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.

## 1 Answer

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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.