The non-squeezing theorem, also called Gromov's non-squeezing theorem, is one of the most important theorems in symplectic geometry.[1] It was first proven in 1985 by Mikhail Gromov.[2] 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.[3] 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.
Note: The choice of axes for the cylinder are not arbitrary given the fixed symplectic form above; namely the circles of the cylinder each lie in a symplectic subspace of .
The non-squeezing theorem tells us that if we can find a symplectic embedding φ : B(R) → Z(r) then R ≤ r.
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.[4] As Maurice A. de Gosson states:
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 , 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?[5]
Similarly:
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.
— Andrea Censi, Symplectic camels and uncertainty analysis[6]
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:
with Q and P the canonical coordinates and var and cov the variance and covariance functions.[7]
References
^Tao, Terence (2006), Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, vol. 106, American Mathematical Society, p. 219, ISBN9780821889503, MR 2233925, This theorem is especially surprising in light of Darboux' theorem ... It is a result of fundamental importance in symplectic geometry.
^Gromov, M. L. (1985). "Pseudo holomorphic curves in symplectic manifolds". Inventiones Mathematicae. 82 (2): 307–347. Bibcode:1985InMat..82..307G. doi:10.1007/BF01388806. S2CID 4983969.
^Stewart, I.: The symplectic camel, Nature 329(6134), 17–18 (1987), doi:10.1038/329017a0. Cited after Maurice A. de Gosson: The Symplectic Camel and the Uncertainty Principle: The Tip of an Iceberg?, Foundations of Physics (2009) 39, pp. 194–214, doi:10.1007/s10701-009-9272-2, therein: p. 196
^Maurice A. de Gosson: The Symplectic Camel and the Uncertainty Principle: The Tip of an Iceberg?, Foundations of Physics (2009) 39, pp. 194–214, doi:10.1007/s10701-009-9272-2, therein: p. 199
^Andrea Censi: Symplectic camels and uncertainty analysis
^Maurice de Gosson: How classical is the quantum universe? arXiv:0808.2774v1 (submitted on 20 August 2008)
Further reading
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
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The non squeezing theorem also called Gromov s non squeezing theorem is one of the most important theorems in symplectic geometry 1 It was first proven in 1985 by Mikhail Gromov 2 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 3 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 Contents 1 Background and statement 2 The symplectic camel 3 References 4 Further readingBackground and statement EditWe start by considering the symplectic spaces R 2 n z x 1 x n y 1 y n displaystyle mathbb R 2n z x 1 ldots x n y 1 ldots y n the ball of radius R B R z R 2 n z lt R displaystyle B R z in mathbb R 2n z lt R and the cylinder of radius r Z r z R 2 n x 1 2 y 1 2 lt r 2 displaystyle Z r z in mathbb R 2n x 1 2 y 1 2 lt r 2 each endowed with the symplectic form w d x 1 d y 1 d x n d y n displaystyle omega dx 1 wedge dy 1 cdots dx n wedge dy n Note The choice of axes for the cylinder are not arbitrary given the fixed symplectic form above namely the circles of the cylinder each lie in a symplectic subspace of R 2 n displaystyle mathbb R 2n The non squeezing theorem tells us that if we can find a symplectic embedding f B R Z r then R r The symplectic camel EditGromov 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 4 As Maurice A de Gosson states 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 x j displaystyle x j p j displaystyle p j 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 5 Similarly 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 Andrea Censi Symplectic camels and uncertainty analysis 6 De Gosson has shown that the non squeezing theorem is closely linked to the Robertson Schrodinger Heisenberg inequality a generalization of the Heisenberg uncertainty relation The Robertson Schrodinger Heisenberg inequality states that v a r Q v a r P c o v 2 Q P ℏ 2 2 displaystyle var Q var P geq 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 7 References Edit Tao Terence 2006 Nonlinear Dispersive Equations Local and Global Analysis CBMS Regional Conference Series in Mathematics vol 106 American Mathematical Society p 219 ISBN 9780821889503 MR 2233925 This theorem is especially surprising in light of Darboux theorem It is a result of fundamental importance in symplectic geometry Gromov M L 1985 Pseudo holomorphic curves in symplectic manifolds Inventiones Mathematicae 82 2 307 347 Bibcode 1985InMat 82 307G doi 10 1007 BF01388806 S2CID 4983969 D McDuff and D Salamon 1996 Introduction to Symplectic Topology Cambridge University Press ISBN 978 0 19 850451 1 Stewart I The symplectic camel Nature 329 6134 17 18 1987 doi 10 1038 329017a0 Cited after Maurice A de Gosson The Symplectic Camel and the Uncertainty Principle The Tip of an Iceberg Foundations of Physics 2009 39 pp 194 214 doi 10 1007 s10701 009 9272 2 therein p 196 Maurice A de Gosson The Symplectic Camel and the Uncertainty Principle The Tip of an Iceberg Foundations of Physics 2009 39 pp 194 214 doi 10 1007 s10701 009 9272 2 therein p 199 Andrea Censi Symplectic camels and uncertainty analysis Maurice de Gosson How classical is the quantum universe arXiv 0808 2774v1 submitted on 20 August 2008 Further reading EditMaurice 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 Retrieved from https en wikipedia org w index php title Non squeezing theorem amp oldid 1127944341, wikipedia, wiki, book, books, library,
