Quantum Mechanics for Mathematicians

are known to physicists as “canonical transformations”, and to mathematicians
as “symplectomorphisms”. We will not try and work out in any more detail
how the exponential map behaves in general. In chapter 16 we will see what
happens forfan order-two homogeneous polynomial in theqj,pj. In that case
the vector fieldXf will take linear functions onM to linear functions, thus
acting onM, in which case its behavior can be studied using the matrix for the
linear transformation with respect to the basis elementsqj,pj.

Digression.The exponential mapexp(tX)can be defined as above on a general
manifold. For a symplectic manifoldM, Hamiltonian vector fieldsXfwill have
the property that they preserve the symplectic form, in the sense that

exp(tXf)∗ω=ω

This is because

LXfω= (diXf+iXfd)ω=diXfω=dω(Xf,·) =ddf= 0 (15.9)

whereLXf is the Lie derivative alongXf.

15.3 Group actions onMand the moment map

Our fundamental interest is in studying the implications of Lie group actions
on physical systems. In classical Hamiltonian mechanics, with a Lie groupG
acting on phase spaceM, such actions are characterized by their derivative,
which takes elements of the Lie algebra to vector fields onM. When these
are Hamiltonian vector fields, equation 15.6 can often be used to instead take
elements of the Lie algebra to functions onM. This is known as the moment
map of the group action, and such functions onM will provide our central
tool to understand the implications of a Lie group action on a physical system.
Quantization then takes such functions to operators which will turn out to be
the important observables of the quantum theory.
Given an action of a Lie groupGon a spaceM, there is a map

L∈g→XL

fromgto vector fields onM. This takesLto the vector fieldXLwhich acts on
functions onMby

XLF(m) =

d

dt

F(etL·m)|t=0 (15.10)

This map however is not a homomorphism (for the Lie bracket 15.1 on vector
fields), but an antihomomorphism. To see why this is, recall that when a group
Gacts on a space, we get a representationπon functionsFon the space by

π(g)F(m) =F(g−^1 ·m)

The derivative of this representation will be the Lie algebra representation

π′(L)F(m) =

d

dt

F(e−tL·m)|t=0=−XLF(m)