Lie algebra calculations, since the Lie algebras ofMp(2,R)andSL(2,R)can
be identified.
Another aspect of the metaplectic representation that is relatively easy to
see in the Bargmann-Fock construction is that the state spaceF is not an
irreducible representation, but is the sum of two irreducible representations
F=Feven⊕FoddwhereFevenconsists of the even functions ofz,Foddof odd functions ofz. On
the subspaceFfin⊂ Fof finite sums of the number eigenstates, these are the
even and odd degree polynomials. Since the generators of the Lie algebra rep-
resentation are degree two combinations of annihilation and creation operators,
they will take even functions to even functions and odd to odd. The separate
irreducibility of these two pieces is due to the fact that (whennandmhave
the same parity), one can get from state|n〉to any another|m〉by repeated
application of the Lie algebra representation operators.
24.2 Intertwining operators in terms ofaanda†
Recall from the discussion in chapter 20 that the metaplectic representation of
Mp(2,R) can be understood in terms of intertwining operators that arise due to
the action of the groupSL(2,R) as automorphisms of the Heisenberg groupH 3.
Such intertwining operators can be constructed by exponentiating quadratic
operators that have the commutation relations with theQ,P operators that
reflect the intertwining relations (see equation 20.3). These quadratic operators
provide the Lie algebra version of the metaplectic representation, discussed in
section 24.1 using the Lie algebrasl(2,R), which is identical to the Lie algebra of
Mp(2,R). In sections 20.3.2 and 20.3.4 these representations were constructed
explicitly forSO(2) andRsubgroups ofSL(2,R) using quadratic combinations
of theQandPoperators. Here we’ll do the same thing using annihilation and
creation operators instead ofQandPoperators.
For theSO(2) subgroup of equation 24.1 (this is the same one discussed in
section 20.3.2), in terms ofzandzcoordinates the moment map will be
μZ=zzand one has
{μZ,(
z
z)
}=
(
i 0
0 −i)(
z
z)
(24.3)
Quantization by annihilation and creation operators gives (see 24.2)
Γ′BF(zz) = Γ′BF(Z) =−
i
2(aa†+a†a)and the quantized analog of 24.3 is
[
−
i
2
(aa†+a†a),(
a
a†)]
=
(
i 0
0 −i)(
a
a†