Your Logo HereDr Maurice de Gosson
Personal Homepage

Introduction | Publications 2017 | Previous Research | Foundations of QM | The Symplectic Camel | Organization of Conferences | Hans G. Feichtinger | Basil Hiley | Jean Leray | Financial Mathematics | Guestbook | Links


The symplectic camel: the tip of an iceberg?





LATEST NEWS!!! My paper "The Symplectic Egg":  http://arxiv.org/pdf/1208.5969.pdf  has appeared in the American Journal of Physics in a slightly revised and augmented form: SymplecticEgg_AJP.pdf
It was the most read/downloaded AJP paper in April 2013!

ERRATUM: Mats Vermeeren (TU, Berlin) has pointed out to me that the proof of teh claim that the section of a symplectic egg by a conjugate plane has constant area is flawed! You can download the Errarum here: symplectic egg erratum.pdf

Here is another short related paper: 

The Symplectic Camel and Quantum Universal Invariants: the Angel of Geometry versus the Demon of Algebra Subm_MdG.pdf


What is the symplectic camel? A Hamiltonian flow is volume preserving: this is Liouville's theorem, one of the best known results from elementary statistical mechanics. Liouville's theorem is perhaps also one of the most understated results of classical mechanics, because in addition of being volume-preserving, Hamiltonian flows have a surprising -I am tempted to say an extraordinary- additional property as soon as the number of degrees of freedom is superior to one. Assume that we are dealing with a large number N of material particles and that the particles are very close to each other. We may in this case approximate that physical with a "cloud" of points in phase space. Suppose that this cloud is, at time t=0 spherical so it is represented by a phase space ball B(r) with center (a,b) and radius r.
The orthogonal projection of that ball on any plane of coordinates will always be a disk with same area as the intersection of the ball with a plane passing through its vcenter. Let us watch the motion of this spherical phase-space cloud as time evolves. It will distort and may take after a while a very different shape, while keeping constant volume. However --and this is the surprising result-- the projections of that deformed ball on any plane of conjugate coordinates x_{j},p_{j}.will never decrease below its original value! If we had chosen, on the contrary, a plane of non-conjugate coordinates then there would be no obstruction for the projection to become arbitrarily small. The property just described is not a physical observation, but a mathematical theorem proved by the 2009 Abel prize winner Mikhail Gromov in 1985. It is also known as "Gromov's non-squeezing theorem", see the Wikipedia article http://en.wikipedia.org/wiki/Non-squeezing_theorem .
 If we choose for r the square root of h-bar (=Planck's constant h divided by 2 pi), then Gromov's theorem says that the projection of the ball on a conjugate plane will always be at least (1/2)h and this is of course strongly reminiscent of the uncertainty principle of quantum mechanics, of which it can be viewed as a classical geometrical version!
But why are we talking about a symplectic "camel"?     Recall that 

"...It is easier for a camel to pass through the eye of a needle than for one who is rich to enter the kingdom of God..."
    The Biblical camel is here the phase space ball and the eye of the needle is the hole in the x_{j},p_{j} plane! 




Quantum Blobs

I introduced the notion of "quantum blob" a few years ago:  quantum blobs are minimum uncertainty units of phase space. They are are different from the usual "quantum cells" used in thermodynamics, and they are -as opposed to these- symplectic invariants which allow a nicely invariant coarse-graining of phase space. 

Quantum blobs are defined in terms of symplectic capacities (and hence related to the Symplectic camel!): they are the images of a phase space ball with radius square root of ℏ (Planck's constant of Plank h divided by 2π) by a linear or affine symplectic transformation. They are phase space ellipsoids which have the strange property that if you cut them with any symplectic plane passing thr4ough their center you will get an ellipse with area exactly  h/2, one half of the quantum of action!
Here is my latest paper on quantum blobs; it appeared in Foundations of Physics, and is part of the Festschrift in honor of Basil Hiley. It is now online and freely available:








© 2005 All Rights Reserved.

Create a Free Website