Non-squeezing theorem

The non-squeezing theorem, also called Gromov's non-squeezing theorem, is one of the most important theorems in symplectic geometry. It was first proven in 1985 by Mikhail Gromov. The theorem states that one cannot embed a ball into a cylinder via a symplectic map unless the radius of the ball is less than or equal to the radius of the cylinder. The theorem is important because formerly very little was known about the geometry behind symplectic maps. One easy consequence of a transformation being symplectic is that it preserves volume. One can easily embed a ball of any radius into a cylinder of any other radius by a volume-preserving transformation: just picture squeezing the ball into the cylinder (hence, the name non-squeezing theorem). Thus, the non-squeezing theorem tells us that, although symplectic transformations are volume-preserving, it is much more restrictive for a transformation to be symplectic than it is to be volume-preserving.

Consider the symplectic spaces

${\displaystyle \mathbb {R} ^{2n}=\{z=(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n})\},}$

${\displaystyle B^{2n}(r)=\{z\in \mathbb {R} ^{2n}:\|z\|<r\},}$

${\displaystyle Z^{2n}(R)=\{z\in \mathbb {R} ^{2n}:x_{1}^{2}+y_{1}^{2}<R^{2}\},}$

each endowed with the symplectic form

${\displaystyle \omega =dx_{1}\wedge dy_{1}+\cdots +dx_{n}\wedge dy_{n}.}$

The space ${\displaystyle B^{2n}(r)}$is called the ball of radius ${\displaystyle r}$and ${\displaystyle Z^{2n}(R)}$is called the cylinder of radius ${\displaystyle R}$. The choice of axes for the cylinder are not arbitrary given the fixed symplectic form above; the circles of the cylinder each lie in a symplectic subspace of ${\displaystyle \mathbb {R} ^{2n}}$.

If ${\displaystyle (M,\eta )}$and ${\displaystyle (N,\nu )}$are symplectic manifolds, a symplectic embedding ${\displaystyle \varphi :(M,\eta )\to (N,\nu )}$is a smooth embedding ${\displaystyle \varphi :M\to N}$such that ${\displaystyle \varphi ^{*}\nu =\eta }$. For ${\displaystyle r\leq R}$, there is a symplectic embedding ${\displaystyle B^{2n}(r)\to Z^{2n}(R)}$which takes ${\displaystyle x\in B^{2n}(r)\subset \mathbb {R} ^{2n}}$to the same point ${\displaystyle x\in Z^{2n}(R)\subset \mathbb {R} ^{2n}}$.

Gromov's non-squeezing theorem says that if there is a symplectic embedding ${\displaystyle \varphi :B^{2n}(r)\to Z^{2n}(R)}$, then ${\displaystyle r\leq R}$.

A symplectic capacity is a map ${\displaystyle c:\{{\text{symplectic manifolds}}\}\to [0,\infty ]}$satisfying

1. (Monotonicity) If there is a symplectic embedding ${\displaystyle (M,\omega )\to (N,\eta )}$and ${\displaystyle \dim M=\dim N}$, then ${\displaystyle c(M,\omega )\leq c(N,\eta )}$,
2. (Conformality) ${\displaystyle c(M,\lambda \omega )=\lambda c(M,\omega )}$,
3. (Nontriviality) ${\displaystyle c(B^{2n}(1))>0}$and ${\displaystyle c(Z^{2n}(1))<\infty }$.

The existence of a symplectic capacity satisfying

${\displaystyle c(B^{2n}(1))=c(Z^{2n}(1))=\pi }$

is equivalent to Gromov's non-squeezing theorem. Given such a capacity, one can verify the non-squeezing theorem, and given the non-squeezing theorem, the Gromov width

${\displaystyle w_{G}(M,\omega )=\sup\{\pi r^{2}:{\text{there exists a symplectic embedding }}B^{2n}(r)\to (M,\omega )\}}$

is such a capacity.

Gromov's non-squeezing theorem has also become known as the principle of the symplectic camel since Ian Stewart referred to it by alluding to the parable of the camel and the eye of a needle. As Maurice A. de Gosson states:

Similarly:

De Gosson has shown that the non-squeezing theorem is closely linked to the Robertson–Schrödinger–Heisenberg inequality, a generalization of the Heisenberg uncertainty relation. The Robertson–Schrödinger–Heisenberg inequality states that:

${\displaystyle \operatorname {var} (Q)\operatorname {var} (P)\geq \operatorname {cov} ^{2}(Q,P)+\left({\frac {\hbar }{2}}\right)^{2}}$

with Q and P the canonical coordinates and var and cov the variance and covariance functions.

• Maurice A. de Gosson: The symplectic egg, arXiv:1208.5969v1, submitted on 29 August 2012 – includes a proof of a variant of the theorem for case of linear canonical transformations
• Dusa McDuff: What is symplectic geometry?, 2009