- Exponentiating the Lie algebra elementE−Fgives rotations of the plane
eθ(E−F)=(
cosθ sinθ
−sinθ cosθ)
Note that the Lie algebra element being exponentiated here isE−F↔
1
2
(p^2 +q^2 )the function studied in section 15.2, which we will later re-encounter as
the Hamiltonian function for the harmonic oscillator in chapter 22.The groupSL(2,R) is non-compact and its representation theory is quite
unlike the case ofSU(2). In particular, all of its non-trivial irreducible unitary
representations are infinite dimensional, forming an important topic in mathe-
matics, but one that is beyond our scope. We will be studying just one such
irreducible representation (the one provided by the quantum mechanical state
space), and it is a representation only of a double cover ofSL(2,R), not of
SL(2,R) itself.
16.1.2 The symplectic group for arbitraryd.
For generald, the symplectic groupSp(2d,R) is the group of linear transfor-
mationsgofMthat leave Ω (see 14.4) invariant, i.e., satisfy
Ω
(
g(
cq
cp)
,g(
c′q
c′p))
= Ω
((
cq
cp)
,
(
c′q
c′p))
wherecq,cpareddimensional vectors. By essentially the same calculation as in
thed= 1 case, we find theddimensional generalization of equation 16.4. This
says thatSp(2d,R) is the group of real 2dby 2dmatricesgsatisfying
gT(
0 1
−1 0
)
g=(
0 1
−1 0
)
(16.10)
where 0 is thedbydzero matrix, 1 thedbydunit matrix.
Again by a similar argument to thed= 1 case where the Lie algebrasp(2,R)
was determined by the condition 16.5,sp(2d,R) is the Lie algebra of 2dby 2d
matricesLsatisfying
LT
(
0 1
−1 0
)
+
(
0 1
−1 0
)
L= 0 (16.11)
Such matrices will be those with the block-diagonal form