representation of a larger Lie algebra, that of all quadratic polynomials on phase
space, a representation that we will continue to denote by Γ′Sand refer to as the
Schr ̈odinger representation. On a basis of homogeneous order two polynomials
we have
Γ′S(
q^2
2)
=−iQ^2
2
=−
i
2
q^2Γ′S
(
p^2
2)
=−iP^2
2
=
i
2d^2
dq^2Γ′S(pq) =−i
2(PQ+QP)
Restricting Γ′Sto linear combinations of these homogeneous order two polyno-
mials (which give the Lie algebrasl(2,R), see theorem 16.1) we get a Lie algebra
representation ofsl(2,R) called the metaplectic representation.
Restricted to the Heisenberg Lie algebra, the Schr ̈odinger representation Γ′S
exponentiates to give a representation ΓSof the corresponding Heisenberg Lie
group (recall section 13.3). As ansl(2,R) representation however, it turns out
that Γ′Shas the same sort of problem as the spinor representation ofsu(2) =
so(3), which was not a representation ofSO(3), but only of its double cover
SU(2) =Spin(3). To get a group representation, one must go to a double cover
of the groupSL(2,R), which will be called the metaplectic group and denoted
Mp(2,R).
For an indication of the problem, consider the element
1
2(q^2 +p^2 )↔E−F=(
0 1
−1 0
)
insl(2,R). Exponentiating this gives a subgroupSO(2)⊂SL(2,R) of clockwise
rotations in theqpplane. The Lie algebra representation operator is
Γ′S
(
1
2
(q^2 +p^2 ))
=−
i
2(Q^2 +P^2 ) =−
i
2(
q^2 −
d^2
dq^2)
which is a second-order differential operator in both the position space and mo-
mentum space representations. As a result, it is not obvious how to exponentiate
this operator.
One can however see what happens on the state
ψ 0 (q) =e−q 22
⊂H=L^2 (R)where one has
−i
2(
q^2 −d^2
dq^2)
ψ 0 (q) =−i
2ψ 0 (q)soψ 0 (q) is an eigenvector of Γ′S(^12 (q^2 +p^2 )) with eigenvalue−i 2. Exponentiating
Γ′S(^12 (q^2 +p^2 )), the representation ΓSacts on this state by multiplication by a
phase. As one goes around the groupSO(2) once (rotating theqpplane by an