Page 241 - Principles and Applications of NanoMEMS Physics

B. BOSONIZATION 231

[b qη ,b + ' ' qη ] δ= ηη ' qq '
δ (B.15)
G
6) The bosonic vacuum states, the ground states given by N , are
0
defined such that,
G
b N = , 0 for all η , q (B.16)
q η
0
and admit a boson-normal-ordering protocol, in which all b are moved to
q η
+ +
the right of all b , so that, for operators A, B, C,... ∈ { b , b }, this is
q η k η k η
represented by,
G G
+
+ ... = ...−
+ ABC + ABC N 0 ABC ... N 0 . (B.17)
G G
7) Every state N in the N -particle Hilbert space, may be generated by
G
acting on the ground state N by a properly chosen bilinear combination
0
G G
of the fermion operators, N = f (c kη c kη ) N , or of boson operators,
+
G G 0
N = f ( ) Nb + .
0

8) There exist raising and lowering (ladder) operators whose action on a
G G
given state N of the N -particle Hilbert space changes the total number of
+
fermions by one. These operators are called Klein factors, denoted F and
η
F , respectively, and obey the following properties, namely,
η
a) They commute with all bosonic operators, i.e.,

[b ,F ] [b= + ,F + ] [b= ,F ] [b= + ,F ] 0= , for all q ,η ' ,η (B.18)
q η ' η q η ' η q η q ' 'η q η ' η
G G
b) Their action on a state N of the N -particle Hilbert space may be

+
b
f
expressed as the product of a particle-hole excitations ( ) acting on the
G G
corresponding N -particle ground state N , in particular,
0
G
+ + + + ˆ
F N ≡ f ()cb N ,...N ,...N ≡ f ()Tb N ,...N + 1 ,...N (B.19)
Nη
η
η
1 +
M
1
0 η 1 η M 0