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?

  • $\begingroup$ Related MO.SE posts: mathoverflow.net/q/355929/13917 , mathoverflow.net/q/339668/13917 $\endgroup$
    – Qmechanic
    Jan 17 at 11:30
  • $\begingroup$ I’ve read some of de Gosson’s papers, as mentioned in one of Qmechanic’s links above. The relationship that de Gosson outlines is exactly the sort of thing I’m curious about. However it still seems like an incomplete explanation. $\endgroup$
    – leob
    Jan 17 at 12:16


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