Showing papers on "Symplectic vector space published in 2006"
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
TL;DR: In this paper, a link between the theory of quasi-state and topological measures has been established, which can be viewed as an algebraic way of packaging certain information contained in Floer theory, and in particular in spectral invariants of Hamiltonian diffeomorphisms introduced by Yong-Geun Oh.
Abstract: We establish a link between symplectic topology and a recently emerg\-ed branch of functional analysis called the theory of quasi-states and quasi-measures (also known as topological measures) In the symplectic context quasi-states can be viewed as an algebraic way of packaging certain information contained in Floer theory, and in particular in spectral invariants of Hamiltonian diffeomorphisms introduced recently by Yong-Geun Oh As a consequence we prove a number of new results on rigidity of intersections in symplectic manifolds This work is a part of a joint project with Paul Biran
175 citations
TL;DR: In this article, the authors define an invariant of oriented links in S 3 using the symplectic geometry of certain spaces which arise naturally in Lie theory and present a knot as the closure of a braid, which in turn views as a loop in configuration space.
Abstract: We define an invariant of oriented links in S 3 using the symplectic geometry of certain spaces which arise naturally in Lie theory. More specifically, we present a knot as the closure of a braid, which in turn we view as a loop in configuration space. Fix an affine subspaceSm of the Lie algebra sl2m(C) which is a transverse slice to the adjoint action at a nilpotent matrix with two equal Jordan blocks. The adjoint quotient map restricted to Sm gives rise to a symplectic fibre bundle over configuration space. An inductive argument constructs a distinguished Lagrangian submani
170 citations
TL;DR: In this article, the notion of the Kodaira dimension for manifolds in dimension 4 was discussed and partial Betti number bounds for 4-manifolds with the dimension zero were given.
Abstract: We discuss the notion of the Kodaira dimension for symplectic manifolds in dimension 4. In particular, we propose and partially verify Betti number bounds for symplectic 4-manifolds with Kodaira dimension zero.
100 citations
TL;DR: In this paper, it was shown that the symplectic 4-manifold Z is minimal unless either one of the Xi contains a ( 1)-sphere disjoint from Fi, or one of them admits a ruling with Fi as a section.
Abstract: Let X1, X2 be symplectic 4-manifolds containing symplectic sur- faces F1, F2 of identical positive genus and opposite squares. Let Z denote the symplectic sum of X1 and X2 along the Fi. Using relative Gromov-Witten theory, we determine precisely when the symplectic 4-manifold Z is minimal (i.e., cannot be blown down); in particular, we prove that Z is minimal unless either: one of the Xi contains a ( 1)-sphere disjoint from Fi; or one of the Xi admits a ruling with Fi as a section. As special cases, this proves a conjecture of Stipsicz asserting the minimality of fiber sums of Lefschetz fibrations, and implies that the non-spin examples constructed by Gompf in his study of the geography problem are minimal. Let (X1,!1), (X2,!2) be symplectic 4-manifolds, and let F1 ⊂ X1, F2 ⊂ X2 be two-dimensional symplectic submanifolds with the same genus whose homology classes satisfy (F1) 2 + (F2) 2 = 0, with the !i normalized to give equal area to the surfaces Fi. For i = 1,2, a neighborhood of Fi is symplectically identified by Weinstein's symplectic neighborhood theorem (19) with the disc normal bundlei of Fi in Xi. Choose a smooth isomorphismof the normal bundle to F1 in X1 (which is a complex line bundle) with the dual of the normal bundle to F2 in X2. According to (2) (and independently (11)), the symplectic sum
88 citations
Book•
26 Jul 2006
TL;DR: Symplectic spinors have been used in many applications, e.g., Lie Derivative and Quantization, Symplectic Connections, Symmlectic Dirac Operators, and Second Order Operators as discussed by the authors.
Abstract: Background on Symplectic Spinors.- Symplectic Connections.- Symplectic Spinor Fields.- Symplectic Dirac Operators.- An Associated Second Order Operator.- The Kahler Case.- Fourier Transform for Symplectic Spinors.- Lie Derivative and Quantization.
67 citations
TL;DR: In this paper, the l-exotic nilpotent cone is introduced for complex symplectic groups, and a character formula and multiplicity formula for simple H-modules are presented.
Abstract: Let G be a complex symplectic group. We introduce a G x (C ^x) ^{l + 1}-variety N_{l}, which we call the l-exotic nilpotent cone. Then, we realize the Hecke algebra H of type C_n ^(1) with three parameters via equivariant algebraic K-theory in terms of the geometry of N_2. This enables us to establish a Deligne-Langlands type classification of "non-critical" simple H-modules. As applications, we present a character formula and multiplicity formulas of H-modules.
63 citations
TL;DR: In this paper, it was shown that if ω is a Kahler form on a complex surface (M,J), then ω(M,ω) agrees with the usual holomorphic Kodaira dimension of (m,J).
Abstract: The Kodaira dimension of a non-minimal manifold is defined to be that of any of its minimal models. It is shown in [12] that, if ω is a Kahler form on a complex surface (M,J), then κ(M,ω) agrees with the usual holomorphic Kodaira dimension of (M,J). It is also shown in [12] that minimal symplectic 4−manifolds with κ = 0 are exactly those with torsion canonical class, thus can be viewed as symplectic Calabi-Yau surfaces. Known examples of symplectic 4−manifolds with torsion canonical class are either Kahler surfaces with (holomorphic) Kodaira dimension zero or T 2−bundles over T 2 ([10], [12]). They all have small Betti numbers and Euler numbers: b+ ≤ 3, b ≤ 19 and b1 ≤ 4; and the Euler number is between 0 and 24. It is speculated in [12] that these are the only ones. In this paper we prove that it is true up to rational homology.
62 citations
TL;DR: In this article, Calabi quasi-morphisms on the universal cover of Ham were constructed for non-monotone manifolds and these quasimorphisms descend to non-trivial homomorphisms.
Abstract: In this work we construct Calabi quasi-morphisms on the universal cover e Ham.M/ of the group of Hamiltonian diffeomorphisms for some non-monotone symplectic manifolds. This complements a result by Entov and Polterovich which applies in the monotone case. Moreover, in contrast to their work, we show that these quasimorphisms descend to non-trivial homomorphisms on the fundamental group of Ham.M/.
60 citations
TL;DR: The concept of pseudo symplectic capacities was introduced in this paper, which is a mild generalization of the Hofer-Zehnder capacity and is used to estimate pseudo-symplectic capacities of Grassmannians and product manifold.
Abstract: We introduce the concept of pseudo symplectic capacities which is a mild generalization of that of symplectic capacities. As a generalization of the Hofer-Zehnder capacity we construct a Hofer-Zehnder type pseudo symplectic capacity and estimate it in terms of Gromov-Witten invariants. The (pseudo) symplectic capacities of Grassmannians and some product symplectic manifolds are computed. As applications we first derive some general nonsqueezing theorems that generalize and unite many previous versions then prove the Weinstein conjecture for cotangent bundles over a large class of symplectic uniruled manifolds (including the uniruled manifolds in algebraic geometry) and also show that any closed symplectic submanifold of codimension two in any symplectic manifold has a small neighborhood whose Hofer-Zehnder capacity is less than a given positive number. Finally, we give two results on symplectic packings in Grassmannians and on Seshadri constants.
59 citations
TL;DR: DifferentDifferential forms on an odd symplectic manifold form a bicomplex: one differential is the wedge product with the symplectic form and the other is de Rham differential.
Abstract: Differential forms on an odd symplectic manifold form a bicomplex: one differential is the wedge product with the symplectic form and the other is de Rham differential. In the corresponding spectral sequence the next differential turns out to be the Batalin–Vilkoviski operator.
TL;DR: In this article, the authors considered a Hilbert space bundle of compatible complex structures on a symplectic vector space and showed that parallel transport along a geodesic in the bundle is a rescaled orthogonal projection or Bogoliubov transformation.
Abstract: In quantum mechanics, the momentum space and position space wave functions are related by the Fourier transform. We investigate how the Fourier transform arises in the context of geometric quantization. We consider a Hilbert space bundle \(\mathcal{H}\) over the space \(\mathcal{J}\) of compatible complex structures on a symplectic vector space. This bundle is equipped with a projectively flat connection. We show that parallel transport along a geodesic in the bundle \(\mathcal{H} \to \mathcal{J}\) is a rescaled orthogonal projection or Bogoliubov transformation. We then construct the kernel for the integral parallel transport operator. Finally, by extending geodesics to the boundary (for which the metaplectic correction is essential), we obtain the Segal-Bargmann and Fourier transforms as parallel transport in suitable limits.
TL;DR: In this article, the theory of Abelian Routh reduction for discrete mechanical systems was developed and applied to the variational integration of mechanical systems with Abelian symmetry, and the reduction of variational Runge-Kutta discretizations was considered, as well as the extent to which symmetry reduction and discretization commute.
Abstract: This paper develops the theory of Abelian Routh reduction for discrete mechanical systems and applies it to the variational integration of mechanical systems with Abelian symmetry. The reduction of variational Runge–Kutta discretizations is considered, as well as the extent to which symmetry reduction and discretization commute. These reduced methods allow the direct simulation of dynamical features such as relative equilibria and relative periodic orbits that can be obscured or difficult to identify in the unreduced dynamics. The methods are demonstrated for the dynamics of an Earth orbiting satellite with a non-spherical J2 correction, as well as the double spherical pendulum. The J2 problem is interesting because in the unreduced picture, geometric phases inherent in the model and those due to numerical discretization can be hard to distinguish, but this issue does not appear in the reduced algorithm, where one can directly observe interesting dynamical structures in the reduced phase space (the cotangent bundle of shape space), in which the geometric phases have been removed. The main feature of the double spherical pendulum example is that it has a non-trivial magnetic term in its reduced symplectic form. Our method is still efficient as it can directly handle the essential non-canonical nature of the symplectic structure. In contrast, a traditional symplectic method for canonical systems could require repeated coordinate changes if one is evoking Darboux' theorem to transform the symplectic structure into canonical form, thereby incurring additional computational cost. Our method allows one to design reduced symplectic integrators in a natural way, despite the non-canonical nature of the symplectic structure.
TL;DR: A family of symplectic splitting methods especially tailored to solve numerically the time-dependent Schrodinger equation are presented, which allows us to build highly efficient symplectic integrators at any order.
Abstract: We present a family of symplectic splitting methods especially tailored to solve numerically the time-dependent Schrodinger equation. When discretized in time, this equation can be recast in the form of a classical Hamiltonian system with a Hamiltonian function corresponding to a generalized high-dimensional separable harmonic oscillator. The structure of the system allows us to build highly efficient symplectic integrators at any order. The new methods are accurate, easy to implement, and very stable in comparison with other standard symplectic integrators.
TL;DR: In this article, the pseudo symplectic capacities of toric manifolds in combinatorial data are given. But they do not consider the impact of symplectic blow-up on the capacities of the polygon spaces.
Abstract: In this paper we give concrete estimations for the pseudo symplectic capacities of toric manifolds in combinatorial data. Some examples are given to show that our estimates can compute their pseudo symplectic capacities. As applications we also estimate the symplectic capacities of the polygon spaces. Other related results are impacts of symplectic blow-up on symplectic capacities, symplectic packings in symplectic toric manifolds, the Seshadri constant of an ample line bundle on toric manifolds, and symplectic capacities of symplectic manifolds with $S^{1}$-action.
TL;DR: In this paper, an analogue of the inflation technique of Lalonde-McDuff was introduced to obtain new symplectic forms from symplectic surfaces of negative self-intersection in symplectic 4-manifolds.
Abstract: We introduce an analogue of the inflation technique of Lalonde--McDuff, allowing us to obtain new symplectic forms from symplectic surfaces of negative self-intersection in symplectic 4-manifolds. We consider the implications of this construction for the symplectic cones of Kaahler surfaces, proving along the way a result which can be used to simplify the intersections of distinct pseudo-holomorphic curves via a perturbation.
TL;DR: In this paper, it was shown that any irreducible admissible representation of the symplectic similitude group GSp(V) decomposes with multiplicity one when restricted to the group Sp(V), where V is an orthogonal space overk.
Abstract: We prove several multiplicity one theorems in this paper. Fork a local field not of characteristic two, andV a symplectic space overk, any irreducible admissible representation of the symplectic similitude group GSp(V) decomposes with multiplicity one when restricted to the symplectic group Sp(V). We prove the analogous result for GO(V) and O(V), whereV is an orthogonal space overk. Whenk is non-archimedean, we prove the uniqueness of Fourier-Jacobi models for representations of GSp(4), and the existence of such models for supercuspidal representations of GSp(4).
TL;DR: In this paper, the determinant of any symplectic matrix is + 1 and the zero patterns compatible with the symplectic structure are also presented, together with some properties of Schur complements.
01 Jan 2006
TL;DR: Symplectic field theory (SFT) as discussed by the authors attempts to approach the theory of holomorphic curves in symplectic manifolds (also called Gromov-Witten theory) in the spirit of a topological fieldtheory.
Abstract: Symplectic field theory (SFT) attempts to approach the theory of holomorphic curves
in symplectic manifolds (also called Gromov-Witten theory) in the spirit of a topological field
theory. This naturally leads to new algebraic structures which seems to have interesting applications
and connections not only in symplectic geometry but also in other areas of mathematics,
e.g. topology and integrable PDE. In this talk we sketch out the formal algebraic structure of
SFT and discuss some current work towards its applications.
TL;DR: The main result of as mentioned in this paper is that a smooth s-cobordism of elliptic 3-manifolds is diffeomorphic to a product (assuming a canonical contact structure on the boundary).
Abstract: The main result of this paper states that a symplectic s-cobordism of elliptic 3-manifolds is diffeomorphic to a product (assuming a canonical contact structure on the boundary). Based on this theorem, we conjecture that a smooth s-cobordism of elliptic 3-manifolds is smoothly a product if its universal cover is smoothly a product. We explain how the conjecture fits naturally into the program of Taubes of constructing symplectic structures on an oriented smooth 4-manifold with b + ≥ 1 from generic self-dual harmonic forms. The paper also contains an auxiliary result of independent interest, which generalizes Taubes’ theorem “SW ⇒ Gr” to the case of symplectic 4-orbifolds.
TL;DR: In this paper, the authors redescribes fundamental problem of the two-dimensional viscoelasticity in a symplectic system and obtain solutions of duality equations with the aid of integral transformation.
TL;DR: The most general action, quadratic in the B fields as well as in the curvature F, having SO(3, 1) or SO(4) as the internal gauge group for a four-dimensional BF theory is presented and its symplectic geometry is displayed in this paper.
Abstract: The most general action, quadratic in the B fields as well as in the curvature F, having SO(3, 1) or SO(4) as the internal gauge group for a four-dimensional BF theory is presented and its symplectic geometry is displayed. It is shown that the space of solutions to the equations of motion for the BF theory can be endowed with symplectic structures alternative to the usual one. The analysis also includes topological terms and cosmological constant. The implications of this fact for gravity are briefly discussed.
TL;DR: In this article, a finite-difference time-domain method for electromagnetic field simulation is proposed, which can successfully solve Maxwell equations involving conductor loss, which cannot be solved by the symplectic integration methods that have been presented in previous works.
Abstract: We introduce a symplectic finite-difference time-domain method for electromagnetic field simulation. Our method can successfully solve Maxwell equations involving conductor loss, which cannot be solved by the symplectic integration methods that have been presented in previous works. A class of high-order symplectic schemes for computing the time-dependent electric and magnetic fields are derived on the basis of an s-stage symplectic partitioned Runge–Kutta method.We present numerical results to illustrate the validity and accuracy of the algorithm.
TL;DR: In this article, the Maslov P-index theory for a symplectic path is defined and various properties of this index theory such as homotopy invariant, symplectic additivity and the relations with other Morse indices are studied.
Abstract: The Maslov P-index theory for a symplectic path is defined. Various properties of this index theory such as homotopy invariant, symplectic additivity and the relations with other Morse indices are studied. As an application, the non-periodic problem for some asymptotically linear Hamiltonian systems is considered.
TL;DR: In this article, it was shown that closed symplectic four-manifolds do not admit any smooth free circle actions with contractible orbits, without assuming that the actions preserve the symplectic forms.
Abstract: We prove that closed symplectic four-manifolds do not admit any smooth free circle actions with contractible orbits, without assuming that the actions preserve the symplectic forms. In higher dimensions such actions by symplectomorphisms do exist, and we give explicit examples based on the constructions of FGM.
TL;DR: In this article, the authors used the minimal coupling procedure of Sternberg and Weinstein and pseudo-symplectic capacity theory to prove that every closed symplectic submanifold in any symplectic manifold has an open neighborhood with finite Hofer-Zehnder capacity.
Abstract: We use the minimal coupling procedure of Sternberg and Weinstein and our pseudo-symplectic capacity theory to prove that every closed symplectic submanifold in any symplectic manifold has an open neighborhood with finite ($\pi_1$-sensitive) Hofer-Zehnder symplectic capacity. Consequently, the Weinstein conjecture holds near closed symplectic submanifolds in any symplectic manifold.
TL;DR: In this article, the authors examine noncommutativity of position coordinates in classical symplectic mechanics and its quantisation and show that the canonical symplectic two-form is omega0 = dq i.
Abstract: In this work we examine noncommutativity of position coordinates in classical symplectic mechanics and its quantisation. In coordinates {q i,p k} the canonical symplectic two-form is omega0 = dq i
TL;DR: In this paper, the authors studied the general case of a lifted action that does not admit a momentum map and used its natural generalization, a cylinder valued momentum map introduced by Condevaux et al.
Abstract: There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three approaches coincide. The goal of this paper is to study the general case of a symplectic action that does not admit a momentum map and one needs to use its natural generalization, a cylinder valued momentum map introduced by Condevaux et al. In this case it will be shown that the three reduced spaces mentioned above do not coincide, in general. More specifically, the Marsden-Weinstein reduced spaces are not symplectic but Poisson and their symplectic leaves are given by the optimal reduced spaces. Foliation reduction produces a symplectic reduced space whose Poisson quotient by a certain Lie group associated to the group of symmetries of the problem equals the Marsden-Weinstein reduced space. We illustrate these constructions with concrete examples, special emphasis being given to the reduction of a magnetic cotangent bundle of a Lie group in the situation when the magnetic term ensures the non-existence of the momentum map for the lifted action. The precise relation of the cylinder valued momentum map with group valued momentum maps for Abelian Lie groups is also given.
TL;DR: The moduli space of solutions to the vortex equations on a Riemann surface is well known to have a symplectic (in fact, Kahler) structure as mentioned in this paper, and a family of prequantum line bundles PΩΨ0 on the modulus space whose curvature is proportional to the symplectic forms ΩΩ 0 is known.
Abstract: The moduli space of solutions to the vortex equations on a Riemann surface are well known to have a symplectic (in fact, Kahler) structure. We show this symplectic structure explictly and proceed to show a family of symplectic (in fact, Kahler) structures ΩΨ0 on the moduli space, parametrized by Ψ0, a section of a line bundle on the Riemann surface. Next, we show that corresponding to these, there is a family of prequantum line bundles PΨ0 on the moduli space whose curvature is proportional to the symplectic forms ΩΨ0.