Page 242 - Principles and Applications of NanoMEMS Physics

232 Appendix B

G
F N ≡ f ()cb + N ,...N ,...N ≡ f ()Tb + ˆ N ,...N − 1 ,...N (B.20)
η
η
Nη
M
1
0 η 1 η M 0
ˆ
where, T is the so-called the phase-counting operator,
η
η −1
−1
T ˆ η ≡ ( ) =1η ¦ ˆ N η (B.21)
which keeps track of the number of signs picked up when acting with a
G G
fermion operator c on a state N to produce a different state N ' ,
k η
0 0
i.e.,
1 N
N η
C
C
C
c kη () ( ) .......CC 1 1 N η N η ( ) N M 0 = T ˆ η () ( ) N η− 1 c kη ( ) ( ) N M 0 . (B.22)
...C
...C
η
η−
M
1
1
M
c) The Klein factors obey commutation relations,
+
{F η , F ' η } 2δ= ηη ' for all η ,η ' , (B.23)
+ +
F η F ' η = F η F η = 1, (B.24)
=
+ +
{F η , F η } {F η , F η } 0= , for η ≠ ' η , (B.25)
+
ˆ
ˆ
[N , F η' + ] δ= ηη' F , [N , F η' ] −= δ ηη' F . (B.26)
η
η
η
η
B.3 Bosonic Field Operators
φ
In analogy with fermion field operators, boson fields operators, () x ,
η
are defined in terms of bosonic operators as follows:
ϕ () −≡x ¦ 1 e −iqx b qη e −aq 2 / , and () −≡xϕ + ¦ 1 e iqx b + qη e − / 2 , (B.27)
aq
η
η
q >0 n q q >0 n q
with
φ η () ≡x ϕ η ()+x ϕ η () −=x ¦ 1 (e −iqx b qη + e iqx b qη )e −aq 2 / , (B.28)
+
+
q >0 n q