16.1 The symplectic group
Recall that the orthogonal group can be defined as the group of linear transfor-
mations preserving an inner product, which is a symmetric bilinear form. We
now want to study the analog of the orthogonal group that comes from replac-
ing the inner product by the antisymmetric bilinear form Ω that determines the
symplectic geometry of phase space. We will define:
Definition(Symplectic group).The symplectic groupSp(2d,R)is the subgroup
of linear transformationsgofM=R^2 dthat satisfy
Ω(gv 1 ,gv 2 ) = Ω(v 1 ,v 2 )forv 1 ,v 2 ∈M
While this definition uses the dual phase spaceMand Ω, it would have been
equivalent to have made the definition usingMandω, since these transforma-
tions preserve the isomorphism betweenMandMgiven by Ω (see equation
14.6). For an action onM
u∈M→gu∈Mthe action on elements ofM(such elements correspond to linear functions Ω(u,·)
onM) is given by
Ω(u,·)∈M→g·Ω(u,·) = Ω(u,g−^1 (·)) = Ω(gu,·)∈M (16.1)Here the first equality uses the definition of the dual representation (see 4.2)
to get a representation on linear functions onMgiven a representation onM,
and the second uses the invariance of Ω.
16.1.1 The symplectic group ford= 1
In order to study symplectic groups as groups of matrices, we’ll begin with the
cased= 1 and the groupSp(2,R). We can write Ω as
Ω
((
cq
cp)
,
(
c′q
c′p))
=cqc′p−cpc′q=(
cq cp)
(
0 1
−1 0
)(
c′q
c′p)
(16.2)
A linear transformationgofMwill be given by
(
cq
cp)
→
(
α β
γ δ)(
cq
cp)
(16.3)
The condition for Ω to be invariant under such a transformation is
(
α β
γ δ)T(
0 1
−1 0
)(
α β
γ δ