The Symplectic Camel in Quantum Mechanics

Gromov's discovery in 1985 of the symplectic non-squeezing theorem,often dubbed "the principle of the symplectic camel", can be viewed as a classical version of theuncertainty principle. We will show that a derived notion, the symplecticcapacity of subsets of phase space, not only allows a symplectically invariantreformulation of the quantum uncertainty principle, but also leads to a newdefinition of quantum indeterminacy. The latter is defined in terms of thenotion of polarity between convex sets, and can be measured using the Hofer-Zehnder capacity. We briefly discuss the relationship between thisnotion of quantum indeterminacy and Hardy's uncertainty principle about thelocalization of a function and its Fourier transform.