Another important special case comes from takingA= 0,B= 1 ,C=− 1 in
equation 16.12, which by equation 16.13 gives
μL=1
2
(|q|^2 +|p|^2 )This generalizes the case ofd= 1 described earlier, and will be the Hamiltonian
function for addimensional harmonic oscillator. Note that exponentiatingL
gives a symplectic action on phase space that mixes position and momentum
coordinates, so this an example that cannot be understood just in terms of a
group action on configuration space.
16.2 The symplectic group and automorphisms of the Heisenberg group
Returning to thed= 1 case, we have found two three dimensional Lie alge-
bras (h 3 andsl(2,R)) as subalgebras of the infinite dimensional Lie algebra of
functions on phase space:
- h 3 , the Lie algebra of linear polynomials onM, with basis 1,q,p.
- sl(2,R), the Lie algebra of order two homogeneous polynomials onM,
with basisq^2 ,p^2 ,qp.
Taking all quadratic polynomials, we get a six dimensional Lie algebra with
basis elements 1,q,p,qp,q^2 ,p^2. This is not the direct product ofh 3 andsl(2,R)
since there are nonzero Poisson brackets
{qp,q}=−q, {qp,p}=p{p^2
2,q}=−p, {q^2
2,p}=q(16.15)
These relations show that operating on a basis of linear functions onMby
taking the Poisson bracket with something insl(2,R) (a quadratic function)
provides a linear transformation onM∗=M.
In this section we will see that this is the infinitesimal version of the fact that
SL(2,R) acts on the Heisenberg groupH 3 by automorphisms. We’ll begin with
a general discussion of what happens when a Lie groupGacts by automorphisms
on a Lie groupH, then turn to two examples: the conjugation action ofGon
itself and the action ofSL(2,R) onH 3.
An action of one group on another by automorphisms means the following:
Definition(Group automorphisms).If an action of elementsgof a groupG
on a groupH
h∈H→Φg(h)∈H
satisfies
Φg(h 1 )Φg(h 2 ) = Φg(h 1 h 2 )