To construct the angular momentum operators in the Bargmann-Fock rep-
resentation, recall that in the Schr ̈odinger representation these were
L 1 =Q 2 P 3 −Q 3 P 2 , L 2 =Q 3 P 1 −Q 1 P 3 , L 3 =Q 1 P 2 −Q 2 P 1
and these operators can be rewritten in terms of annihilation and creation op-
erators. Alternatively, theorem 25.2 can be used, for Lie algebra basis elements
lj∈so(3)⊂u(3)⊂gl(3,C) which are (see chapter 6)
l 1 =
0 0 0
0 0 − 1
0 1 0
, l 2 =
0 0 1
0 0 0
−1 0 0
, l 3 =
0 −1 0
1 0 0
0 0 0
to calculate
−iLj=Ul′j=
∑^3
m,n=1
a†m(lj)mnan
This gives
Ul′ 1 =a† 3 a 2 −a† 2 a 3 , Ul′ 2 =a† 1 a 3 −a† 3 a 1 , Ul′ 3 =a† 2 a 1 −a† 1 a 2
Exponentiating these operators gives a representation of the rotation group
SO(3) on the state spaceF 3 , commuting with the Hamiltonian, so acting on
energy eigenspaces (which will be the homogeneous polynomials of fixed degree).
25.5 Normal ordering and the anomaly in finite dimensions
ForA∈u(d)⊂sp(2d,R) we have seen that we can construct the Lie algebra
version of the metaplectic representation as
U ̃A′ =^1
2
∑
j,k
Ajk(a†jak+aka†j)
which gives a representation that extends tosp(2d,R), or we can normal order,
getting
UA′ = :U ̃A′: =
∑
j,k
a†jAjkak=U ̃A′−
1
2
∑d
j=1
Ajj 1
To see that this normal ordered version does not extend tosp(2d,R), observe
that basis elements ofsp(2d,R) that are not inu(d) are the linear combinations
ofzjzkandzjzkthat correspond to real-valued functions. These are given by
1
2
∑
jk
(Bjkzjzk+Bjkzjzk)