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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
A Generalization of the Weyl Algebra 287
Example 8.28 (Weyl algebra) Let δ(V )=h, that is, UV VU = h. Iterating δ(V
m
)=
mhV
m1
, one finds δ
r
(V
m
)=
m!
(mr)!
h
r
V
mr
. Inserting this into (8.14),oneobtains
U
n
V
m
=
n
k=0
n
k

m
n k
(n k)!h
nk
V
m(nk)
U
k
.
Switching to l = n k, one recovers (6.2).
Example 8.29 (Meromorphic Weyl algebra) Let δ(V )=V
2
, that is, UV VU =
V
2
. A simple induction shows that
δ
r
(V
m
)=(1)
r
(m + r 1)!
(m 1)!
V
m+r
, (8.15)
implying that (8.14) reduces for the case at hand to (7.71).
The following analog of Proposition 8.25 was given in [87, Equation (1.3)] and had
appeared earlier in [577, Equation (4)].
Proposition 8.30 (Irving; Benkart, ...
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Publisher Resources

ISBN: 9781466579897