Symplectic Area Research Articles

Disk-Like Surfaces of Section and Symplectic Capacities

We prove that the cylindrical capacity of a dynamically convex domain in \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\mathbb{R}}^{4}$\\end{document} agrees with the least symplectic area of a disk-like global surface of section of the Reeb flow on the boundary of the domain. Moreover, we prove the strong Viterbo conjecture for all convex domains in \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\mathbb{R}}^{4}$\\end{document} which are sufficiently C3 close to the round ball. This generalizes a result of Abbondandolo-Bramham-Hryniewicz-Salomão establishing a systolic inequality for such domains.

A note on the Gromov width of toric manifolds

The Gromov width of a uniruled projective Kähler manifold can be bounded from above by the symplectic area of its minimal curves. We apply this result to toric varieties and thus get in this case upper bounds expressed in toric combinatorial invariants.

Exploration of solitons and analytical solutions by sub-ODE and variational integrators to Klein-Gordon model

In this paper, we use the sub-ODE method to analyze soliton solutions for the renowned nonlinear Klein-Gordon model (NLKGM). This method provides a variety of soliton solutions, including three positive solitons, three Jacobian elliptic function solutions, bright solitons, dark solitons, periodic solitons, rational solitons and hyperbolic function solutions. Applications for these solitons can be found in optical communication, fiber optic sensors, plasma physics, Bose-Einstein condensation and other areas. We also study some numerical solutions by using forward, backward, and central difference techniques. Moreover, we discuss variational integrators (VIs) using the projection technique for NLKGM. We develop a numerical solution for NLKGM using the discrete Euler lagrange equation, the Lagrangian and the Euler lagrange equation. At the end, in various dimensions, covering 3D, 2D, and contour, we will also plot several graphs for the obtained NLKGM solutions. A contour plot is a type of graphic representation that displays a three-dimensional surface on a two-dimensional plane by using contour lines. Each contour line in the plotted function represents one of the function's constant values, mapping the function's value across the plane. This model has been studied across multiple soliton solutions using various methods in the open literature, but this model for VIs and finite deference scheme (FDS) is the first time it has been studied. Within the various numerical techniques accessible for solving Hamiltonian systems, variational integrators distinguish themselves because of their symplectic quality. Here are some of the symplectic properties: symplectic orthogonality, energy conservation, area preservation, and structure preservation.

The Gromov Width of Bott-Samelson Varieties

We prove that the Gromov width of any Bott-Samelson variety associated to a reduced expression and equipped with a rational Kähler form equals the symplectic area of a minimal curve. From this, we derive an estimate for the Seshadri constants of ample line bundles on Bott-Samelson varieties.

Geometrical interpretation of the argument of weak values of general observables in N-level quantum systems

Observations in quantum weak measurements are determined by complex numbers called weak values. We present a geometrical interpretation of the argument of weak values of general Hermitian observables in N-dimensional quantum systems in terms of geometric phases. We formulate an arbitrary weak value in terms of three real vectors on the unit sphere in N 2 − 1 dimensions, . These vectors are linked to the initial and final states, and to the weakly measured observable, respectively. We express pure states in the complex projective space of N − 1 dimensions, , which has a non-trivial representation as a 2N − 2 dimensional submanifold of (a generalization of the Bloch sphere for qudits). The argument of the weak value of a projector on a pure state of an N-level quantum system describes a geometric phase associated to the symplectic area of the geodesic triangle spanned by the vectors representing the pre-selected state, the projector and the post-selected state in . We then proceed to show that the argument of the weak value of a general observable is equivalent to the argument of an effective Bargmann invariant. Hence, we extend the geometrical interpretation of projector weak values to weak values of general observables. In particular, we consider the generators of SU(N) given by the generalized Gell–Mann matrices. Finally, we study in detail the case of the argument of weak values of general observables in two-level systems and we illustrate weak measurements in larger dimensional systems by considering projectors on degenerate subspaces, as well as Hermitian quantum gates. To conclude, we discuss the interpretation and usefulness of geometric phases in weak values in connection to weak measurements.

Zindler-type hypersurfaces in [formula omitted

In this paper the definition of Zindler-type hypersurfaces is introduced in R4 as a generalization of planar Zindler curves. After recalling some properties of planar Zindler curves, it is shown that Zindler hypersurfaces satisfy similar properties. Techniques from quaternions and symplectic geometry are used. Moreover, each Zindler hypersurface is fibrated by space Zindler curves that correspond, in the convex case, to some space curves of constant width lying on the associated hypersurface of constant width and with the same symplectic area.

On the minimal symplectic area of Lagrangians

On the minimal symplectic area of Lagrangians

The evolution of geometric Robertson–Schrödinger uncertainty principle for spin 1 system

Geometric Quantum Mechanics is a mathematical framework that shows how quantum theory may be expressed in terms of Hamiltonian phase-space dynamics. The states are points in complex projective Hilbert space, the observables are real valued functions on the space, and the Hamiltonian flow is specified by the Schrödinger equation in this framework. The quest to express the uncertainty principle in geometrical language has recently become the focus of significant research in geometric quantum mechanics. One has demonstrated that the Robertson–Schrödinger uncertainty principle, which is a stronger version of the uncertainty relation, can be defined in terms of symplectic form and Riemannian metric. On the basis of this formulation, we study the dynamical behavior of the uncertainty relation for the spin 1 system in this work. We show that under Hamiltonian flow, the Robertson–Schrödinger uncertainty principles are not invariant. This is because, unlike the symplectic area, the Riemannian metric is not invariant under Hamiltonian flow throughout the evolution process.

THE DYNAMICAL EVOLUTION OF GEOMETRIC UNCERTAINTY PRINCIPLE FOR SPIN 1/2 SYSTEM

Geometric Quantum Mechanics is a formulation that demonstrates how quantum theory may be casted in the language of Hamiltonian phase-space dynamics. In this framework, the states are referring to points in complex projective Hilbert space, the observables are real valued functions on the space and the Hamiltonian flow is defined by Schr{\"o}dinger equation. Recently, the effort to cast uncertainty principle in terms of geometrical language appeared to become the subject of intense study in geometric quantum mechanics. One has shown that the stronger version of uncertainty relation i.e. the Robertson-Schr{\"o}dinger uncertainty relation can be expressed in terms of the symplectic form and Riemannian metric. In this paper, we investigate the dynamical behavior of the uncertainty relation for spin $\frac{1}{2}$ system based on this formulation. We show that the Robertson-Schr{\"o}dinger uncertainty principle is not invariant under Hamiltonian flow. This is due to the fact that during evolution process, unlike symplectic area, the Riemannian metric is not invariant under the flow.

Real and complex hedgehogs, their symplectic area, curvature and evolutes

Classical (real) hedgehogs can be regarded as the geometrical realiza-tions of formal di¤erences of convex bodies in R n+1. Like convex bodies, hedgehogs can be identi…ed with their support functions. Adopting a pro-jective viewpoint, we prove that any holomorphic function h : C n ! C can be regarded as the 'support function' of a complex hedgehog H h in C n+1. In the same vein, we introduce the notion of evolute of such a hedgehog H h in C 2 , and a natural (but apparently hitherto unknown) notion of complex curvature, which allows us to interpret this evolute as the locus of the centers of complex curvature. It is of course permissible to think that the development of a 'Brunn-Minkowski theory for complex hedgehogs' (replacing Euclidean volumes by symplectic ones) might be a promising way of research. We give …rst two results in this direction. We next return to real hedgehogs in R 2n endowed with a linear complex structure. We introduce and study the notion of evolute of a hedgehog. We particularly focus our attention on R 4 endowed with a linear Kahler structure determined by the datum of a pure unit quaternion. In parallel, we study the symplectic area of the images of the oriented Hopf circles under hedgehog parametrizations and introduce a quaternionic curvature function for such an image. Finally, we consider brie ‡y the convolution of hedgehogs, and the particular case of hedgehogs in R 4n regarded as a hyperkahler vector space.

The symplectic area of a geodesic triangle in a Hermitian symmetric space of compact type

Let M be an irreducible Hermitian symmetric space of compact type, and let $$\omega $$ be its Kahler form. For a triplet $$(p_1,p_2,p_3)$$ of points in M we study conditions under which a geodesic triangle $$\mathcal T(p_1,p_2,p_3)$$ with vertices $$p_1,p_2,p_3$$ can be unambiguously defined. We consider the integral $$A(p_1,p_2,p_3)=\int _\Sigma \omega $$ , where $$\Sigma $$ is a surface filling the triangle $$\mathcal T(p_1,p_2,p_3)$$ and discuss the dependence of $$A(p_1,p_2,p_3)$$ on the surface $$\Sigma $$ . Under mild conditions on the three points, we prove an explicit formula for $$A(p_1,p_2,p_3)$$ analogous to the known formula for the symplectic area of a geodesic triangle in a non-compact Hermitian symmetric space.

Introducing symplectic billiards

In this article we introduce a simple dynamical system called symplectic billiards. As opposed to usual/Birkhoff billiards, where length is the generating function, for symplectic billiards symplectic area is the generating function. We explore basic properties and exhibit several similarities, but also differences of symplectic billiards to Birkhoff billiards.

Embedded surfaces for symplectic circle actions

The purpose of this article is to characterize symplectic and Hamiltonian circle actions on symplectic manifolds in terms of symplectic embeddings of Riemann surfaces. More precisely, it is shown that (1) if (M, ω) admits a Hamiltonian S1-action, then there exists a two-sphere S in M with positive symplectic area satisfying ‹c1(M, ω), [S]› > 0, and (2) if the action is non-Hamiltonian, then there exists an S1-invariant symplectic 2-torus T in (M, ω) such that ‹c1(M, ω), [T]› = 0. As applications, the authors give a very simple proof of the following well-known theorem which was proved by Atiyah-Bott, Lupton-Oprea, and Ono: Suppose that (M, ω) is a smooth closed symplectic manifold satisfying c1(M, ω) = λ·[ω] for some λ ∈ R and G is a compact connected Lie group acting effectively on M preserving ω. Then (1) if λ 0, then the G-action is Hamiltonian.

Conley Conjecture Revisited

We show that whenever a closed symplectic manifold admits a Hamiltonian diffeomorphism with finitely many simple periodic orbits, the manifold has a spherical homology class of degree two with positive symplectic area and positive integral of the first Chern class. This theorem encompasses all known cases of the Conley conjecture (symplectic CY and negative monotone manifolds) and also some new ones (e.g., weakly exact symplectic manifolds with non-vanishing first Chern class). The proof hinges on a general Lusternik-Schnirelmann type result that, under some natural additional conditions, the sequence of mean spectral invariants for the iterations of a Hamiltonian diffeomorphism never stabilizes. We also show that for the iterations of a Hamiltonian diffeomorphism with finitely many periodic orbits the sequence of action gaps between the largest and the smallest spectral invariants remains bounded and, as consequence, establish some new cases of the $C^\infty$-generic existence of infinitely many simple periodic orbits.

Non-Lie top tunneling and quantum bilocalization in planar Penning trap

We describe how a top-like quantum Hamiltonian over a non-Lie algebra appears in the model of the planar Penning trap under the breaking of its axial symmetry (inclination of the magnetic field) and tuning parameters (electric voltage, magnetic field strength and inclination angle) at double resonance. For eigenvalues of the quantum non-Lie top, under a specific variation of the voltage on the trap electrode, there exists an avoided crossing effect and a corresponding effect of bilocalization of quantum states on pairs of closed trajectories belonging to common energy levels. This quantum tunneling happens on the symplectic leaves of the symmetry algebra, and hence it generates a tunneling of quantum states of the electron between the 3D-tori in the whole 6D-phase space. We present a geometric formula for the leading term of asymptotics of the tunnel energy-splitting in terms of symplectic area of membranes bounded by invariantly defined instantons.

On Hofer energy of $J$-holomorphic curves for asymptotically cylindrical $J$

In this paper, we provide a bound for the generalized Hofer energy of punctured J-holomorphic curves in almost complex manifolds with asymptotically cylindrical ends. As an application, we prove a version of Gromov’s Monotonicity Theorem with multiplicity. Namely, for a closed symplectic manifold (M,!0) with a compatible almost complex structure J and a ball B in M, there exists a constant ~ > 0, such that any J-holomorphic curve ũ passing through the center of B for k times (counted with multiplicity) with boundary mapped to @B has symplectic area ũ 1(B) ũ ⇤!0 > k~, where the constant ~ depends only on (M,!0, J) and the radius of B. As a consequence, the number of times that any closed J-holomorphic curve in M passes through a point is bounded by a constant depending only on (M,!0, J)1 and the symplectic area of ũ. Here J is any !0 compatible smooth almost complex structure on M . In particular, we do not require J to be integrable.

Four remarks on spin coherent states

We discuss how to recognize the constellations seen in the Majorana representation of quantum states. Then we give explicit formulae for the metric and symplectic form on orbits containing general number states. Their metric and symplectic areas differ unless the states are coherent. Finally we discuss some patterns that arise from the Lieb–Solovej map, and for dimensions up to nine we find the location of those states that maximize the Wehrl–Lieb entropy.

The action homomorphism, quasimorphisms and moment maps on the space of compatible almost complex structures

We extend the definition of Weinstein’s action homomorphism to Hamiltonian actions with equivariant moment maps of (possibly infinite-dimensional) Lie groups on symplectic manifolds, and show that under conditions including a uniform bound on the symplectic areas of geodesic triangles the resulting homomorphism extends to a quasimorphism on the universal cover of the group. We apply these principles to finite-dimensional Hermitian Lie groups like the linear symplectic group, reinterpreting the Guichardet–Wigner quasimorphisms, and to the infinite-dimensional groups of Hamiltonian diffeomorphisms of closed symplectic manifolds that act on the space of compatible almost complex structures with an equivariant moment map given by the theory of Donaldson and Fujiki. We show that the quasimorphism on the universal cover of the Hamiltonian group obtained in the second case is symplectically conjugation-invariant and compute its restrictions to the fundamental group via a homomorphism introduced by Lalonde–McDuff–Polterovich, answering a question of Polterovich; to the subgroup of Hamiltonian biholomorphisms via the Futaki invariant; and to subgroups of diffeomorphisms supported in an embedded ball via the Barge–Ghys average Maslov quasimorphism, the Calabi homomorphism and the average Hermitian scalar curvature. We show that when the first Chern class vanishes this quasimorphism is proportional to a quasimorphism of Entov and when the symplectic manifold is monotone, it is proportional to a quasimorphism due to Py. As an application we show that a Sobolev distance on the universal cover of the Hamiltonian group is unbounded, similarly to the results of Eliashberg–Ratiu.

Geometric phase and spinorial representation of mixed state

Abstract A novel approach to geometric phase of mixed state is proposed by using the normalized spinorial representation in connecting the density matrix with mixed state vector. We find that though the spinor involves N separate U ( 1 ) phases correspondingly to any N-level decomposition of the density matrix, both geometric phase and density matrix of mixed state are holonomy ⨂ k = 1 N U ( 1 ) gauge invariants. This noncyclic invariant is conceptually useful in analyzing geometric phase and CP violation of open system. Under a quasicyclic case, the geometric phase depends only on the symplectic area spanned in a given closed evolving curve with the classical probabilities relating to the Bloch radius, in which quantifies mixed degree of open system, in the Bloch sphere structure.

Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds

We prove that the group of Hamiltonian automorphisms of a symplectic $4$-manifold $(M,\omega)$, Ham$(M,\omega)$, contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group Symp$(M,\omega)$. We also consider the set of conjugacy classes of\/ $2$-tori in Ham$(M,\omega)$ with respect to Hamiltonian conjugation and show that its finiteness is equivalent to the finiteness of the symplectic mapping class group $\pi_{0}$(Symp$(M,\omega)$). Finally, we extend to rational and ruled manifolds a result of Kedra which asserts that if $(M,\omega)$ is a simply connected symplectic $4$-manifold with $b_{2}\geq 3$, and if $(\widetilde{M},\widetilde{\omega}_{\delta})$ denotes a symplectic blow-up of $(M,\omega)$ of small enough capacity $\delta$, then the rational cohomology algebra of the Hamiltonian group Ham($\widetilde{M},\widetilde{\omega}_{\delta})$ is not finitely generated. Our results are based on the fact that in a symplectic $4$-manifold endowed with any tamed almost structure $J$, exceptional classes of minimal symplectic area are $J$-indecomposable.