Quantum Mechanics for Mathematicians

so we see that it is the map

L→π′(L) =−XL

that will be a homomorphism.
When the vector fieldXLis a Hamiltonian vector field, we can define:

Definition(Moment map). Given an action ofGon phase spaceM, a Lie
algebra homomorphism
L→μL

fromgto functions onMis said to be a moment map if

XL=XμL

Equivalently, for functionsF onM,μLsatisfies

{μL,F}(m) =−XLF=

d

dt

F(e−tL·m)|t=0 (15.11)

This is sometimes called a “co-moment map”, with the term “moment map”
referring to a repackaged form of the same information, the map

μ:M→g∗

where
(μ(m))(L) =μL(m)

A conventional physical terminology for the statement 15.11 is that “the function
μLgenerates the symmetryL”, giving its infinitesimal action on functions.
Only for certain actions ofGonMwill theXLbe Hamiltonian vector fields
and an identityXL=XμLpossible. A necessary condition is thatXLsatisfy
equation 15.7
XL{g 1 ,g 2 }={XLg 1 ,g 2 }+{g 1 ,XLg 2 }

Even when a functionμLexists such thatXμL=XL, it is only unique up to a
constant, sinceμLandμL+Cwill give the same vector field. To get a moment
map, we need to be able to choose these constants in such a way that the map

L→μL

is a Lie algebra homomorphism fromgto the Lie algebra of functions onM.
When this is possible, theG-action is said to be a HamiltonianG-action. When
such a choice of constants is not possible, theG-action on the classical phase
space is said to have an “anomaly”.

Digression.For the case ofMa general symplectic manifold, the moment map
can still be defined, whenever one has a Lie groupGacting onM, preserving
the symplectic formω. The infinitesimal condition for such aGaction is (see
equation 15.9)
LXω= 0