Why do we construct Lagrangian submanifolds after symplectic reductions

Question

I am learning about Hamilton-Jacobi actions, symplectic reductions and Lagrangian submanifolds and I am trying to understand the relation between these concepts.

I have read that Lagrangian submanifolds are physically interesting as they can be thought of the space of all momenta at fixed coordinate, locally. Moreover, they allow us to recover the variational form of the Hamiltonian mechanics of the system we are dealing with.

As I understand it, symplectic reductions arose from the interest of taking quotients of symplectic manifolds under group actions and the Hamilton-Jacobi group action is the one that makes it possible (via the momentum map $\mu$) as it satisfies several requirements as the dimension and the symplectic structure of the manifold. In particular, the Mardsen-Weinstein-Meyer theorem states that the quotient $\mu^{-1}(0) / G$ is a symplectic manifold. Finally, I have read that this latter quotient captures the original Hamiltonian mechanics.

From this, most of the documents I have read naturally turn to the wish to recover Lagrangian submanifolds from symplectic reductions. For this, they use the level sets of the momentum map.

It is a rough summary and obviously any correction/precision will be appreciated.

My confusion comes from the wish to construct Lagrangian manifolds from symplectic reductions. First, $\mu^{-1}(0) / G$ seems to provide some physical insights on our system and so do Lagrangian submanifolds. So, either they do it in a more interesting/accurate way, or I am missing the point in doing it "twice". Then, the problem is probably that I am a bit short on basic knowledge here but why are we interested in many level sets of $\mu$ and not only the $0$ one? Is it because the variational form of the Hamiltonian mechanics is local?

Answer 1

You're seeing people first doing symplectic reduction and then looking for Lagrangian submanifolds because that's what you have to do to quantize a gauge theory.

There are two a priori distinct notions here:

A Lagrangian submanifold is, in physical terms and the right coordinates, just the space of canonical momenta at some fixed canonical coordinates (or vice versa). If you have $n$ constants of motion $f_i(q,p)$, then $f^{-1}(c)$ for any $c\in\mathbb{R}$ that's a regular value of $f = (f_1,\dots,f_n)$ is such a Lagrangian submanifold (this is Arnold-Liouville, see e.g. this MO answer). The connection to the first formulation is that these two notions become the same by choosing action-angle variables where the constants of motion are your coordinates/momenta.

Symplectic reduction constructs, for the momentum map $\mu$ of a particular symmetry $G$, a "quotient" $\mu^{-1}(0)/G$, the reduced phase space. While $\mu^{-1}(0)$ is a Lagrangian submanifold since the momenta of symmetries are constants of motion, this isn't really the point here: Symplectic reduction is a technique applied to constrained Hamiltonian dynamics, where the $G$ is a gauge symmetry and so the generators of $G$ (in the Lie algebraic sense) are the constraints of our theory, so $\mu^{-1}(0)$ is the constraint surface to which all physical motions are confined. The quotient by $G$ gets rid of the rest of the gauge theory, it shrinks all the gauge orbits (the orbits of points on the surface under the transformations of $G$) to a single point in the quotient so that every point $\mu^{-1}(0)/G$ really does correspond to a meaningful and distinguishable physical state.

Now, on this reduced phase space, the same motivations you might have to find Lagrangian submanifold on a general phase space still apply - this is now a constraint-free Hamiltonian phase space (or at least one with fewer constraints). In particular, if you are doing geometric quantization you want to first get rid of the gauge theory and then quantize, and quantization requires a choice of Lagrangian foliation called the polarization. You can think of this as having to choose the arguments of the wavefunction - the classical phase space has both $x$ and $p$, but the quantum mechanical wavefunction is just a function of only $x$ (or only $p$, via Fourier transform). The idea is that the Lagrangian foliation tells you what the arguments to the wavefunction are, since as we discussed at the beginning a Lagrangian submanifold corresponds to having all the coordinates or all the momenta at fixed values, so something that's a function on such a submanifold is just a function of only the coordinates or only the momenta.

So the process "first symplectically reduce, then find a Lagrangian foliation" is the mathematical formulation of the geometric quantization of gauge theories.