Quantum Mechanics for Mathematicians

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)