
218 Commutation Relations, Normal Ordering, and Stirling Numbers
6.6.1 The Bosonic Case
In the general case where r modes are present, the creation resp. annihilation operators
ˆa
†
k
resp. ˆa
k
(with k =1,...,r) satisfy the commutation relations (5.33). To simplify the
notation, we consider only two modes and denote the operators by {ˆa, ˆa
†
,
ˆ
b,
ˆ
b
†
}.Allof
the operators {ˆa, ˆa
†
} commute with all of the operators {
ˆ
b,
ˆ
b
†
}, and each of these two
sets represents the creation resp. annihilation operator of a harmonic oscillator. From a
physical point of view, the two modes do not interact (that is, they are not coupled). Let
us also introduce the number operators ...