I would like to know if there is some notion of classical uncertainty which quantizes to give quantum uncertainty?
For instance, suppose we have a classical system whose phase space is given by a symplectic manifold $M$. Applying methods like geometric quantization we can often construct a corresponding quantum theory with a Hilbert space $\mathcal{H}$. Here we find the quantum uncertainty principle, which states that Hermitian operators $A, B$ satisfy the inequality $$ \sigma^2_A \sigma^2_B \geq \lvert \frac{1}{2} \langle \{ A, B \} \rangle - \langle A \rangle \langle B \rangle \rvert^2 + \lvert \frac{1}{2i} \langle [A, B] \rangle \rvert^2 $$ with the standard definition of variance.
For instance: symplectic capacity and Gromov's non-squeezing theorem describes one notion of classical uncertainty. Does this (or some other notion of classical uncertainty) quantize to give the familiar quantum uncertainty principle?
