Quantum Footprints of Symplectic Rigidity

Abstract

In this note, we discuss an interaction between symplectic topology and quantum mechanics. The interaction goes in both directions. On one hand, ideas from quantum mechanics give rise to new notions and structures on the symplectic side and, furthermore, quantum mechanical insights lead to useful symplectic predictions when topological intuition fails. On the other hand, some phenomena discovered within symplectic topology admit a meaningful translation into the language of quantum mechanics, thus revealing quantum footprints of symplectic rigidity. What is . . . symplectic?A symplectic manifold is an even-dimensional manifold M 2n equipped with a closed differential 2-form ω that can be written as n i=1 dp i ∧ dq i in appropriate local coordinates (p, q). For an oriented surface Σ ⊂ M, the integral Σ ω plays the role of a generalised area, which, in contrast to the Riemannian area, can be negative or vanish.To have some interesting examples in mind, think of surfaces with an area form and their products, as well as complex projective spaces equipped with the Fubini-Study form, and their complex submanifolds.Symplectic manifolds model the phase spaces of systems of classical mechanics. Observables (i.e. physical quantities such as energy, momentum, etc.) are represented by functions on M. The states of the system are encoded by Borel probability measures µ on M. The simplest states are given by the Dirac measure δ z concentrated at a point z ∈ M.The laws of motion are governed by the Poisson bracket, a canonical operation on smooth functions on M, given by. The evolution of the system is determined by its energy, a time-dependent function f t : M → R called its Hamiltonian. Hamilton's famous equation describing the motion of a system is given, in the Heisenberg picture, byġ t = { f t , g t }, where g t = g • φ −1 t stands for the time evolution of an observable function g on M under the Hamiltonian flow φ t . The maps φ t are called Hamiltonian diffeomorphisms. They preserve the symplectic form ω and constitute a group with respect to composition.In the 1980s, new methods, such as Gromov's theory of pseudo-holomorphic curves and the Floer-Morse theory on loop spaces, gave birth to "hard" symplectic topology. It detected surprising symplectic rigidity phenomena involving symplectic manifolds, their subsets and diffeomorphisms. A number of recent advances show that there is yet another manifestation of symplectic rigidity taking place in function spaces associated to a symplectic manifold. Its study forms the subject of function theory on symplectic manifolds, a rapidly evolving area whose development has led to the interactions with quantum mechanics described below. 2The non-displaceable fiber theoremIn 1990, Hofer [21] introduced an intrinsic "small scale" on a symplectic manifold: a subset X ⊂ M is called displaceable if there exists a Hamiltonian diffeomorphism φ such that φ(X) ∩ X = ∅.

Inferring topology of quantum phase space

“…While reconstructing the symplectic structure seems to be out of reach, one can detect certain quantum footprints of symplectic rigidity, see e.g. [21].…”

The Symplectic Camel and Poincaré Superrecurrence: Open Problems

“…Gromov’s Two Balls Theorem’ whose classical consequences we discussed above is related to the notion of dislocation of quantum states as discussed by Polterovich [ 20 ] and Charles and Polterovich [ 21 ].…”