Chapter 15

Hamiltonian Vector Fields

and the Moment Map

A basic feature of Hamiltonian mechanics is that, for any functionfon phase
spaceM, there are parametrized curves in phase space that solve Hamilton’s
equations

q ̇j=

∂f

∂pj

p ̇j=−

∂f

∂qj

and the tangent vectors of these parametrized curves provide a vector field on
phase space. Such vector fields are called Hamiltonian vector fields. There is a
distinguished choice off, the Hamiltonian functionh, which gives the velocity
vector fields for time evolution trajectories in phase space.
More generally, when a Lie groupGacts on phase spaceM, the infinitesimal
action of the group associates to each elementL∈ga vector fieldXLon phase
space. When these are Hamiltonian vector fields, there is (up to a constant) a
corresponding functionμL. The map fromL∈gto the functionμLonMis
called the moment map, and such functions play a central role in both classical
and quantum mechanics. For the case of the action ofG=R^3 onM=R^6 by
spatial translations, the components of the momentum arise in this way, for the
action ofG=SO(3) by rotations, the angular momentum.
Conventional physics discussions of symmetry in quantum mechanics focus
on group actions on configuration space that preserve the Lagrangian, using
Noether’s theorem to provide corresponding conserved quantities (see chapter
35). In the Hamiltonian formalism described here, these same conserved quan-
tities appear as moment map functions. The operator quantizations of these
functions provide quantum observables and (modulo the problem of indetermi-
nacy up to a constant) a unitary representation ofGon the state spaceH. The
use of moment map functions rather than Lagrangian-derived conserved quanti-
ties allows one to work with cases whereGacts not on configuration space, but
on phase space, mixing position and momentum coordinates. It also applies to
cases where the group action is not a “symmetry” (i.e., does not commute with