Non-squeezing Theorem Research Articles

On the partial saturation of the uncertainty relations of a mixed Gaussian state

Let ρ the density matrix of a mixed Gaussian state. Assuming that one of the Robertson–Schrödinger uncertainty inequalities is saturated by ρ, e.g. , we show that there exists a unique pure Gaussian state whose Wigner distribution is dominated by that of ρ and having the same variances and covariance ΔρX1, ΔρP1 and Δρ(X1, P1) as ρ. This property can be viewed as an analytic version of Gromov’s non-squeezing theorem in the linear case, which implies that the intersection of a symplectic ball by a single plane of conjugate coordinates determines the radius of this ball. We conclude by giving a short geometric proof of the fact that pure Gaussian states are the only quantum states saturating the Robertson–Schrödinger uncertainty inequalities.

Hartogs figure and symplectic non-squeezing

We solve a problem on filling by Levi-flat hypersurfaces for a class of totally real 2-tori in a real 4-manifold with an almost complex structure tamed by an exact symplectic form. As an application, we obtain a simple proof of Gromov’s non-squeezing theorem in dimension 4 and new results on rigidity of symplectic structures.

Equivariant homology for generating functions and orderability of lens spaces

In her PhD thesis [M08] Milin developed a Z k-equivariant version of the contact homology groups constructed in [EKP06] and used it to prove a Z k-equivariant contact non-squeezing theorem. In this article, we re-obtain the same result in the setting of generating functions, starting from the homology groups studied in [S11]. As Milin showed, this result implies orderability of lens spaces.

THE SHARP ENERGY-CAPACITY INEQUALITY

Using the Oh–Schwarz spectral invariants and some arguments of Frauenfelder, Ginzburg and Schlenk, we show that the π1-sensitive Hofer–Zehnder capacity of any subset of a closed symplectic manifold is less than or equal to its displacement energy. This estimate is sharp, and implies some new extensions of the Non-Squeezing Theorem.

A symplectic non-squeezing theorem for BBM equation

We study the initial value problem for the BBM equation: $$\left\{\begin{array}{l} u_t+u_x+uu_x-u_{txx}=0 \qquad x\in \T, t \in \R u(0,x)=u_0(x) \end{array} \right. .$$ We prove that the BBM equation is globaly well-posed on $H^s(\T)$ for $s\geq0$ and a symplectic non-squeezing theorem on $H^{1/2}(\T)$. That is to say the flow-map $u_0 \mapsto u(t)$ that associates to initial data $u_0 \in H^{1/2}(\T)$ the solution $u$ cannot send a ball into a symplectic cylinder of smaller width.

Symplectic embeddings and continued fractions: a survey

As has been known since the time of Gromov’s Non-squeezing Theorem, symplectic embedding questions lie at the heart of symplectic geometry. After surveying some of the most important ways of measuring the size of a symplectic set, these notes discuss some recent developments concerning the question of when a 4-dimensional ellipsoid can be symplectically embedded in a ball. This problem turns out to have unexpected relations to the properties of continued fractions and of exceptional curves in blow ups of the complex projective plane. It is also related to questions of lattice packing of planar triangles.

Erratum to “Geometry of contact transformations and domains: orderability versus squeezing”

Gromov's famous non-squeezing theorem (1985) states that the standard symplectic ball cannot be symplectically squeezed into any cylinder of smaller radius. Does there exist an analogue of this result in contact geometry? Our main finding is that the answer depends on the sizes of the domains in question: We establish contact non-squeezing on large scales, and show that it disappears on small scales. The algebraic counterpart of the (non)-squeezing problem for contact domains is the question of existence of a natural partial order on the universal cover of the contactomorphisms group of a contact manifold. In contrast to our earlier beliefs, we show that the answer to this question is very sensitive to the topology of the manifold. For instance, we prove that the standard contact sphere is non-orderable while the real projective space is known to be orderable. Our methods include a new embedding technique in contact geometry as well as a generalized Floer homology theory which contains both cylindrical contact homology and Hamiltonian Floer homology. We discuss links to a number of miscellaneous topics such as topology of free loops spaces, quantum mechanics and semigroups. An erratum is attached whose purpose is to is to correct a number of inconsistencies in the main paper. These are related to the grading of generalized Floer homology and do not affect formulations and proofs of the main results of the paper.

Fundamental Constraints on Uncertainty Evolution in Hamiltonian Systems

A realization of Gromov's nonsqueezing theorem and its applications to uncertainty analysis in Hamiltonian systems are studied in this note. Gromov's nonsqueezing theorem describes a fundamental property of symplectic manifolds, however, this theorem is usually started in terms of topology and its physical meaning is vague. In this note we introduce a physical interpretation of the linear symplectic width, which is the lower bound in the nonsqueezing theorem, in terms of the eigenstructure of a positive-definite, symmetric matrix. Since uncertainty is often represented in terms of a positive definite, symmetric matrix in control theory, our study can be applied to uncertainty analysis by applying the nonsqueezing theorem to the uncertainty ellipsoid. We find a fundamental inequality for the evolving uncertainty in a linear dynamical system and provide some numerical examples

Fundamental limits on spacecraft orbit uncertainty and distribution propagation

In this paper we present and review a number of fundamental constraints that exist on the propagation of orbit uncertainty and phase volume flows in astrodynamics. These constraints arise due to the Hamiltonian nature of spacecraft dynamics. First we review the role of integral invariants and their connection to orbit uncertainty, and show how they can be used to formally solve the diffusion-less Fokker-Plank equation for a spacecraft probability density function. Then, we apply Gromov’s Non-Squeezing Theorem, a recent advance in symplectic topology, to find a previously unrecognized fundamental constraint that exists on general, nonlinear mappings of orbit distributions. Specifically, for a given orbit distribution, it can be shown that the projection of future orbit uncertainties in each coordinate-momentum pair describing the system must be greater than or equal to a fundamental limit, called the symplectic width. This implies that there is always a fundamental limit to which we can know a spacecraft’s future location in its coordinate and conjugate momentum space when mapped forward in time from an initial covariance distribution. This serves as an “uncertainty” principle for spacecraft uncertainty distributions.

Geometry of contact transformations and domains: orderability versus squeezing

Gromov’s famous non-squeezing theorem (1985) states that the standard symplectic ball cannot be symplectically squeezed into any cylinder of smaller radius. Does there exist an analogue of this result in contact geometry? Our main finding is that the answer depends on the sizes of the domains in question: We establish contact nonsqueezing on large scales, and show that it disappears on small scales. The algebraic counterpart of the (non)-squeezing problem for contact domains is the question of existence of a natural partial order on the universal cover of the contactomorphisms group of a contact manifold. In contrast to our earlier beliefs, we show that the answer to this question is very sensitive to the topology of the manifold. For instance, we prove that the standard contact sphere is non-orderable while the real projective space is known to be orderable. Our methods include a new embedding technique in contact geometry as well as a generalized Floer homology theory which contains both cylindrical contact homology and Hamiltonian Floer homology. We discuss links to a number of miscellaneous topics such as topology of free loops spaces, quantum mechanics and semigroups.

Notes on symplectic non-squeezing of the KdV flow

We prove two finite dimensional approximation results and a symplectic non-squeezing property for the Korteweg-de Vries (KdV) flow on the circle 𝕋. The nonsqueezing result relies on the aforementioned approximations and the finite-dimensional nonsqueezing theorem of Gromov [14]. Unlike the work of Kuksin [22] which initiated the investigation of non-squeezing results for infinite dimensional Hamiltonian systems, the nonsqueezing argument here does not construct a capacity directly. In this way our results are similar to those obtained for the NLS flow by Bourgain [3]. A major difficulty here though is the lack of any sort of smoothing estimate which would allow us to easily approximate the infinite dimensional KdV flow by a finite-dimensional Hamiltonian flow. To resolve this problem we invert the Miura transform and work on the level of the modified KdV (mKdV) equation, for which smoothing estimates can be established.

An extension of Biran’s Lagrangian barrier theorem

We use the Gromov-Witten invariants and a nonsqueezing theorem by the author to affirm a conjecture by P. Biran on the Lagrangian barriers.

On a problem related to a non-squeezing theorem

In this paper we give a solution to a problem of Kulpa about the interior of the image of certain continuous maps f : X → R n where X is a compact subset of R n with non-empty interior. Moreover we show that the image of every continuous map f : X → R 2 where X is a non-empty compact subset of R 2 and diam f −1 f ( x ) < 3 a X for every x ∈ Fr X , has non-empty interior.

The $lquot$symplectic camel principle$rquot$ and semiclassical mechanics

Gromov's nonsqueezing theorem, aka the property of the symplectic camel, leads to a very simple semiclassical quantiuzation scheme by imposing that the only "physically admissible" semiclassical phase space states are those whose symplectic capacity (in a sense to be precised) is nh + (1/2)h where h is Planck's constant. We the construct semiclassical waveforms on Lagrangian submanifolds using the properties of the Leray-Maslov index, which allows us to define the argument of the square root of a de Rham form.

The symplectic camel and phase space quantization

We show that a result of symplectic topology, Gromov's non-squeezing theorem, also known as the `principle of the symplectic camel', can be used to quantize phase space in cells. That quantization scheme leads to the correct energy levels for integrable systems and to Maslov quantization of Lagrangian manifolds by purely topological arguments. We finally show that the argument leading to the proof of the non-squeezing theorem leads to a classical form of Heisenberg's inequalities.

Geometric Invariants of the Hofer Norm

This note discusses some geometrically defined seminorms on the group Ham(M, !) of Hamiltonian diffeomorphisms of a closed symplectic manifold (M, !), giving conditions under which they are nondegenerate and explaining their relation to the Hofer norm. As a consequence we show that if an element in Ham(M, !) is sufficiently close to the identity in theC 2 -topology then it may be joined to the identity by a path whose Hofer length is minimal among all paths, not just among paths in the same homotopy class relative to endpoints. Thus, true geodesics always exist for the Hofer norm. The main step in the proof is to show that a “weighted” version of the nonsqueezing theorem holds for all fibrations over S 2 generated by sufficiently short loops. Further, an example isgiven showing that the Hofer norm may differ from the sum of the one sided seminorms.

Total mean curvature and closed geodesics

The proofs and applications are based on a Riemannian version of Gromov’s non-squeezing theorem and classical integral geometry. Given a convex surface Σ ⊂ R and a point q in the unit sphere S we denote by UΣ(q) the perimeter of the orthogonal projection of Σ onto a plane perpendicular to q. We obtain a function UΣ on the sphere which is clearly continuous, even, and positive. Let us denote the minimum value for this function by uΣ. The analogue of the non-squeezing theorem we wish to present is the following result.

Hofer’s L∞-geometry: energy and stability of Hamiltonian flows, part I

In this paper we first show that the necessary condition introduced in our previous paper is also a sufficient condition for a path to be a geodesic in the group Ham c (M) of compactly supported Hamiltonian symplectomorphisms. This applies with no restriction onM. We then discuss conditions which guarantee that such a path minimizes the Hofer length. Our argument relies on a general geometric construction (the gluing of monodromies) and on an extension of Gromov's non-squeezing theorem both to more general manifolds and to more general capacities. The manifolds we consider are quasi-cylinders, that is spaces homeomorphic toM×D2 which are symplectically ruled overD2. When we work with the usual capacity (derived from embedded balls), we can prove the existence of paths which minimize the length among all homotopic paths, provided thatM is semi-monotone. (This restriction occurs because of the well-known difficulty with the theory ofJ-holomorphic curves in arbitraryM.) However, we can only prove the existence of length-minimizing paths (i.e. paths which minimize length amongstall paths, not only the homotopic ones) under even more restrictive conditions onM, for example whenM is exact and convex or of dimension 2. The new difficulty is caused by the possibility that there are non-trivial and very short loops in Ham c (M). When such lengthminimizing paths do exist, we can extend the Bialy-Polterovich calculation of the Hofer norm on a neighbourhood of the identity (Cl-flatness).

Hofer'sL ?-geometry: energy and stability of Hamiltonian flows, part I

In this paper we first show that the necessary condition introduced in our previous paper is also a sufficient condition for a path to be a geodesic in the group Ham c (M) of compactly supported Hamiltonian symplectomorphisms. This applies with no restriction onM. We then discuss conditions which guarantee that such a path minimizes the Hofer length. Our argument relies on a general geometric construction (the gluing of monodromies) and on an extension of Gromov's non-squeezing theorem both to more general manifolds and to more general capacities. The manifolds we consider are quasi-cylinders, that is spaces homeomorphic toM×D2 which are symplectically ruled overD2. When we work with the usual capacity (derived from embedded balls), we can prove the existence of paths which minimize the length among all homotopic paths, provided thatM is semi-monotone. (This restriction occurs because of the well-known difficulty with the theory ofJ-holomorphic curves in arbitraryM.) However, we can only prove the existence of length-minimizing paths (i.e. paths which minimize length amongstall paths, not only the homotopic ones) under even more restrictive conditions onM, for example whenM is exact and convex or of dimension 2. The new difficulty is caused by the possibility that there are non-trivial and very short loops in Ham c (M). When such lengthminimizing paths do exist, we can extend the Bialy-Polterovich calculation of the Hofer norm on a neighbourhood of the identity (Cl-flatness).

Hofer's L?-geometry: energy and stability of Hamiltonian flows, part II

In this paper we first show that the necessary condition introduced in our previous paper is also a sufficient condition for a path to be a geodesic in the group $\Ham^c(M)$ of compactly supported Hamiltonian symplectomorphisms. This applies with no restriction on $M$. We then discuss conditions which guarantee that such a path minimizes the Hofer length. Our argument relies on a general geometric construction (the gluing of monodromies) and on an extension of Gromov's non-squeezing theorem both to more general manifolds and to more general capacities. The manifolds we consider are quasi-cylinders, that is spaces homeomorphic to $M \times D^2$ which are symplectically ruled over $D^2$. When we work with the usual capacity (derived from embedded balls), we can prove the existence of paths which minimize the length among all homotopic paths, provided that $M$ is semi-monotone. (This restriction occurs because of the well-known difficulty with the theory of $J$-holomorphic curves in arbitrary $M$.) However, we can only prove the existence of length-minimizing paths (i.e. paths which minimize length amongst {\it all} paths, not only the homotopic ones) under even more restrictive conditions on $M$, for example when $M$ is exact and convex or of dimension $2$. The new difficulty is caused by the possibility that there are non-trivial and very short loops in $\Ham^c(M)$. When such length-minimizing paths do exist, we can extend the Bialy--Polterovich calculation of the Hofer norm on a neighbourhood of the identity ($C^1$-flatness).