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Commutation Relations, Normal Ordering, and Stirling Numbers
book

Commutation Relations, Normal Ordering, and Stirling Numbers

by Toufik Mansour, Matthias Schork
September 2015
Intermediate to advanced content levelIntermediate to advanced
528 pages
19h 34m
English
Chapman and Hall/CRC
Content preview from Commutation Relations, Normal Ordering, and Stirling Numbers
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)
a+z)=ˆa
ˆa+zˆa
zˆa+|z|
2
was considered in
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Publisher Resources

ISBN: 9781466579897