representation Γ′BFcomes from a Lie group representation ΓBFofSL(2,R), we
have
ΓBF(eθZ) =eθΓ
′BF(Z)where
Γ′BF(Z) =−i(
a†a+1
2
)
=−i(
N+
1
2
)
so
ΓBF(eθZ)|n〉=e−iθ(n+
(^12) )
|n〉
Takingθ= 2π, this gives an inconsistency
ΓBF( 1 )|n〉=−|n〉
This is the same phenomenon first described in the context of the Schr ̈odinger
representation in section 17.1.
As remarked there, it is the same sort of problem we found when studying the
spinor representation of the Lie algebraso(3). Just as in that case, the problem
indicates that we need to consider not the groupSL(2,R), but a double cover,
the metaplectic groupMp(2,R). The behavior here is quite a bit more subtle
than in theSpin(3) double cover case, whereSpin(3) was the groupSU(2),
and topologically the only non-trivial cover ofSO(3) was theSpin(3) one since
π 1 (SO(3)) =Z 2. Hereπ 1 (SL(2,R)) =Z, and each extra time one goes around
theU(1) subgroup we are looking at, one gets a topologically different non-
contractible loop in the group. As a result,SL(2,R) has lots of non-trivial
covering groups, of which only one interests us, the double coverMp(2,R). In
particular, there is an infinite-sheeted universal coverSL ̃(2,R), but that plays
no role here.
Digression.This groupMp(2,R)is quite unusual in that it is a finite dimen-
sional Lie group, but does not have any sort of description as a group of finite
dimensional matrices. This is due to the fact that all its finite dimensional
irreducible representations are the same as those ofSL(2,R), which has the
same Lie algebra (these are representations on homogeneous polynomials in two
variables, those first studied in chapter 8, which areSL(2,C)representations
which can be restricted toSL(2,R)). These finite dimensional representations
factor throughSL(2,R)so their matrices don’t distinguish between two different
elements ofMp(2,R)that correspond asSL(2,R)elements.
There are no faithful finite dimensional representations ofMp(2,R)itself
which could be used to identifyMp(2,R)with a group of matrices. The only
faithful irreducible representation available is the infinite dimensional one we are
studying. Note that the lack of a matrix description means that this is a case
where the definition we gave of a Lie algebra in terms of the matrix exponential
does not apply. The more general geometric definition of the Lie algebra of
a group in terms of the tangent space at the identity of the group does apply,
although to do this one really needs a construction of the double coverMp(2,R),
which is quite non-trivial and not done here. This is not a problem for purely