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Symplectic vector space
About: Symplectic vector space is a research topic. Over the lifetime, 2048 publications have been published within this topic receiving 53456 citations.
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01 Feb 1976
TL;DR: In this article, Liberman et al. gave a construction of closed symplectic manifolds with no Kaehler structure, and showed that such manifolds do not have even odd Betti numbers.
Abstract: This is a construction of closed symplectic manifolds with no Kaehler structure. A symplectic manifold is a manifold of dimension 2k with a closed 2-form a such that ak is nonsingular. If M2k is a closed symplectic manifold, then the cohomology class of a is nontrivial, and all its powers through k are nontrivial. M also has an almost complex structure associated with a, up-to homotopy. It has been asked whether every closed symplectic manifold has also a Kaehler structure (the converse is immediate). A Kaehler manifold has the property that its odd dimensional Betti numbers are even. H. Guggenheimer claimed [1], [2] that a symplectic manifold also has even odd Betti numbers. In the review [3] of [1], Liberman noted that the proof was incomplete. We produce elementary examples of symplectic manifolds which are not Kaehler by constructing counterexamples to Guggenheimer's assertion. There is a representation p of Z E Z in the group of diffeomorphisms of T2 defined by (1, 0) -P4 id, (0,1 I 0o 1l where [81 ]" denotes the transformation of T2 covered by the linear transformation of R2. This representation determines a bundle M4 over T with fiber T2: M4 = T2 XZ9Z T2, where Z E Z acts on T2 by covering transformations, and on T2 by p (M4 can also be seen as R4 modulo a group of affine transformations). Let Q1 be the standard volume form for T2. Since p preserves 21, this defines a closed 2-form i2 on M4 which is nonsingular on each fiber. Let p be projection to the base: then it can be checked that S21 + P*' 1 is a symplectic form. (It is, in general, true that "'j + Kp* 21 is a symplectic form, for any closed Q'1 which is a volume form for each fiber, and K sufficiently large.) But H1 (M4) = Z @ Z @ Z, so M4 is not a Kaehler manifold. Many more examples can be constructed. In the same vein, if M2k is a closed symplectic manifold, and if N2k+2 fibers over M2k with the fundamental class of the fiber not homologous to zero in N, then N is also a symplectic manifold. If, for instance, the Euler characteristic of the fiber is not zero, this Received by the editors July 31, 1974. AMS (MOS) subject classifications (1970). Primary 57D15, 58H05. C American Mathematical Society 1976
498 citations
TL;DR: In this paper, the authors define a subset of the path space whose trajectories are given by the solutions of the Cauchy-Riemann equation with respect to a suitable almost complex structure on a symplectic manifold.
Abstract: The symplectic action can be defined on the space of smooth paths in a symplectic manifold P which join two Lagrangian submanifolds of P. To pursue a new approach to the variational theory of this function, we define on a subset of the path space the flow whose trajectories are given by the solutions of the Cauchy-Riemann equation with respect to a suitable almost complex structure on P. In particular, we prove compactness and transversality results for the set of bounded trajectories.
410 citations
Book•
18 May 2006
TL;DR: In this paper, a phase space Weyl Calculus is introduced and the uncertainty principle of the Density Operator is defined. But the complexity of the Weyl calculus is not discussed.
Abstract: Symplectic Geometry.- Symplectic Spaces and Lagrangian Planes.- The Symplectic Group.- Multi-Oriented Symplectic Geometry.- Intersection Indices in Lag(n) and Sp(n).- Heisenberg Group, Weyl Calculus, and Metaplectic Representation.- Lagrangian Manifolds and Quantization.- Heisenberg Group and Weyl Operators.- The Metaplectic Group.- Quantum Mechanics in Phase Space.- The Uncertainty Principle.- The Density Operator.- A Phase Space Weyl Calculus.
375 citations
Book•
09 Dec 2010
TL;DR: In this paper, Symplectic Algebra and Geometry Preliminaries of Differential Manifolds are presented, as well as Symmlectic Difference Schemes for Hamiltonian Systems.
Abstract: Preliminaries of Differential Manifolds.- Symplectic Algebra and Geometry Preliminaries.- Hamiltonian Mechanics and Symplectic Geometry.- Symplectic Difference Schemes for Hamiltonian Systems.- The Generating Function Method.- The Calculus of Generating Function and Formal Energy.- Symplectic Runge-Kutta Methods.- Composition Scheme.- Formal Power Series and B-Series.- Volume-Preserving Methods for Source-Free Systems.- Free Systems.- Contact Algorithms for Contact Dynamic Systems.- Poisson Bracket and Lie-Poisson Schemes.- KAM Theorem of Symplectic Algorithms.- Lee-Variational Integrator.- Structure Preserving Schemes for Birkhoff Systems.- Multisymplectic and Variational Integrators.
353 citations
TL;DR: In this article, the Gray map from a corresponding kerdock code to its binary image was shown to induce a geometric map from symplectic to orthogonal spreads, leading to the construction, for any odd composite $m, of large numbers of kerdock codes.
Abstract: When $m$ is odd, spreads in an orthogonal vector space of type $\Omega^+ (2m+2,2)$ are related to binary Kerdock codes and extremal line-sets in $\RR^{2^{m+1}}$ with prescribed angles. Spreads in a $2m$-dimensional binary symplectic vector space are related to Kerdock codes over $\ZZ_4$ and extremal line-sets in $\CC^{2^m}$ with prescribed angles. These connections involve binary, real and complex geometry associated with extraspecial 2-groups. A geometric map from symplectic to orthogonal spreads is shown to induce the Gray map from a corresponding $\ZZ_4$-Kerdock code to its binary image. These geometric considerations lead to the construction, for any odd composite $m$, of large numbers of $\ZZ_4$-Kerdock codes. They also produce new $\ZZ_4$-linear Kerdock and Preparata codes. 1991 Mathematics Subject Classification: primary 94B60; secondary 51M15, 20C99.
336 citations