Electrical impedance myography

2012-08-21T02:27:16Z

76.118.180.130: /* Biological relevance */

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 the winner of the 2009 [[Abel Prize]], [[Mikhail Gromov (mathematician)|Mikhail Gromov]].<ref>{{cite journal|title=Pseudo holomorphic curves in symplectic manifolds|journal=Inventiones Mathematicae|year=1985|first=M. L. |last=Gromov|coauthors=|volume=82|issue=|pages=307–347|id= |url=|format=|accessdate=|doi=10.1007/BF01388806|bibcode=1985InMat..82..307G}}</ref> The theorem states that one cannot embed a sphere into a cylinder via a [[symplectomorphisms|symplectic map]] unless the radius of the sphere is less than or equal to the radius of the cylinder. The importance of this theorem is as follows: very little was known about the geometry behind [[symplectomorphisms|symplectic transformations]]. One easy consequence of a transformation being symplectic is that it preserves volume.<ref>D. McDuff and D. Salamon''Introduction to Symplectic Topology'', Cambridge University Press (1996), ISBN 978-0-19-850451-1.</ref> Since 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 ==
We start by considering the symplectic spaces

: <math> \mathbb{R}^{2n} = \{z = (x_1, \ldots , x_n, y_1, \ldots , y_n )\}, </math>

the ball of radius ''R'': <math>B(R) = \{z \in \mathbb{R}^{2n} | \|z \| < R \}, </math>

and the cylinder of radius ''r'': <math>Z(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 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 (mathematician)|Ian Stewart]] referred to it by alluding to the parable of the ''camel and the [[eye of the needle]]''.<ref>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?'', Foundation of Physics (2009) 39, pp. 194–214, {{DOI|10.1007/s10701-009-9272-2}}, therein: p. 196</ref> 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 transformation]]s 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>Maurice A. de Gosson: ''The Symplectic Camel and the Uncertainty Principle: The Tip of an Iceberg?'', Foundation of Physics (2009) 39, pp. 194–214, {{DOI|10.1007/s10701-009-9272-2}}, therein: p. 199</ref>}}
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 (Hamiltonian)|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<ref>Andrea Censi: [http://www.cds.caltech.edu/~marsden/wiki/uploads/projects/geomech/Censi.pdf ''Symplectic camels and uncertainty analysis'']</ref>}}

De Gosson has shown that the non-squeezing theorem is closely linked to the ''Robertson–Schrödinger–Heisenberg inequality'', a generalization of the [[Uncertainty principle|Heisenberg uncertainty relation]]. The ''Robertson–Schrödinger–Heisenberg inequality'' states that:
:<math>var(Q) var(P) \geq cov^2(Q,P) + (\frac{\hbar}{2})^2</math>
with Q and P the [[canonical coordinates]] and ''var'' and ''cov'' the variance and covariance functions.<ref>Maurice de Gosson: ''How classical is the quantum universe?'' [http://arxiv.org/abs/0808.2774v1 arXiv:0808.2774v1] (submitted on 20 August 2008)</ref>

== References ==
{{reflist}}

== Further reading ==
* [[Maurice A. de Gosson]]: ''The symplectic egg'', [http://arxiv.org/abs/1208.5969v1 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]]: [http://www.math.sunysb.edu/~dusa/ewmcambrevjn23.pdf What is symplectic geometry?], 2009

[[Category:Symplectic geometry]]
[[Category:Theorems in geometry]]

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Electrical impedance myography