Non-squeezing Theorem Research Articles

Advanced fuel fusion, phase space engineering, and structure-preserving geometric algorithms

Non-thermal advanced fuel fusion trades the requirement of a large amount of recirculating tritium in the system for that of large recirculating power. Phase space engineering technologies utilizing externally injected electromagnetic fields can be applied to meet the challenge of maintaining non-thermal particle distributions at a reasonable cost. The physical processes of the phase space engineering are studied from a theoretical and algorithmic perspective. It is emphasized that the operational space of phase space engineering is limited by the underpinning symplectic dynamics of charged particles. The phase space incompressibility according to the Liouville theorem is just one of many constraints, and Gromov's non-squeezing theorem determines the minimum footprint of the charged particles on every conjugate phase space plane. In this sense and level of sophistication, the mathematical abstraction of phase space engineering is symplectic topology. To simulate the processes of phase space engineering, such as the Maxwell demon and electromagnetic energy extraction, and to accurately calculate the minimum footprints of charged particles, recently developed structure-preserving geometric algorithms can be used. The family of algorithms conserves exactly, on discretized spacetime, symplecticity and thus incompressibility, non-squeezability, and symplectic capacities. The algorithms apply to the dynamics of charged particles under the influence of external electromagnetic fields as well as the charged particle–electromagnetic field system governed by the Vlasov–Maxwell equations.

Gromov width of symplectic toric manifolds associated with graphs

We give an explicit formula for the Gromov width for a class of symplectic toric manifolds constructed from simple graphs. As a corollary, we show a version of non-squeezing theorem with respect to the inclusion of connected graphs.

On the local systolic optimality of Zoll contact forms

We prove a normal form for contact forms close to a Zoll one and deduce that Zoll contact forms on any closed manifold are local maximizers of the systolic ratio. Corollaries of this result are: (1) sharp local systolic inequalities for Riemannian and Finsler metrics close to Zoll ones, (2) the perturbative case of a conjecture of Viterbo on the symplectic capacity of convex bodies, (3) a generalization of Gromov’s non-squeezing theorem in the intermediate dimensions for symplectomorphisms that are close to linear ones.

On the relation between cosymplectic and symplectic structures

In this paper we study some aspects of the relationship between cosymplectic and symplectic structures. In particular, we show that similar to the symplectic case, one can prove a nonsqueezing theorem for cosymplectomorphisms on R2m+1. Using this nonsqueezing theorem, we can define the concept of “cosymplectic capacity”. We show that the set of all cosymplectic capacities on cosymplectic manifolds of dimension 2m+1 has a close relationship with the set of all symplectic capacities on symplectic manifolds of dimension 2m. Furthermore, we study the relationship between fixed points of cosymplectomorphisms and symplectomorphisms.

The role of geometric and dynamical phases in the Dirac–Bohm picture

In this paper we explore a different but unitarily equivalent picture to the standard Schrödinger and Heisenberg pictures, namely the Dirac–Bohm picture introduced by Hiley and Dennis. This is not merely another interpretation as it allows us to examine the unfolding quantum process in terms of the algebraic structure of the dynamical variables, which in turn allows us to explore the geometric implications of quantum phenomena. Furthermore this enables us to exploit the double valued nature of the orthogonal and symplectic symmetry groups and show how they profoundly affect the nature of the conjugate momentum by enabling the introduction of classes of vector potentials into Hamiltonian systems. This throws more light on the significance of gauge invariance both in classical and quantum mechanics.The connection with geometric aspects of quantum phenomena allows us to introduce Clifford algebras in a natural way so that it is possible to extend the Dirac–Bohm approach to include spin and relativity, offering new geometric insights into quantum phenomena. In particular the bi-vector aspects allow us to introduce the Gromov non-squeezing theorem in a new way, ultimately offering the possibility of relating our work to the non-commutative geometry of M-theory.

Coarse-graining and symplectic non-squeezing

We address aspects of coarse-graining in classical Statistical Physics from the viewpoint of the symplectic non-squeezing theorem. We make some comments regarding the implications of the symplectic non-squeezing theorem for the BBGKY hierarchy. We also see the cubic cells appearing in coarse-graining as a direct consequence of the uniqueness of Hofer’s metric on the group of Hamiltonian diffeomorphisms of the phase space.

Coisotropic Hofer–Zehnder capacities and non-squeezing for relative embeddings

We introduce the notion of a symplectic capacity relative to a coisotropic submanifold of a symplectic manifold, and we construct two examples of such capacities through modifications of the Hofer-Zehnder capacity. As a consequence, we obtain a non-squeezing theorem for symplectic embeddings relative to coisotropic constraints and existence results for leafwise chords on energy surfaces.

Gromov’s theorem in n-symplectic geometry on LRn

If M is an n-dimensional manifold, then the associated bundle of linear frames LM of M supports the canonically defined Rn-valued soldering 1-form θ^. The pair (LM,dθ^) is an n-symplectic manifold, where dθ^ is the n-symplectic 2-form. We adapt the proofs of de Gosson and McDuff and Salamon of Gromov’s non-squeezing theorem on R2n to give a proof of Gromov’s theorem for affine n-symplectomorphisms on LRn.

Time irreversibility from symplectic non-squeezing

The issue of how time reversible microscopic dynamics gives rise to macroscopic irreversible processes has been a recurrent issue in Physics since the time of Boltzmann whose ideas shaped, and essentially resolved, such an apparent contradiction. Following Boltzmann’s spirit and ideas, but employing Gibbs’s approach, we advance the view that macroscopic irreversibility of Hamiltonian systems of many degrees of freedom can be also seen as a result of the symplectic non-squeezing theorem.

Sharp systolic inequalities for Reeb flows on the three-sphere

The systolic ratio of a contact form $\alpha$ on the three-sphere is the quantity \[ \rho_{\mathrm{sys}}(\alpha) = \frac{T_{\min}(\alpha)^2}{\mathrm{vol}(S^3,\alpha\wedge d\alpha)}, \] where $T_{\min}(\alpha)$ is the minimal period of closed Reeb orbits on $(S^3,\alpha)$. A Zoll contact form is a contact form such that all the orbits of the corresponding Reeb flow are closed and have the same period. Our first main result is that $\rho_{\mathrm{sys}}\leq 1$ in a neighbourhood of the space of Zoll contact forms on $S^3$, with equality holding precisely at Zoll contact forms. This implies a particular case of a conjecture of Viterbo, a local middle-dimensional non-squeezing theorem, and a sharp systolic inequality for Finsler metrics on the two-sphere which are close to Zoll ones. Our second main result is that $\rho_{\mathrm{sys}}$ is unbounded from above on the space of tight contact forms on $S^3$.

Symplectic nonsqueezing in Hilbert space and discrete Schrödinger equations

We prove a generalization of Gromov’s symplectic nonsqueezing theorem for the case of Hilbert spaces. Our approach is based on filling almost complex Hilbert spaces by complex discs partially extending Gromov’s results on existence of J-complex curves. We apply our result to the flow of the discrete nonlinear Schrodinger equation.

Global well-posedness and nonsqueezing property for the higher-order KdV-type flow

In this paper, we prove that the periodic higher-order KdV-type equation{∂tu+(−1)j+1∂x2j+1u+12∂x(u2)=0,(t,x)∈R×T,u(0,x)=u0(x),u0∈Hs(T), is globally well-posed in Hs for s≥−j2, j≥3. The proof is based on “I-method” introduced by Colliander et al. [4]. We also prove the nonsqueezing property of the periodic higher-order KdV-type equation. The proof relies on Gromov's nonsqueezing theorem for the finite dimensional Hamiltonian system and an approximation argument for the solution flow. More precisely, after taking the frequency truncation to the solution flow, we apply the nonsqueezing theorem. By using the approximation argument, we extend this result to the infinite dimensional system. This argument was introduced by Kuksin [14] and made concretely by Bourgain [2] for the 1D cubic NLS flow, and Colliander et al. [5] for the KdV flow. One of our observations is that the higher-order KdV-type equation has the better modulation effect from the non-resonant interaction than that the KdV equation has. Hence, unlike the work of Colliander et al. [5], we can get the nonsqueezing property for the solution flow without the Miura transform.

Two geometric interpretations of the multidimensional Hardy uncertainty principle

Hardy's uncertainty principle says that a square integrable function and its Fourier transform cannot be simultaneously arbitrarily sharply localized. We show that a multidimensional version of this uncertainty principle can be best understood in geometrical terms using the fruitful notion of symplectic capacity, which was introduced in the mid-eighties following unexpected advances in symplectic topology (Gromov's non-squeezing theorem). In this geometric formulation, the notion of Fourier transform is replaced with that of polar duality, well-known from convex geometry. • We state a multidimensional version of Hardy's uncertainty principle. • The Hardy uncertainty principle is equivalent to a statement about the symplectic capacity of the Hardy ellipsoid. • We express this result in terms of the projections of the Hardy ellipsoid on the x - and p -spaces.

Bi-invariant metrics on the contactomorphism groups

Contact manifolds are odd-dimensional smooth manifolds endowed with a maximally non-integrable field of hyperplanes. They are intimately related to symplectic manifolds, i.e. even-dimensional smooth manifolds endowed with a closed non-degenerate 2-form. Although in symplectic topology a famous bi-invariant metric, the Hofer metric, has been studied since more than 20 years ago, it is only recently that some somehow analogous bi-invariant metrics have been discovered on the group of diffeomorphisms that preserve a contact structure. In this expository article I will review some constructions of bi-invariant metrics on the contactomorphism group, and how these metrics are related to some other global rigidity phenomena in contact topology which have been discovered in the last few years, in particular the notion of orderability (due to Eliashberg and Polterovich) and an analogue in contact topology (due to Eliashberg, Kim and Poltorovich) of Gromov's symplectic non-squeezing theorem.

A non-squeezing theorem for convex symplectic images of the Hilbert ball

We prove that the non-squeezing theorem of Gromov holds for symplectomorphisms on an infinite-dimensional symplectic Hilbert space, under the assumption that the image of the ball is convex. The proof is based on the construction by duality methods of a symplectic capacity for bounded convex neighbourhoods of the origin. We also discuss the role of infinite-dimensional non-squeezing results in the study of Hamiltonian PDEs and show some examples of symplectomorphisms on infinite-dimensional spaces exhibiting behaviours which would be impossible in finite dimensions.

Gromov's Non-Squeezing Theorem and Beltrami Type Equation

We introduce a method for constructing J-complex discs. The method only uses the standard scheme for solving the Beltrami equation and the Schauder principle. As an application, we give a short self-contained proof of Gromov's Non-Squeezing Theorem.

The Symplectic Camel and Quantum Universal Invariants: The Angel of Geometry versus the Demon of Algebra

A positive de nite symmetric matrix quali es as a quantum mechanical covariance matrix if and only if + 12 i~ 0 where is the standard symplectic matrix. This well-known condition is a strong version of the uncertainty principle, which can be reinterpreted in terms of the topological notion of symplectic capacity, closely related to Gromov’s non-squeezing theorem. We show that a recent re nement of the latter leads to a new class of geometric invariants. These are the volumes of the orthogonal projections of the covariance ellipsoid on symplectic subspaces of the phase space. We compare these geometric invariants to the algebraic “universal quantum invariants”of Dodonov and Sera ni.

Applications of Symplectic Topology to Orbit Uncertainty and Spacecraft Navigation

Gromov’s symplectic nonsqueezing theorem, a fundamental property from symplectic topology, is applied to the study of uncertainty analysis in Hamiltonian Dynamical systems with a particular emphasis on spacecraft trajectory uncertainty. Previous results published in the literature are rederived and shown to be similar to the uncertainty principle of quantum mechanics. The application of Gromov’s Theorem to uncertainty distributions in Hamiltonian Dynamical systems are discussed, including the effect of time mapping and measurement updates. Finally, we provide constraint relations on the phase volume of a distribution and the Gromov width.

HOW LARGE IS THE SHADOW OF A SYMPLECTIC BALL?

Consider the image of the 2n-dimensional unit ball by a symplectic embedding into the standard symplectic vector space of dimension 2n. Its 2k-dimensional shadow is its orthogonal projection onto a complex subspace of real dimension 2k. Is it true that the volume of this 2k-dimensional shadow is at least the volume of the unit 2k-dimensional ball? This statement is trivially true when k = n, and when k = 1 it is a reformulation of Gromov's non-squeezing theorem. Therefore, this question can be considered as a middle-dimensional generalization of the non-squeezing theorem. We investigate the validity of this statement in the linear, nonlinear and perturbative setting.

NILPOTENCE AND THE GENERALIZED UNCERTAINTY PRINCIPLE (S)

We point out that some of the proposed generalized/modified uncertainty principles originate from solvable, or nilpotent at appropriate limits, &#34deformations&#34 of Lie algebras. We briefly comment on formal aspects related to the well-posedness of one of these algebras. We point out a potential relation of such algebras with Classical Mechanics in the spirit of the symplectic non-squeezing theorem. We also point out their relation to a hierarchy of generalized measure theories emerging in a covariant formalism of quantum gravity.