The '''non-squeezing theorem''', also called '''Gromov's non-squeezing theorem''', is one of the most important theorems in symplectic geometry.<ref>{{cite book | last = Tao | first = Terence | mr = 2233925 | page = 219 | publisher = American Mathematical Society | quote = This theorem is especially surprising in light of Darboux' theorem ... It is a result of fundamental importance in symplectic geometry | series = CBMS Regional Conference Series in Mathematics | title = Nonlinear Dispersive Equations: Local and Global Analysis | url = https://books.google.com/books?id=Uc1WXBWv-EsC&pg=PA219 | volume = 106 | year = 2006| isbn = 9780821889503 }}.</ref> It was first proven in 1985 by Mikhail Gromov.<ref>{{cite journal|title=Pseudo holomorphic curves in symplectic manifolds|journal=Inventiones Mathematicae|year=1985|first=M. L. |last=Gromov|volume=82|issue=2 |pages=307–347|doi=10.1007/BF01388806|bibcode=1985InMat..82..307G|s2cid=4983969 }}</ref> 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.<ref name="McDuff">{{cite book |last1=McDuff |first1=Dusa |last2=Salamon |first2=Dietmar |year=2017 |title=Introduction to Symplectic Topology |publisher=Oxford University Press |series=Oxford Graduate Texts in Mathematics}}</ref> 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.
== Background and statement == Consider the symplectic spaces : <math>\mathbb{R}^{2n} = \{z = (x_1, \ldots , x_n, y_1, \ldots , y_n) \},</math> : <math>B^{2n}(r) = \{z \in \mathbb{R}^{2n} : \| z \| < r \},</math> : <math>Z^{2n}(R) = \{z \in \mathbb{R}^{2n} : x_1^2 + y_1^2 < R^2 \},</math> each endowed with the symplectic form : <math> \omega = dx_1 \wedge dy_1 + \cdots + dx_n \wedge dy_n.</math> The space <math>B^{2n}(r)</math> is called the ball of radius <math>r</math> and <math>Z^{2n}(R)</math> is called the cylinder of radius <math>R</math>. 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 <math>\mathbb{R}^{2n}</math>.
If <math>(M,\eta)</math> and <math>(N,\nu)</math> are symplectic manifolds, a ''symplectic embedding'' <math>\varphi : (M,\eta) \to (N,\nu)</math> is a smooth embedding <math>\varphi : M \to N</math> such that <math>\varphi^* \nu = \eta</math>. For <math>r \leq R</math>, there is a symplectic embedding <math>B^{2n}(r) \to Z^{2n}(R)</math> which takes <math>x \in B^{2n}(r) \subset \mathbb R^{2n}</math> to the same point <math>x \in Z^{2n}(R) \subset \mathbb R^{2n}</math>.
''Gromov's non-squeezing theorem'' says that if there is a symplectic embedding <math>\varphi : B^{2n}(r) \to Z^{2n}(R)</math>, then <math>r \leq R</math>.<ref name="McDuff" />
== Symplectic capacities == A ''symplectic capacity'' is a map <math>c : \{ \text{symplectic manifolds} \} \to [0,\infty]</math> satisfying # (Monotonicity) If there is a symplectic embedding <math>(M, \omega) \to (N,\eta)</math> and <math>\dim M = \dim N</math>, then <math>c(M,\omega) \leq c(N,\eta)</math>, # (Conformality) <math>c(M, \lambda \omega) = \lambda c(M, \omega)</math>, # (Nontriviality) <math>c(B^{2n}(1)) > 0</math> and <math>c(Z^{2n}(1)) < \infty</math>.<ref name="McDuff" />
The existence of a symplectic capacity satisfying : <math>c(B^{2n}(1)) = c(Z^{2n}(1)) = \pi</math> 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'' : <math>w_G(M, \omega) = \sup \{ \pi r^2 : \text{there exists a symplectic embedding } B^{2n}(r) \to (M,\omega) \}</math> is such a capacity.<ref name="McDuff" />
== The “symplectic camel” == 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''.<ref>Stewart, I.: ''The symplectic camel'', Nature 329(6134), 17–18 (1987), {{doi|10.1038/329017a0}}. Cited after {{cite journal |last1=De Gosson |first1=Maurice A. |title=The Symplectic Camel and the Uncertainty Principle: The Tip of an Iceberg? |journal=Foundations of Physics |date=2009 |volume=39 |issue=2 |pages=194–214 [196]|doi=10.1007/s10701-009-9272-2 |bibcode=2009FoPh...39..194D }}</ref> As Maurice A. de Gosson states: {{Quote|Now, why do we refer to a symplectic camel in the title of this paper? This is because one can restate Gromov’s theorem in the following way: there is no way to deform a phase space ball using canonical transformations in such a way that we can make it pass through a hole in a plane of conjugate coordinates <math>x_j</math> , <math>p_j</math> if the area of that hole is smaller than that of the cross-section of that ball.|Maurice A. de Gosson|The Symplectic Camel and the Uncertainty Principle: The Tip of an Iceberg?<ref>{{cite journal |last1=De Gosson |first1=Maurice A. |title=The Symplectic Camel and the Uncertainty Principle: The Tip of an Iceberg? |journal=Foundations of Physics |date=2009 |volume=39 |issue=2 |pages=194–214 [199]|doi=10.1007/s10701-009-9272-2 |bibcode=2009FoPh...39..194D }}</ref>}} Similarly: {{Quote|Intuitively, a volume in phase space cannot be stretched with respect to one particular symplectic plane more than its “symplectic width” allows. In other words, it is impossible to squeeze a symplectic camel into the eye of a needle, if the needle is small enough. This is a very powerful result, which is intimately tied to the Hamiltonian nature of the system, and is a completely different result than Liouville's theorem, which only interests the overall volume and does not pose any restriction on the ''shape''<!--italics in original-->.|Andrea Censi|Symplectic camels and uncertainty analysis<ref>Andrea Censi: [http://www.cds.caltech.edu/~marsden/wiki/uploads/projects/geomech/Censi.pdf ''Symplectic camels and uncertainty analysis'']</ref>}} Although the term "symplectic camel" is sometimes used loosely to describe Gromov's non-squeezing theorem in its static form, experts in symplectic topology reserve it for the parametric version, which concerns the impossibility of moving a symplectic ball from one side of a hyperplane {{math|''H''}} to the other via a one-parameter family of symplectic embeddings, in such a way that the symplectic reductions of the intersections of the balls with the hyperplane {{math|''H''}} are always contained in a cylinder of smaller capacity.<ref>{{cite journal |last=Eliashberg |first=Yakov |author2=Mikhail Gromov |year=1991 |title=Convex symplectic manifolds |journal=Proceedings of Symposia in Pure Mathematics |volume=52 |issue=pt. 2 |pages=135–162 |doi=10.1090/pspum/052.2}}</ref><ref>{{cite journal |last=Viterbo |first=Claude |year=1992 |title=Symplectic topology as the geometry of generating functions |journal=Mathematische Annalen |volume=292 |issue=4 |pages=685–710 |doi=10.1007/BF01444643}}</ref><ref>{{cite journal |last=McDuff |first=Dusa |author2=Lisa Traynor |year=1994 |title=The 4-dimensional symplectic camel and related results |journal=London Mathematical Society Lecture Note Series |volume=192 |pages=169–182 |publisher=Cambridge University Press |doi=10.1017/CBO9780511526343.010 |isbn=978-0-521-44699-0 }}</ref>
== Further work == 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: :<math>\operatorname{var}(Q) \operatorname{var}(P) \geq \operatorname{cov}^2(Q,P) + \left(\frac{\hbar}{2}\right)^2</math> with Q and P the canonical coordinates and ''var'' and ''cov'' the variance and covariance functions.<ref>{{Cite journal |last1=Costa Dias |first1=Nuno |last2=de Gosson |first2=Maurice |last3=Nuno Prata |first3= João |title=A Refinement of the Robertson–Schrödinger Uncertainty Principle and a Hirschman–Shannon Inequality for Wigner Distributions |journal=Journal of Fourier Analysis and Applications |volume=25 |date=2019 |issue=1 |pages=210–241 [212] |doi=10.1007/s00041-018-9602-x|pmid=30872908 |pmc=6383836 |arxiv=1712.09475 |bibcode=2019JFAA...25..210D }}</ref><ref>{{Cite journal |last1=de Gosson |first1=Maurice |last2=Luef |first2=Franz |title=Symplectic capacities and the geometry of uncertainty: The irruption of symplectic topology in classical and quantum mechanics |journal=Physics Reports |volume=484 |number=5 |date=December 2009 |pages=131–179 |doi=10.1016/j.physrep.2009.08.001 |bibcode=2009PhR...484..131D }}</ref><ref>{{cite arXiv | author1=Maurice de Gosson | title=How classical is the quantum universe? | date=2008 | class=quant-ph | eprint=0808.2774 }}</ref>
== References == {{reflist}}
== Further reading == * {{cite arXiv |last1=De Gosson |first1=Maurice A. |author-link=Maurice A. de Gosson |title=The Symplectic Egg |date=2012 |eprint=1208.5969 |class=math-ph}} – includes a proof of a variant of the theorem for case of ''linear'' canonical transformations * Dusa McDuff: [http://www.math.sunysb.edu/~dusa/ewmcambrevjn23.pdf What is symplectic geometry?], 2009
Category:Symplectic geometry Category:Theorems in geometry