using it to identify Sym(n, N) with itself and composing it with the dual of the Lie algebra isomorphism with , our system becomes a Mischenko- Fomenko system directly on Sym(n, N). If N is invertible and has distinct eigenvalues, it is shown that this geodesic flow on Sym(n, N) is linearized on the Prym subvariety of the Jacobian of the spectral curve associated to a Lax pair formulation with parameter of the system. If, on the other hand, N has nullity one and distinct eigenvalues, in spite of the fact that the system is completely integrable, it is shown that the flow does not linearize on the Jacobian of the spectral curve. Research partially supported by NSF grants CMS-0408542 and DMS-0604307. Research partially supported by the Swiss SCOPES grant IB7320-110721/1, 2005-2008, and MEdC Contract 2-CEx 06-11-22/25.07.2006. Research partially supported by the California Institute of Technology and NSF-ITR Grant ACI-0204932. Research partially supported by the Swiss NSF and the Swiss SCOPES grant IB7320-110721/1.     

Geometry of Vlasov kinetic moments: A bosonic Fock space for the symmetric Schouten bracket

The dynamics of Vlasov kinetic moments is shown to be Lie-Poisson on the dual Lie algebra of symmetric contravariant tensor fields. The corresponding Lie bracket is identified with the symmetric Schouten bracket and the moment Lie algebra is related with a bundle of bosonic Fock spaces, where creation and annihilation operators are used to construct the cold plasma closure. Kinetic moments are also shown to define a momentum map, which is infinitesimally equivariant. This momentum map is the dual of a Lie algebra homomorphism, defined through the Schouten bracket. Finally the moment Lie-Poisson bracket is extended to anisotropic interactions.     

Deformation quantization of the Heisenberg group

A *-product compatible with the comultiplication of the Hopf algebra of the functions on the Heisenberg group is determined by deforming a coboundary Lie-Poisson structure defined by a classicalr-matrix satisfying the modified Yang-Baxter equation. The corresponding quantum group is studied and itsR-matrix is explicitly calculated.     

Generalised Chern–Simons actions for 3d gravity and κ-Poincaré symmetry

We consider Chern–Simons theories for the Poincaré, de Sitter and anti-de Sitter groups in three dimensions which generalise the Chern–Simons formulation of 3d gravity. We determine conditions under which κ-Poincaré symmetry and its de Sitter and anti-de Sitter analogues can be associated to these theories as quantised symmetries. Assuming the usual form of those symmetries, with a timelike vector as deformation parameter, we find that such an association is possible only in the de Sitter case, and that the associated Chern–Simons action is not the gravitational one. Although the resulting theory and 3d gravity have the same equations of motion for the gauge field, they are not equivalent, even classically, since they differ in their symplectic structure and the coupling to matter. We deduce that κ-Poincaré symmetry is not associated to either classical or quantum gravity in three dimensions. Starting from the (non-gravitational) Chern–Simons action we explain how to construct a multi-particle model which is invariant under the classical analogue of κ-de Sitter symmetry, and carry out the first steps in that construction.     

Taylor and Padé analysis of the level spacing distributions of random-matrix ensembles

We construct the distributionP(S) of nearest-neighbor level spacings for the orthogonal, unitary, and symplectic ensembles of (Hermitian and unitary) random matrices in the limit of large dimension. The Taylor expansion ofP(S) aroundS=0 is given explicitly to arbitrarily high orders. By employing a diagonal Padé approximation we interpolate between the small-S behavior given by the Taylor expansion and the rigorously known asymptotic form at largeS.     

Coherent states in geometric quantization

In this paper we study overcomplete systems of coherent states associated to compact integral symplectic manifolds by geometric quantization. Our main goals are to give a systematic treatment of the construction of such systems and to collect some recent results. We begin by recalling the basic constructions of geometric quantization in both the Kähler and non-Kähler cases. We then study the reproducing kernels associated to the quantum Hilbert spaces and use them to define symplectic coherent states. The rest of the paper is dedicated to the properties of symplectic coherent states and the corresponding Berezin–Toeplitz quantization. Specifically, we study overcompleteness, symplectic analogues of the basic properties of Bargmann’s weighted analytic function spaces, and the ‘maximally classical’ behavior of symplectic coherent states. We also find explicit formulas for symplectic coherent states on compact Riemann surfaces.     

Dynamical origin of quantum mechanics

A new quantization method is proposed to obtain the equation of motion for a quantum system as the Lie-Poisson equation for a probability current on the S¹-bundle over a physical space.     

The Geometry of Recursion Operators

We study the fields of endomorphisms intertwining pairs of symplectic structures. Using these endomorphisms we prove an analogue of Moser’s theorem for simultaneous isotopies of two families of symplectic forms. We also consider the geometric structures defined by pairs and triples of symplectic forms for which the squares of the intertwining endomorphisms are plus or minus the identity. For pairs of forms we recover the notions of symplectic pairs and of holomorphic symplectic structures. For triples we recover the notion of a hypersymplectic structure, and we also find three new structures that have not been considered before. One of these is the symplectic formulation of hyper-Kähler geometry, which turns out to be a strict generalization of the usual definition in terms of differential or Kähler geometry.     

Chaos in classical cosmology (II)

We present a new dynamical calculation about the Friedman-Robertson-Walker universe considered as an autonomous Hamiltonian. The time evolution of this Hamiltonian presents numerical instabilities so we apply a symplectic integration via infinitesimal canonical transformations of the phase space time evolution that preserves the Poincaré invariant. In this way, we have also obtained a sensitive improvement in the accuracy of the Hamiltonian constraint, as well as in the computing time. We confirm our previous results; in a spatially closed universe, the route to chaos is reached by sucessive breakage of the resonant tori due to the action of 11 resonances.     

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.