
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
1−thV
.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 ...