  We consider the cubic nonlinear Schr\"odinger equation (NLS) in any spatial
dimension, which is a well-known example of an infinite-dimensional Hamiltonian
system. Inspired by the knowledge that the NLS is an effective equation for a
system of interacting bosons as the particle number tends to infinity, we
provide a derivation of the Hamiltonian structure, which is comprised of both a
Hamiltonian functional and a weak symplectic structure, for the nonlinear
Schr\"odinger equation from quantum many-body systems. Our geometric
constructions are based on a quantized version of the Poisson structure
introduced by Marsden, Morrison and Weinstein for a system describing the
evolution of finitely many indistinguishable classical particles.
