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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
356 Commutation Relations, Normal Ordering, and Stirling Numbers
We now generalize this. Recalling from Proposition 9.3 that a representation of variables
(U, V ) satisfying UV = qV U + hV
s
is given by (E
s;h|q
,X)whereE
s;h|q
= hE
s;1|q
= hX
s
D
q
,
we can write (9.17) equivalently as
(X + hE
s;1|q
)
n
=
n
j=0
n
j
q
H
(s)
n;j
(X, h; q)h
j
E
j
s;1|q
, (9.32)
wherewedefined
H
(s)
n;j
(X, h; q)=
n
j
1
q
X
nj
nj1
i=0
d
(s)
n
(j, i; q)h
i
X
i(s2)
. (9.33)
To be more concrete, we record here the first few cases which may be derived explicitly
using the commutation relation E
s;1|q
X
k
= q
k
X
k
E
s;1|q
+[k]
q
X
k1+s
.Intheq-deformed
case, these expressions quickly become rather messy:
(X + hE
s;1|q
)
1
=X + h
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Publisher Resources

ISBN: 9781466579897