Definition(Intertwining operator).If(π 1 ,V 1 ),(π 2 ,V 2 )are two representations
of a groupG, an intertwining operator between these two representations is an
operatorUsuch that
π 2 (g)U=Uπ 1 (g) ∀g∈G
In our caseV 1 =V 2 is the Schr ̈odinger representation state spaceHandUk:
H→His an intertwining operator between ΓSand ΓS,kfor eachk∈Sp(2d,R).
Since
ΓS,k 1 k 2 =Uk 1 k 2 ΓSUk− 11 k 2
one might expect that theUkshould satisfy the group homomorphism property
Uk 1 k 2 =Uk 1 Uk 2and give us a representation of the groupSp(2d,R) onH. This is what would
follow from the general principle that a group action on the classical phase space
after quantization becomes a unitary representation on the quantum state space.
The problem with this argument is that theUkare not uniquely defined.
Schur’s lemma tells us that since the representation onHis irreducible, the
operators commuting with the representation operators are just the complex
scalars. These give a phase ambiguity in the definition of the unitary operators
Uk, which then give a representation ofSp(2d,R) onHonly up to a phase, i.e.,
Uk 1 k 2 =Uk 1 Uk 2 eiφ(k^1 ,k^2 )for some real-valued functionφof pairs of group elements. In terms of corre-
sponding Lie algebra representation operatorsUL′, this ambiguity appears as an
unknown constant times the identity operator.
The question then arises whether the phases of theUkcan be chosen so
as to satisfy the homomorphism property (i.e., can phases be chosen so that
φ(k 1 ,k 2 ) =N 2 πforNintegral?). It turns out that this cannot quite be done,
sinceNmay have to be half-integral, giving the homomorphism property only
up to a sign. Just as in theSO(d) case where a similar sign ambiguity showed
the need to go to a double coverSpin(d) to get a true representation, here
one needs to go to a double cover ofSp(2d,R), called the metaplectic group
Mp(2d,R). The nature of this sign ambiguity and double cover is quite subtle,
and unlike for theSpin(d) case, we will not provide an actual construction of
Mp(2d,R). For more details on this, see [56] or [37]. In section 20.3.2 we will
show by computation one aspect of the double cover.
Since this is just a sign ambiguity, it does not appear infinitesimally: the
ambiguous constants in the Lie algebra representation operators can be chosen
so that the Lie algebra homomorphism property is satisfied. However, this will
no longer necessarily be true for infinite dimensional phase spaces, a situation
that is described as an “anomaly” in the symmetry. This phenomenon will be
examined in more detail in chapter 39.