  The dynamics of quantum expectation values is considered in a geometric
setting. First, expectation values of the canonical operators are shown to be
equivariant momentum maps for the action of the Heisenberg group on quantum
states. Then, the Hamiltonian structure of Ehrenfest's theorem is shown to be
Lie-Poisson for a semidirect-product Lie group, named the `Ehrenfest group'.
The underlying Poisson structure produces classical and quantum mechanics as
special limit cases. In addition, quantum dynamics is expressed in the frame of
the expectation values, in which the latter undergo canonical Hamiltonian
motion. In the case of Gaussian states, expectation values dynamics couples to
second-order moments, which also enjoy a momentum map structure. Eventually,
Gaussian states are shown to possess a Lie-Poisson structure associated to
another semidirect-product group, which is called the Jacobi group. This
structure produces the energy-conserving variant of a class of Gaussian moment
models previously appeared in the chemical physics literature.
