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Journal articles on the topic 'Symplectic and Poisson geometry'

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Author: Grafiati

Published: 11 March 2023

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1

BOJOWALD, MARTIN, and THOMAS STROBL. "POISSON GEOMETRY IN CONSTRAINED SYSTEMS." Reviews in Mathematical Physics 15, no. 07 (September 2003): 663–703. http://dx.doi.org/10.1142/s0129055x0300176x.

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Associated to a constrained system with closed constraint algebra there are two Poisson manifolds P and Q forming a symplectic dual pair with respect to the original, unconstrained phase space: P is the image of the constraint map (equipped with the algebra of constraints) and Q the Poisson quotient with respect to the orbits generated by the constraints (the orbit space is assumed to be a manifold). We provide sufficient conditions so that the reduced phase space of the constrained system may be identified with a symplectic leaf of Q. By these methods, a second class constrained system with closed algebra is reformulated as an abelian first class system in an extended phase space. While any Poisson manifold (P,Π) has a symplectic realization (Karasev, Weinstein 87), it does not always permit a leafwise symplectic embedding into a symplectic manifold (M,ω). For regular P, it is seen that such an embedding exists, iff the characteristic form-class of Π, a certain element of the third relative cohomology of P, vanishes. A tubular neighborhood of the constraint surface of a general second class constrained system equipped with the Dirac bracket provides a physical example for such an embedding into the original symplectic manifold. In contrast, a leafwise symplectic embedding of e.g. (the maximal regular part of) a Poisson Lie manifold associated to a compact, semisimple Lie algebra does not exist.

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2

Contreras, Ivan, and Nicolás Martínez Alba. "Poly-symplectic geometry and the AKSZ formalism." Reviews in Mathematical Physics 33, no. 09 (May 31, 2021): 2150030. http://dx.doi.org/10.1142/s0129055x21500306.

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In this paper, we extend the AKSZ formulation of the Poisson sigma model to more general target spaces, and we develop the general theory of graded geometry for poly-symplectic and poly-Poisson structures. In particular, we prove a Schwarz-type theorem and transgression for graded poly-symplectic structures, recovering the action functional and the poly-symplectic structure of the reduced phase space of the poly-Poisson sigma model, from the AKSZ construction.

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3

Cahen, Michel, and LORENZ J. SCHWACHH�FER. "Special Symplectic Connections and Poisson Geometry." Letters in Mathematical Physics 69, no. 1-3 (July 2004): 115–37. http://dx.doi.org/10.1007/s11005-004-0474-4.

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4

Crooks, Peter, and Markus Röser. "The $\log$ symplectic geometry of Poisson slices." Journal of Symplectic Geometry 20, no. 1 (2022): 135–90. http://dx.doi.org/10.4310/jsg.2022.v20.n1.a4.

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5

Guillemin, Victor, Eva Miranda, and Ana Rita Pires. "Symplectic and Poisson geometry on b-manifolds." Advances in Mathematics 264 (October 2014): 864–96. http://dx.doi.org/10.1016/j.aim.2014.07.032.

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6

Ortega, Juan-Pablo, and Judor S. Ratiu. "Symmetry Reduction in Symplectic and Poisson Geometry." Letters in Mathematical Physics 69, no. 1-3 (July 2004): 11–60. http://dx.doi.org/10.1007/s11005-004-0898-x.

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7

Ivancevic, V., and C. E. M. Pearce. "Poisson manifolds in generalised Hamiltonian biomechanics." Bulletin of the Australian Mathematical Society 63, no. 3 (June 2001): 515–26. http://dx.doi.org/10.1017/s0004972700019584.

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In this paper the generalised Hamiltonian approach to the modelling of dynamical systems is developed no via the standard formalism of symplectic geometry but rather via Poisson manifolds and evolution equations. This alternative approach has the merit of being available in a wider context than the former. Application is made to three biomechanical models, one in which the symplectic–geometry approach also applies (the motion of a body segment) and two in which it does not (Schwan's model of blood and lymph circulation and Davydov's molecular model of muscle contraction).

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8

ASADI, E., and J. A. SANDERS. "INTEGRABLE SYSTEMS IN SYMPLECTIC GEOMETRY." Glasgow Mathematical Journal 51, A (February 2009): 5–23. http://dx.doi.org/10.1017/s0017089508004746.

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AbstractQuaternionic vector mKDV equations are derived from the Cartan structure equation in the symmetric space=Sp(n+1)/Sp(1) ×Sp(n). The derivation of the soliton hierarchy utilizes a moving parallel frame and a Cartan connection 1-form ω related to the Cartan geometry onmodelled on$(\mk{sp}_{n+1}, \mk{sp}_{1}\,{\times}\, \mk{sp}_{n})$. The integrability structure is shown to be geometrically encoded by a Poisson–Nijenhuis structure and a symplectic operator.

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LAVROV, P. M., and O. V. RADCHENKO. "SYMPLECTIC GEOMETRIES ON SUPERMANIFOLDS." International Journal of Modern Physics A 23, no. 09 (April 10, 2008): 1337–50. http://dx.doi.org/10.1142/s0217751x08039426.

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Extension of symplectic geometry on manifolds to the supersymmetric case is considered. In the even case it leads to the even symplectic geometry (or, equivalently, to the geometry on supermanifolds endowed with a nondegenerate Poisson bracket) or to the geometry on an even Fedosov supermanifolds. It is proven that in the odd case there are two different scalar structures (namely, an odd closed differential 2-form and the antibracket) which can be used for construction of two different types of odd geometries on supermanifolds.

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10

Frejlich, Pedro, and Ioan Mărcuț. "The Homology Class of a Poisson Transversal." International Mathematics Research Notices 2020, no. 10 (May 23, 2018): 2952–76. http://dx.doi.org/10.1093/imrn/rny105.

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Abstract This note is devoted to the study of the homology class of a compact Poisson transversal in a Poisson manifold. For specific classes of Poisson structures, such as unimodular Poisson structures and Poisson manifolds with closed leaves, we prove that all their compact Poisson transversals represent nontrivial homology classes, generalizing the symplectic case. We discuss several examples in which this property does not hold, as well as a weaker version of this property, which holds for log-symplectic structures. Finally, we extend our results to Dirac geometry.

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11

Lanius, Melinda. "Symplectic, Poisson, and contact geometry on scattering manifolds." Pacific Journal of Mathematics 310, no. 1 (January 26, 2021): 213–56. http://dx.doi.org/10.2140/pjm.2021.310.213.

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12

Barbaresco, Frédéric, and François Gay-Balmaz. "Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics." Entropy 22, no. 5 (April 25, 2020): 498. http://dx.doi.org/10.3390/e22050498.

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In this paper, we describe and exploit a geometric framework for Gibbs probability densities and the associated concepts in statistical mechanics, which unifies several earlier works on the subject, including Souriau’s symplectic model of statistical mechanics, its polysymplectic extension, Koszul model, and approaches developed in quantum information geometry. We emphasize the role of equivariance with respect to Lie group actions and the role of several concepts from geometric mechanics, such as momentum maps, Casimir functions, coadjoint orbits, and Lie-Poisson brackets with cocycles, as unifying structures appearing in various applications of this framework to information geometry and machine learning. For instance, we discuss the expression of the Fisher metric in presence of equivariance and we exploit the property of the entropy of the Souriau model as a Casimir function to apply a geometric model for energy preserving entropy production. We illustrate this framework with several examples including multivariate Gaussian probability densities, and the Bogoliubov-Kubo-Mori metric as a quantum version of the Fisher metric for quantum information on coadjoint orbits. We exploit this geometric setting and Lie group equivariance to present symplectic and multisymplectic variational Lie group integration schemes for some of the equations associated with Souriau symplectic and polysymplectic models, such as the Lie-Poisson equation with cocycle.

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13

Crainic, Marius, Rui Loja Fernandes, and David Martínez Torres. "Poisson manifolds of compact types (PMCT 1)." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 756 (November 1, 2019): 101–49. http://dx.doi.org/10.1515/crelle-2017-0006.

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AbstractThis is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompasses several classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamental properties and constructions of PMCTs. For instance, we show that their Poisson cohomology behaves very much like the de Rham cohomology of a compact manifold (Hodge decomposition, non-degenerate Poincaré duality pairing, etc.) and that the Moser trick can be adapted to PMCTs. More important, we find unexpected connections between PMCTs and symplectic topology: PMCTs are related with the theory of Lagrangian fibrations and we exhibit a construction of a non-trivial PMCT related to a classical question on the topology of the orbits of a free symplectic circle action. In subsequent papers, we will establish deep connections between PMCTs and integral affine geometry, Hamiltonian G-spaces, foliation theory, orbifolds, Lie theory and symplectic gerbes.

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14

Barbaresco, Frédéric. "Symplectic Foliation Structures of Non-Equilibrium Thermodynamics as Dissipation Model: Application to Metriplectic Nonlinear Lindblad Quantum Master Equation." Entropy 24, no. 11 (November 9, 2022): 1626. http://dx.doi.org/10.3390/e24111626.

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The idea of a canonical ensemble from Gibbs has been extended by Jean-Marie Souriau for a symplectic manifold where a Lie group has a Hamiltonian action. A novel symplectic thermodynamics and information geometry known as “Lie group thermodynamics” then explains foliation structures of thermodynamics. We then infer a geometric structure for heat equation from this archetypal model, and we have discovered a pure geometric structure of entropy, which characterizes entropy in coadjoint representation as an invariant Casimir function. The coadjoint orbits form the level sets on the entropy. By using the KKS 2-form in the affine case via Souriau’s cocycle, the method also enables the Fisher metric from information geometry for Lie groups. The fact that transverse dynamics to these symplectic leaves is dissipative, whilst dynamics along these symplectic leaves characterize non-dissipative phenomenon, can be used to interpret this Lie group thermodynamics within the context of an open system out of thermodynamics equilibrium. In the following section, we will discuss the dissipative symplectic model of heat and information through the Poisson transverse structure to the symplectic leaf of coadjoint orbits, which is based on the metriplectic bracket, which guarantees conservation of energy and non-decrease of entropy. Baptiste Coquinot recently developed a new foundation theory for dissipative brackets by taking a broad perspective from non-equilibrium thermodynamics. He did this by first considering more natural variables for building the bracket used in metriplectic flow and then by presenting a methodical approach to the development of the theory. By deriving a generic dissipative bracket from fundamental thermodynamic first principles, Baptiste Coquinot demonstrates that brackets for the dissipative part are entirely natural, just as Poisson brackets for the non-dissipative part are canonical for Hamiltonian dynamics. We shall investigate how the theory of dissipative brackets introduced by Paul Dirac for limited Hamiltonian systems relates to transverse structure. We shall investigate an alternative method to the metriplectic method based on Michel Saint Germain’s PhD research on the transverse Poisson structure. We will examine an alternative method to the metriplectic method based on the transverse Poisson structure, which Michel Saint-Germain studied for his PhD and was motivated by the key works of Fokko du Cloux. In continuation of Saint-Germain’s works, Hervé Sabourin highlights the, for transverse Poisson structures, polynomial nature to nilpotent adjoint orbits and demonstrated that the Casimir functions of the transverse Poisson structure that result from restriction to the Lie–Poisson structure transverse slice are Casimir functions independent of the transverse Poisson structure. He also demonstrated that, on the transverse slice, two polynomial Poisson structures to the symplectic leaf appear that have Casimir functions. The dissipative equation introduced by Lindblad, from the Hamiltonian Liouville equation operating on the quantum density matrix, will be applied to illustrate these previous models. For the Lindblad operator, the dissipative component has been described as the relative entropy gradient and the maximum entropy principle by Öttinger. It has been observed then that the Lindblad equation is a linear approximation of the metriplectic equation

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15

Deriglazov, Alexei A. "Basic Notions of Poisson and Symplectic Geometry in Local Coordinates, with Applications to Hamiltonian Systems." Universe 8, no. 10 (October 17, 2022): 536. http://dx.doi.org/10.3390/universe8100536.

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This work contains a brief and elementary exposition of the foundations of Poisson and symplectic geometries, with an emphasis on applications for Hamiltonian systems with second-class constraints. In particular, we clarify the geometric meaning of the Dirac bracket on a symplectic manifold and provide a proof of the Jacobi identity on a Poisson manifold. A number of applications of the Dirac bracket are described: applications for proof of the compatibility of a system consisting of differential and algebraic equations, as well as applications for the problem of the reduction of a Hamiltonian system with known integrals of motion.

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16

Alekseev, Anton, Benjamin Hoffman, Jeremy Lane, and Yanpeng Li. "Concentration of symplectic volumes on Poisson homogeneous spaces." Journal of Symplectic Geometry 18, no. 5 (2020): 1197–220. http://dx.doi.org/10.4310/jsg.2020.v18.n5.a1.

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17

Treloar, Thomas. "The Symplectic Geometry of Polygons in the 3-Sphere." Canadian Journal of Mathematics 54, no. 1 (February 1, 2002): 30–54. http://dx.doi.org/10.4153/cjm-2002-002-1.

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AbstractWe study the symplectic geometry of the moduli spaces Mr = Mr() of closed n-gons with fixed side-lengths in the 3-sphere. We prove that these moduli spaces have symplectic structures obtained by reduction of the fusion product of n conjugacy classes in SU(2) by the diagonal conjugation action of SU(2). Here the fusion product of n conjugacy classes is a Hamiltonian quasi-Poisson SU(2)-manifold in the sense of [AKSM]. An integrable Hamiltonian system is constructed on Mr in which the Hamiltonian flows are given by bending polygons along a maximal collection of nonintersecting diagonals. Finally, we show the symplectic structure on Mr relates to the symplectic structure obtained from gauge-theoretic description of Mr. The results of this paper are analogues for the 3-sphere of results obtained for Mr(), the moduli space of n-gons with fixed side-lengths in hyperbolic 3-space [KMT], and for Mr(), the moduli space of n-gons with fixed side-lengths in [KM1].

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18

Beltiţă, D., and T. S. Ratiu. "Symplectic leaves in real banach Lie–Poisson spaces." GAFA Geometric And Functional Analysis 15, no. 4 (August 2005): 753–79. http://dx.doi.org/10.1007/s00039-005-0524-9.

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19

Vorobjev, Yurii. "Coupling tensors and Poisson geometry near a single symplectic leaf." Banach Center Publications 54 (2001): 249–74. http://dx.doi.org/10.4064/bc54-0-14.

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20

ISIDRO, JOSÉ M. "DIRAC BRACKETS FROM MAGNETIC BACKGROUNDS." International Journal of Geometric Methods in Modern Physics 04, no. 04 (June 2007): 523–32. http://dx.doi.org/10.1142/s0219887807002181.

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In symplectic mechanics, the magnetic term describing the interaction between a charged particle and an external magnetic field has to be introduced by hand. On the contrary, in generalized complex geometry, such magnetic terms in the symplectic form arise naturally by means of B-transformations. Here we prove that, regarding classical phase space as a generalized complex manifold, the transformation law for the symplectic form under the action of a weak magnetic field gives rise to Dirac's prescription for Poisson brackets in the presence of constraints.

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21

DOLAN, BRIAN P. "SYMPLECTIC GEOMETRY AND HAMILTONIAN FLOW OF THE RENORMALIZATION GROUP EQUATION." International Journal of Modern Physics A 10, no. 18 (July 20, 1995): 2703–32. http://dx.doi.org/10.1142/s0217751x95001273.

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It is argued that renormalization group flow can be interpreted as a Hamiltonian vector flow on a phase space which consists of the couplings of the theory and their conjugate “momenta,” which are the vacuum expectation values of the corresponding composite operators. The Hamiltonian is linear in the conjugate variables and can be identified with the vacuum expectation value of the trace of the energy-momentum operator. For theories with massive couplings the identity operator plays a central role and its associated coupling gives rise to a potential in the flow equations. The evolution of any quantity, such as N-point Green functions, under renormalization group flow can be obtained from its Poisson bracket with the Hamiltonian. Ward identities can be represented as constants of the motion which act as symmetry generators on the phase space via the Poisson bracket structure.

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22

Crooks, Peter, and Maxence Mayrand. "Symplectic reduction along a submanifold." Compositio Mathematica 158, no. 9 (September 2022): 1878–934. http://dx.doi.org/10.1112/s0010437x22007710.

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We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex algebraic varieties, and has an interpretation in terms of derived stacks in shifted symplectic geometry. It also encompasses Marsden–Weinstein–Meyer reduction, Mikami–Weinstein reduction, the pre-images of Poisson transversals under moment maps, symplectic cutting, symplectic implosion, and the Ginzburg–Kazhdan construction of Moore–Tachikawa varieties in topological quantum field theory. A key feature of our construction is a concrete and systematic association of a Hamiltonian $G$ -space $\mathfrak {M}_{G, S}$ to each pair $(G,S)$ , where $G$ is any Lie group and $S\subseteq \mathrm {Lie}(G)^{*}$ is any submanifold satisfying certain non-degeneracy conditions. The spaces $\mathfrak {M}_{G, S}$ satisfy a universal property for symplectic reduction which generalizes that of the universal imploded cross-section. Although these Hamiltonian $G$ -spaces are explicit and natural from a Lie-theoretic perspective, some of them appear to be new.

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23

BIELIAVSKY, PIERRE, and PHILIPPE BONNEAU. "ON THE GEOMETRY OF THE CHARACTERISTIC CLASS OF A STAR PRODUCT ON A SYMPLECTIC MANIFOLD." Reviews in Mathematical Physics 15, no. 02 (April 2003): 199–215. http://dx.doi.org/10.1142/s0129055x0300159x.

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The characteristic class of a star product on a symplectic manifold appears as the class of a deformation of a given symplectic connection, as described by Fedosov. In contrast, one usually thinks of the characteristic class of a star product as the class of a deformation of the Poisson structure (as in Kontsevich's work). In this paper, we present, in the symplectic framework, a natural procedure for constructing a star product by directly quantizing a deformation of the symplectic structure. Basically, in Fedosov's recursive formula for the star product with zero characteristic class, we replace the symplectic structure by one of its formal deformations in the parameter ℏ. We then show that every equivalence class of star products contains such an element. Moreover, within a given class, equivalences between such star products are realized by formal one-parameter families of diffeomorphisms, as produced by Moser's argument.

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24

Mori, Atsuhide. "Symplectic/Contact Geometry Related to Bayesian Statistics." Proceedings 46, no. 1 (November 17, 2019): 13. http://dx.doi.org/10.3390/ecea-5-06665.

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In the previous work, the author gave the following symplectic/contact geometric description of the Bayesian inference of normal means: The space H of normal distributions is an upper halfplane which admits two operations, namely, the convolution product and the normalized pointwise product of two probability density functions. There is a diffeomorphism F of H that interchanges these operations as well as sends any e-geodesic to an e-geodesic. The product of two copies of H carries positive and negative symplectic structures and a bi-contact hypersurface N. The graph of F is Lagrangian with respect to the negative symplectic structure. It is contained in the bi-contact hypersurface N. Further, it is preserved under a bi-contact Hamiltonian flow with respect to a single function. Then the restriction of the flow to the graph of F presents the inference of means. The author showed that this also works for the Student t-inference of smoothly moving means and enables us to consider the smoothness of data smoothing. In this presentation, the space of multivariate normal distributions is foliated by means of the Cholesky decomposition of the covariance matrix. This provides a pair of regular Poisson structures, and generalizes the above symplectic/contact description to the multivariate case. The most of the ideas presented here have been described at length in a later article of the author.

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25

Chu, Chong-Sun, and Pei-Ming Ho. "Poisson Algebra of Differential Forms." International Journal of Modern Physics A 12, no. 31 (December 20, 1997): 5573–87. http://dx.doi.org/10.1142/s0217751x97002929.

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We give a natural definition of a Poisson differential algebra. Consistency conditions are formulated in geometrical terms. It is found that one can often locally put the Poisson structure on the differential calculus in a simple canonical form by a coordinate trans-formation. This is in analogy with the standard Darboux's theorem for symplectic geometry. For certain cases there exists a realization of the exterior derivative through a certain canonical one-form. All the above are carried out similarly for the case of a complex Poisson differential algebra. The case of one complex dimension is treated in detail and interesting features are noted. Conclusions are made in the last section.

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26

Chruściński, Dariusz, and Giuseppe Marmo. "Remarks on the GNS Representation and the Geometry of Quantum States." Open Systems & Information Dynamics 16, no. 02n03 (September 2009): 157–77. http://dx.doi.org/10.1142/s1230161209000128.

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It is shown how to introduce a geometric formulation of the algebraic approach to the standard non-relativistic quantum mechanics. It turns out that the GNS representation provides not only symplectic but even Hermitian realization of a 'quantum Poisson algebra'. We discuss alternative Hamiltonian structures emerging out of different GNS representations which provide a natural setting for quantum bi-Hamiltonian systems.

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27

Mayrand, Maxence. "Stratification of singular hyperkähler quotients." Complex Manifolds 9, no. 1 (January 1, 2022): 261–84. http://dx.doi.org/10.1515/coma-2021-0140.

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Abstract Hyperkähler quotients by non-free actions are typically singular, but are nevertheless partitioned into smooth hyperkähler manifolds. We show that these partitions are topological stratifications, in a strong sense. We also endow the quotients with global Poisson structures which recover the hyperkähler structures on the strata. Finally, we give a local model which shows that these quotients are locally isomorphic to linear complex-symplectic reductions in the GIT sense. These results can be thought of as the hyperkähler analogues of Sjamaar–Lerman’s theorems for singular symplectic reduction. They are based on a local normal form for the underlying complex-Hamiltonian manifold, which may be of independent interest.

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28

Khesin, Boris, Gerard Misiołek, and Klas Modin. "Geometric hydrodynamics and infinite-dimensional Newton’s equations." Bulletin of the American Mathematical Society 58, no. 3 (June 2, 2021): 377–442. http://dx.doi.org/10.1090/bull/1728.

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We revisit the geodesic approach to ideal hydrodynamics and present a related geometric framework for Newton’s equations on groups of diffeomorphisms and spaces of probability densities. The latter setting is sufficiently general to include equations of compressible and incompressible fluid dynamics, magnetohydrodynamics, shallow water systems and equations of relativistic fluids. We illustrate this with a survey of selected examples, as well as with new results, using the tools of infinite-dimensional information geometry, optimal transport, the Madelung transform, and the formalism of symplectic and Poisson reduction.

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29

Schmidt, J. R. "The Darboux coordinate system and Holstein–Primakoff representations on Kähler manifolds." Canadian Journal of Physics 84, no. 10 (October 1, 2006): 891–904. http://dx.doi.org/10.1139/p06-081.

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The Kahler geometry of minimal coadjoint orbits of classical Lie groups is exploited to construct Darboux coordinates, a symplectic two-form and a Lie–Poisson structure on the dual of the Lie algebra. Canonical transformations cast the generators of the dual into Dyson or Holstein–Primakoff representations.PACS Nos.: 02.20.Sv, 02.30.Ik, 02.40.Tt

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SÄMANN, CHRISTIAN, and RICHARD J. SZABO. "GROUPOIDS, LOOP SPACES AND QUANTIZATION OF 2-PLECTIC MANIFOLDS." Reviews in Mathematical Physics 25, no. 03 (April 2013): 1330005. http://dx.doi.org/10.1142/s0129055x13300057.

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We describe the quantization of 2-plectic manifolds as they arise in the context of the quantum geometry of M-branes and non-geometric flux compactifications of closed string theory. We review the groupoid approach to quantizing Poisson manifolds in detail, and then extend it to the loop spaces of 2-plectic manifolds, which are naturally symplectic manifolds. In particular, we discuss the groupoid quantization of the loop spaces of ℝ3, 𝕋3and S3, and derive some interesting implications which match physical expectations from string theory and M-theory.

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31

Dufour, Jean-Paul, and Aïssa Wade. "On the local structure of Dirac manifolds." Compositio Mathematica 144, no. 3 (May 2008): 774–86. http://dx.doi.org/10.1112/s0010437x07003272.

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AbstractWe give a local normal form for Dirac structures. As a consequence, we show that the dimensions of the pre-symplectic leaves of a Dirac manifold have the same parity. We also show that, given a point m of a Dirac manifold M, there is a well-defined transverse Poisson structure to the pre-symplectic leaf P through m. Finally, we describe the neighborhood of a pre-symplectic leaf in terms of geometric data. This description agrees with that given by Vorobjev for the Poisson case.

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32

JEJJALA, VISHNU, DJORDJE MINIC, and CHIA-HSIUNG TZE. "TOWARDS A BACKGROUND INDEPENDENT QUANTUM THEORY OF GRAVITY." International Journal of Modern Physics D 13, no. 10 (December 2004): 2307–13. http://dx.doi.org/10.1142/s0218271804006371.

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Any canonical quantum theory can be understood to arise from the compatibility of the statistical geometry of distinguishable observations with the canonical Poisson structure of Hamiltonian dynamics. This geometric perspective offers a novel, background independent non-perturbative formulation of quantum gravity. We invoke a quantum version of the equivalence principle, which requires both the statistical and symplectic geometries of canonical quantum theory to be fully dynamical quantities. Our approach sheds new light on such basic issues of quantum gravity as the nature of observables, the problem of time, and the physics of the vacuum. In particular, the observed numerical smallness of the cosmological constant can be rationalized in this approach.

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33

Mehta, Rajan Amit. "On homotopy Poisson actions and reduction of symplectic Q-manifolds." Differential Geometry and its Applications 29, no. 3 (June 2011): 319–28. http://dx.doi.org/10.1016/j.difgeo.2011.03.002.

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34

Lane, Jeremy. "The Geometric Structure of Symplectic Contraction." International Mathematics Research Notices 2020, no. 12 (June 8, 2018): 3521–39. http://dx.doi.org/10.1093/imrn/rny122.

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Abstract We show that the symplectic contraction map of Hilgert–Manon–Martens [9], a symplectic version of Popov’s horospherical contraction, is simply the quotient of a Hamiltonian manifold $M$ by a “stratified null foliation” that is determined by the group action and moment map. We also show that the quotient differential structure on the symplectic contraction of $M$ supports a Poisson bracket. We end by proving a very general description of the topology of fibers of Gelfand–Zeitlin (also spelled Gelfand–Tsetlin or Gelfand–Cetlin) systems on multiplicity-free Hamiltonian $U(n)$ and $SO(n)$ manifolds.

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35

FRYDRYSZAK, ANDRZEJ. "NILPOTENT CLASSICAL MECHANICS." International Journal of Modern Physics A 22, no. 14n15 (June 20, 2007): 2513–33. http://dx.doi.org/10.1142/s0217751x07036749.

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The formalism of nilpotent mechanics is introduced in the Lagrangian and Hamiltonian form. Systems are described using nilpotent, commuting coordinates η. Necessary geometrical notions and elements of generalized differential η-calculus are introduced. The so-called s-geometry, in a special case when it is orthogonally related to a traceless symmetric form, shows some resemblances to the symplectic geometry. As an example of an η-system the nilpotent oscillator is introduced and its supersymmetrization considered. It is shown that the R-symmetry known for the graded superfield oscillator also present here for the supersymmetric η-system. The generalized Poisson bracket for (η, p)-variables satisfies modified Leibniz rule and has nontrivial Jacobiator.

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36

Wolpert, Scott A. "Products of twists, geodesic lengths and Thurston shears." Compositio Mathematica 151, no. 2 (October 9, 2014): 313–50. http://dx.doi.org/10.1112/s0010437x1400757x.

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AbstractThurston introduced shear deformations (cataclysms) on geodesic laminations–deformations including left and right displacements along geodesics. For hyperbolic surfaces with cusps, we consider shear deformations on disjoint unions of ideal geodesics. The length of a balanced weighted sum of ideal geodesics is defined and the Weil–Petersson (WP) duality of shears and the defined length is established. The Poisson bracket of a pair of balanced weight systems on a set of disjoint ideal geodesics is given in terms of an elementary$2$-form. The symplectic geometry of balanced weight systems on ideal geodesics is developed. Equality of the Fock shear coordinate algebra and the WP Poisson algebra is established. The formula for the WP Riemannian pairing of shears is also presented.

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37

Schnitzer, Jonas, and Luca Vitagliano. "The Local Structure of Generalized Contact Bundles." International Mathematics Research Notices 2020, no. 20 (February 25, 2019): 6871–925. http://dx.doi.org/10.1093/imrn/rnz009.

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Abstract Generalized contact bundles are odd-dimensional analogues of generalized complex manifolds. They have been introduced recently and very little is known about them. In this paper we study their local structure. Specifically, we prove a local splitting theorem similar to those appearing in Poisson geometry. In particular, in a neighborhood of a regular point, a generalized contact bundle is either the product of a contact and a complex manifold or the product of a symplectic manifold and a manifold equipped with an integrable complex structure on the gauge algebroid of the trivial line bundle.

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38

Spies, Alexander. "Poisson-geometric Analogues of Kitaev Models." Communications in Mathematical Physics 383, no. 1 (March 9, 2021): 345–400. http://dx.doi.org/10.1007/s00220-021-03992-5.

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AbstractWe define Poisson-geometric analogues of Kitaev’s lattice models. They are obtained from a Kitaev model on an embedded graph $$\Gamma $$ Γ by replacing its Hopf algebraic data with Poisson data for a Poisson-Lie group G. Each edge is assigned a copy of the Heisenberg double $${\mathcal {H}}(G)$$ H ( G ) . Each vertex (face) of $$\Gamma $$ Γ defines a Poisson action of G (of $$G^*$$ G ∗ ) on the product of these Heisenberg doubles. The actions for a vertex and adjacent face form a Poisson action of the double Poisson-Lie group D(G). We define Poisson counterparts of vertex and face operators and relate them via the Poisson bracket to the vector fields generating the actions of D(G). We construct an isomorphism of Poisson D(G)-spaces between this Poisson-geometrical Kitaev model and Fock and Rosly’s Poisson structure for the graph $$\Gamma $$ Γ and the Poisson-Lie group D(G). This decouples the latter and represents it as a product of Heisenberg doubles. It also relates the Poisson-geometrical Kitaev model to the symplectic structure on the moduli space of flat D(G)-bundles on an oriented surface with boundary constructed from $$\Gamma $$ Γ .

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39

Bove, Antonio, and David S. Tartakoff. "Analytic Hypoellipticity at Non-Symplectic Poisson-Treves Strata for Certain Sums of Squares of Vector Fields." Journal of Geometric Analysis 18, no. 4 (July 10, 2008): 1002–21. http://dx.doi.org/10.1007/s12220-008-9043-x.

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40

Suárez-Serrato, P., J. Torres Orozco, and R. Vera. "Poisson and near-symplectic structures on generalized wrinkled fibrations in dimension 6." Annals of Global Analysis and Geometry 55, no. 4 (February 13, 2019): 777–804. http://dx.doi.org/10.1007/s10455-019-09651-2.

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Entov, Michael, and Leonid Polterovich. "Rigid subsets of symplectic manifolds." Compositio Mathematica 145, no. 03 (May 2009): 773–826. http://dx.doi.org/10.1112/s0010437x0900400x.

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AbstractWe show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P. Albers and P. Biran-O. Cornea) as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-simplicity of the quantum homology of symplectic manifolds. The proofs are based on the Floer-theoretical machinery of partial symplectic quasi-states.

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42

Marle, Charles-Michel. "The Inception of Symplectic Geometry: the Works of Lagrange and Poisson During the Years 1808–1810." Letters in Mathematical Physics 90, no. 1-3 (October 17, 2009): 3–21. http://dx.doi.org/10.1007/s11005-009-0347-y.

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43

Vaisman, Izu. "On the geometric quantization of the symplectic leaves of Poisson manifolds." Differential Geometry and its Applications 7, no. 3 (September 1997): 265–75. http://dx.doi.org/10.1016/s0926-2245(96)00056-3.

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44

Marle, Charles-Michel. "From Tools in Symplectic and Poisson Geometry to J.-M. Souriau’s Theories of Statistical Mechanics and Thermodynamics." Entropy 18, no. 10 (October 19, 2016): 370. http://dx.doi.org/10.3390/e18100370.

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45

Vitagliano, Luca. "Partial Differential Hamiltonian Systems." Canadian Journal of Mathematics 65, no. 5 (October 1, 2013): 1164–200. http://dx.doi.org/10.4153/cjm-2012-055-0.

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AbstractWe define partial differential (PD in the following), i.e., field theoretic analogues ofHamiltonian systems on abstract symplectic manifolds and study their main properties, namely, PD Hamilton equations, PD Noether theorem, PD Poisson bracket, etc. Unlike the standard multisymplectic approach to Hamiltonian field theory, in our formalism, the geometric structure (kinematics) and the dynamical information on the “phase space” appear as just different components of one single geometric object.

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46

Caticha, Ariel. "The Entropic Dynamics Approach to Quantum Mechanics." Entropy 21, no. 10 (September 26, 2019): 943. http://dx.doi.org/10.3390/e21100943.

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Entropic Dynamics (ED) is a framework in which Quantum Mechanics is derived as an application of entropic methods of inference. In ED the dynamics of the probability distribution is driven by entropy subject to constraints that are codified into a quantity later identified as the phase of the wave function. The central challenge is to specify how those constraints are themselves updated. In this paper we review and extend the ED framework in several directions. A new version of ED is introduced in which particles follow smooth differentiable Brownian trajectories (as opposed to non-differentiable Brownian paths). To construct ED we make use of the fact that the space of probabilities and phases has a natural symplectic structure (i.e., it is a phase space with Hamiltonian flows and Poisson brackets). Then, using an argument based on information geometry, a metric structure is introduced. It is shown that the ED that preserves the symplectic and metric structures—which is a Hamilton-Killing flow in phase space—is the linear Schrödinger equation. These developments allow us to discuss why wave functions are complex and the connections between the superposition principle, the single-valuedness of wave functions, and the quantization of electric charges. Finally, it is observed that Hilbert spaces are not necessary ingredients in this construction. They are a clever but merely optional trick that turns out to be convenient for practical calculations.

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Grillo, Sergio, and Edith Padrón. "A Hamilton–Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds." Journal of Geometry and Physics 110 (December 2016): 101–29. http://dx.doi.org/10.1016/j.geomphys.2016.07.010.

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48

Moraru, Ruxandra. "Integrable Systems Associated to a Hopf Surface." Canadian Journal of Mathematics 55, no. 3 (June 1, 2003): 609–35. http://dx.doi.org/10.4153/cjm-2003-025-3.

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AbstractA Hopf surface is the quotient of the complex surface by an infinite cyclic group of dilations of . In this paper, we study the moduli spaces of stable -bundles on a Hopf surface , from the point of view of symplectic geometry. An important point is that the surface is an elliptic fibration, which implies that a vector bundle on can be considered as a family of vector bundles over an elliptic curve. We define a map that associates to every bundle on a divisor, called the graph of the bundle, which encodes the isomorphism class of the bundle over each elliptic curve. We then prove that the map G is an algebraically completely integrable Hamiltonian system, with respect to a given Poisson structure on . We also give an explicit description of the fibres of the integrable system. This example is interesting for several reasons; in particular, since the Hopf surface is not Kähler, it is an elliptic fibration that does not admit a section.

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DEGIOVANNI, L., and G. MAGNANO. "TRI–HAMILTONIAN VECTOR FIELDS, SPECTRAL CURVES AND SEPARATION COORDINATES." Reviews in Mathematical Physics 14, no. 10 (October 2002): 1115–63. http://dx.doi.org/10.1142/s0129055x0200151x.

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We show that for a class of dynamical systems, Hamiltonian with respect to three distinct Poisson brackets (P0,P1,P2), separation coordinates are provided by the common roots of a set of bivariate polynomials. These polynomials, which generalise those considered by E. Sklyanin in his algebro-geometric approach, are obtained from the knowledge of: (i) a common Casimir function for the two Poisson pencils (P1-λP0) and (P2-μP0); (ii) a suitable set of vector fields, preserving P0 but transversal to its symplectic leaves. The framework is applied to Lax equations with spectral parameter, for which not only it establishes a theoretical link between the separation techniques of Sklyanin and of Magri, but also provides a more efficient "inverse" procedure to obtain separation variables, not involving the extraction of roots.

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Bascone, Francesco, Franco Pezzella, and Patrizia Vitale. "Topological and Dynamical Aspects of Jacobi Sigma Models." Symmetry 13, no. 7 (July 5, 2021): 1205. http://dx.doi.org/10.3390/sym13071205.

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The geometric properties of sigma models with target space a Jacobi manifold are investigated. In their basic formulation, these are topological field theories—recently introduced by the authors—which share and generalise relevant features of Poisson sigma models, such as gauge invariance under diffeomorphisms and finite dimension of the reduced phase space. After reviewing the main novelties and peculiarities of these models, we perform a detailed analysis of constraints and ensuing gauge symmetries in the Hamiltonian approach. Contact manifolds as well as locally conformal symplectic manifolds are discussed, as main instances of Jacobi manifolds.

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