20.2 Constructing intertwining operators
The method we will use to construct the intertwining operatorsUkis to find
a solution to the differentiated version of equation 20.1 and then getUk by
exponentiation. Differentiating 20.1 fork=etLatt= 0 gives
[UL′,Γ′S(X)] = Γ′S(L·X) (20.2)where
L·X=d
dtΦetL(X)|t=0and we have used equation 5.1 on the left-hand side.
In terms ofQjandPjoperators, which areitimes the Γ′S(X) forXa basis
vectorqj,pj, equation 20.2 is
[
UL′,(
Q
P
)]
=LT
(
Q
P
)
(20.3)
We can findUL′ by quantizing the moment map functionμL, which satisfies
{
μL,(
q
p)}
=LT
(
q
p)
(20.4)
Recall from 16.1.2 that theμLare quadratic polynomials in theqj,pj. We saw
in section 17.3 that the Schr ̈odinger representation Γ′Scould be extended from
the Heisenberg Lie algebra to the symplectic Lie algebra, by taking a product of
Qj,Pjoperators corresponding to the product inμL. The ambiguity in ordering
for non-commuting operators is resolved by quantizingqjpjusing
Γ′S(qjpj) =−
i
2(QjPj+PjQj)We thus take
UL′= Γ′S(μL)and this will satisfy 20.3 as desired. It will also satisfy the Lie algebra homo-
morphism property
[UL′ 1 ,UL′ 2 ] =U[′L 1 ,L 2 ] (20.5)
If one shiftsUL′ by a constant operator, it will still satisfy 20.3, but in general
will no longer satisfy 20.5. Exponentiating thisUL′ will give us ourUk, and thus
the intertwining operators that we want.
This method will be our fundamental way of producing observable operators.
They come from an action of a Lie group on phase space preserving the Poisson
bracket. For an elementLof the Lie algebra, we first use the moment map to
findμL, the classical observable, then quantize to get the quantum observable
UL′.