This question arises when studying semi-classical analysis. Since Weinstein's creed claims that "everything is Lagrangian", where a point in the phase space of classical mechanics is just a cotangent fiber, hence Lagrangian, what can we say about canonical relations, which is a Lagrangian submanifold of the twisted direct product of two symplectic manifolds? As far as I know, if we quantize the canonical relations we get unbounded operators between two Hilbert spaces. Why so? I just want someone to figure this out for me, just as the way we can regard a point in phase space as Lagrangian of the cotangent bundle. Thanks.



What is the physical interpretation of the canonical relations?
I think I have figure this out. Consider two symplectic manifolds $M\_1$ and $M\_2$, and the canonical relation $\Gamma: M\_1\twoheadrightarrow M\_2$. Then if one consider the "point" in symplectic category, i.e. the Lagrangian submanifold $\Lambda\_i$ of $M\_i, (i=1,2)$, $\Gamma$ just maps one "point" to another "point" via$$
\Gamma(p)=q\in Morph(pt., M\_2)
$$ 
where $p\in Morph(pt., M\_1)$. In geometric quantization, this is equivalent to say that canonical relations are quantized to be unbounded operators between Hilbert spaces. 


