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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 297
Example 8.55 (Shift algebra) If f(V )=hV , one considers the extended shift algebra
ˆ
A
1;h
where UV VU = hV .From(8.39) one obtains that
ˆ
V (t)=Ve
ht
.Since
λ
0
Ve
ht
dt =
e
λh
1
h
V , (8.38) becomes e
λ(U+V )
= e
e
λh
1
h
V
e
λU
, that is, the Kirzhnits–Sack formula (5.74).
Example 8.56 (Meromorphic Weyl algebra) If f(V )=hV
2
, one considers the ex-
tended meromorphic Weyl algebra
ˆ
A
2;h
where UV VU = hV
2
.From(8.39) one obtains
that
ˆ
V (t)=(V
1
th)
1
=
V
1thV
.Since
λ
0
(V
1
th)
1
dt =ln
'
(1 λhV )
1/h
(
, (8.38)
becomes e
λ(U+V )
=(1 λhV )
1/h
e
λU
, that is, Berry’s identity (7.78).
The above examples motivate the consideration of
ˆ
A
s;h
where ...
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Publisher Resources

ISBN: 9781466579897