
The Weyl Algebra, Quantum Theory, and Normal Ordering 179
Example 5.43 Let us consider ˆq = zˆa
†
+¯zˆa, z ∈ C. The Baker–Campbell–Hausdorff
formula (see Appendix E) implies that e
itˆq
= e
itzˆa
†
e
−
1
2
t
2
|z|
2
e
it¯zˆa
. Thus, the vacuum expec-
tation value is given by 0|e
itˆq
|0 = e
−
1
2
t
2
|z|
2
. A nice discussion and pictorial representa-
tion was given in [495]; see also [494]. The odd moments vanish and the even moments
are given by 0|ˆq
2n
|0 =
(2n)!
2
n
n!
|z|
2
. The combinatorial factor counts the number of ways
to partition the 2n (time-ordered) vertices into n (contraction) pairs. In a similar fashion,
ˆ
W =(ˆa+z)
†
(ˆa+z)=ˆa
†
ˆa+zˆa
†
+¯zˆa+|z|
2
was considered in