
The Weyl Algebra, Quantum Theory, and Normal Ordering 169
This follows also from the operator relation [756, 904, 905]
:ˆn
k
:= ˆn(ˆn − 1) ···(ˆn − k +1)=(ˆn)
k
, (5.49)
which we recognize in the representation ˆn → XD = x
d
dx
as Boole’s relation (1.13). Recall
from (1.37) that Katriel showed
ˆn
m
=
m
k=1
S(m, k)(ˆa
†
)
k
ˆa
k
=
m
k=1
S(m, k):ˆn
k
:, (5.50)
implying due to n|ˆn
m
|n = n
m
and (5.48) that n
m
=
k
S(m, k)(n)
k
, which is precisely
(1.3). See [731] for an early physical application of (5.50). The antinormal ordering analog
of (5.49) is given by
.
.
.ˆn
k
.
.
.=(ˆn +1)···(ˆn + k)=ˆn +1
k
,
implying as analog of (5.48) the relation n|
.
.
.ˆn
k
.
.
.|n = n +1
k
. In [453],