
Normal Ordering in the Weyl Algebra – Further Aspects 213
the operators act (to the right). For example, one has
1
A
2
B = BA, (
1
A +
2
B)
2
=(
1
A)
2
+2
1
A
2
B +(
2
B)
2
= A
2
+2BA + B
2
.
Note that only the relative size of the indices is important, and instead of the pair (1, 2) one
could also use (4, 13); there exist several rules for dealing with these indices [779, 862,863].
For a polynomial (or also formal series) one indicates in the same fashion the order of the
operator arguments. For example, if p(x, ξ)=
n
k=0
a
k
(x)ξ
k
denotes a polynomial in x and
ξ,then
P
⎛
⎝
2
x,
1
d
dx
⎞
⎠
=
n
k=0
a
k
(x)
d
k
dx
k
denotes the associated differential operator in standard form (and p(x, ξ) denotes