102
G. Vinod
where the operators ˜
N satisfies the following commutation relations:
˜
N , ˜
a
= −˜ a;
˜
N , ˜
a
†
= ˜
a
†
(65)
The commutation relations (63) and (64) are equivalent and they can be written
alternatively as [18]
˜
a ˜
a
†
− q ˜
a
†
˜
a = q
− ˜
N
(66)
From (65),
˜
aq
±r N
= q
±r q
r N
˜
a
˜
a
† q
±r N
= q
∓r q
r N
˜
a
†
(67)
where r is a rational number. If we make the substitutions
˜
a = q
−
1
4
˜
N a; ˜
a
†
= a
† q
−
1
4
˜
N
(68)
and use Eqs. (65), (63) can be written as
aa
†
− qa
† a = 1
(69)
whose multimode generalisation is the quon algebra proposed by Greenberg [19].
a i a
†
j − qa
†
j a i = δ i j
(70)
We can construct the representation of (70) in the Fock space spanned by the orthonormalised eigenstates |n of N .
|n =
a
†
n
[n] q !
; N |n = n |n ; a |0 = 0
(71)
In view of (32), (70) can be had with the following relations:
aa
†
= [N + 1] q ; a
† a = [N ] q
(72)
The q-commutator (70) can be written as [20]
a, a
†
= f (N )
(73)
with
f (N ) = q
N
f or q = 0
f (N ) = θ (1 − N ) f or q = 0
( 7 4 )