the quadratic functionμLthat satisfies
μL,
q 1
q 2
p 1
p 2
=LT
q 1
q 2
p 1
p 2
=
0 1 0 0
−1 0 0 0
0 0 0 1
0 0 −1 0
q 1
q 2
p 1
p 2
=
q 2
−q 1
p 2
−p 1
is
μL=−q·(
0 − 1
1 0
)
p=q 1 p 2 −q 2 p 1which is just the formula for the angular momentum
l=q 1 p 2 −q 2 p 1ind= 2.
Quantization gives a representation of the Lie algebraso(2) with
UL′=−i(Q 1 P 2 −Q 2 P 1 )satisfying [
UL′,(
Q 1
Q 2
)]
=
(
Q 2
−Q 1
)
,
[
UL′,
(
P 1
P 2
)]
=
(
P 2
−P 1
)
Exponentiating gives a representation ofSO(2)
UeθL=e−iθ(Q^1 P^2 −Q^2 P^1 )with conjugation byUeθL rotating linear combinations of theQ 1 ,Q 2 (or the
P 1 ,P 2 ) each by an angleθ.
UeθL(cq 1 Q 1 +cq 2 Q 2 )Ue−θL^1 =c′q 1 Q 1 +c′q 2 Q 2where (
c′q 1
c′q 2
)
=
(
cosθ −sinθ
sinθ cosθ)(
cq 1
cq 2)
These representation operators are exactly the ones found in (section 19.1)
the discussion of the representation ofE(2) corresponding to the quantum free
particle in two dimensions. There we saw that on position space wavefunctions
this is just the representation induced from rotations of the position space. It
also comes from the Schr ̈odinger representation, by taking a specific quadratic
combination of theQj,Pj operators, the one corresponding to the quadratic
functionl. Note that there is no ordering ambiguity in this case since one does
not multiplyqjandpjwith the same value ofj. Also note that for thisSO(2)
the double cover is trivial: as one goes around the circle inSO(2) once, the
operatorUeθL is well-defined and returns to its initial value. As far as this
subgroup ofSp(4,R) is concerned, there is no need to consider the double cover
Mp(4,R) to get a well-defined representation.