1
$\begingroup$

It is well known that the coadjoint orbits of the Heisenberg group (with a suitable choice of coordinate system) are the planes $z=c\ne 0$ parallel to the $xy$-plane, and the points in the $xy$-plane. In the former case, the canonical symplectic form of the coadjoint orbit is a given by $$\mu_c=\frac{dx\wedge dy}{c}$$ which is a scaling of the the Lebesgue measure.

My question is whether there are any mathematical (or physical) interpretations of the limiting cases of $\mu_c$ as $c\to 0$ and $c\to \infty$.

Although the derivation of $\mu_c$ is quite easy, I find it difficult to come up with a satisfactory explanation about the way the symplectic measure changes as we move on coadjoint orbits.

Any explanation is very much appreciated.

Improve this question
$\endgroup$
1
  • $\begingroup$ Not sure about the $z \to \infty$ case, but the case $z \to 0$ is best viewed by considering the linear Poisson structure on the dual of the Lie algebra of the Heisenberg group. The corresponding Poisson bivector is (up to sign) $z \partial_x \wedge \partial_y$ -- this is "just" the inverse of the leafwise symplectic form that you wrote. This bivector vanishes on the $z=0$ plane, which is just saying that each point on this plane is a coadjoint orbit. $\endgroup$
    – Daniele Sepe
    Commented Jul 19, 2015 at 23:57

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .