- Space translation (q→q+a). On states one has
|ψ〉→e−iaP|ψ〉
which in the Schr ̈odinger representation is
e−ia(−i
dqd)
ψ(q) =e−a
dqd
ψ(q) =ψ(q−a)
So, the Lie algebra action is given by the operator−iP=−dqd. Note that
this has opposite sign to the time translation. On operators one has
O(a) =eiaPOe−iaP
or infinitesimally
d
da
O(a) = [O,−iP]
- The classical expressions for angular momentum quadratic inqj,pj, for
example
l 1 =q 2 p 3 −q 3 p 2
under quantization go to the self-adjoint operator
L 1 =Q 2 P 3 −Q 3 P 2
and−iL 1 will be the skew-adjoint operator giving a unitary representation
of the Lie algebraso(3). The three such operators will satisfy the Lie
bracket relations ofso(3), for instance
[−iL 1 ,−iL 2 ] =−iL 3
A.4 Complex structures and Bargmann-Fock quan-
tization
We define complex coordinates on phase space by
zj=
1
√
2
(qj−ipj), zj=
1
√
2
(qj+ipj)
The standard choice of complex structure on phase spaceMis given by
J 0
∂
∂qj
=−
∂
∂pj
, J 0
∂
∂pj
=
∂
∂qj
and on coordinate basis vectorsqj,pjof the dual spaceMby
J 0 qj=pj, J 0 pj=−qj
The complex coordinates satisfy
J 0 zj=izj, J 0 zj=−izj