Page 239 - Principles and Applications of NanoMEMS Physics

B. BOSONIZATION 229

−
where x ∈ [ L , 2 / L ] 2 / , but may be allowed to go to infinity (L ĺ’), at
the conclusion of the procedure, if necessary. The fields ψ η () x and the
variable x are, in general, mathematical constructs which result from the
development undertaken to formulate the model in terms of the operators
c . In particular, for discrete k, ψ () x obeys the following properties:
k η η
iπδ
ψ η (x + ) 2 / L = e ψ η (x − ) 2 / L , (B.6)
b
where δ b = 0 for the periodicity condition and δ b = 1 for anti-periodicity.
3) The fermionic number representation (Fock) space is reorganized so
that the Fock space of states spanned by the operators c is rendered as a
k η
direct sum, F = ¦ H G over the Hilbert spaces H G characterized by a
G N N
⊕N
G
fixed particle number N , within each of which all excitations are bosonic,
i.e., particle-hole-like. The first step towards accomplishing this is to define
the vacuum state 0 by,

c 0 ≡ 0 for >k , 0 (n > ) 0 , (B.7)
k η k

+
k
c 0 ≡ 0 for ≤ , 0 (n ≤ ) 0 . (B.8)
k η k
(B.7) signifies that states above k=0 are empty, therefore, none may be
destroyed, and states below k=0 are all occupied, therefore, none may be
populated. The occupation of all other states in Fock space are defined
relative to the vacuum, particularly the operation of fermion normal ordering
with respect to it. A function is said to be in fermion-normal-order form
+
when all c with k>0, and all c with ≤k 0 are positioned to the right of
k η k η
+
all other operators c with k>0 and c with k ≤ 0 . Thus, for operators
η
η
k
k
A, B, C,... ∈ { c ; c + }, this is represented by,
k η k η
+
+ ... = ...−
+ ABC + ABC 0 ABC ... 0 . (B.9)
ˆ
4) The number operator N η possesses eigenvalues
G G
N = (N , N ,..., N ) Z∈ M , whose aggregate makes up the N -particle
1 2 M
ˆ
Hilbert space H G . In particular, N counts the number of electrons of
N η
species η , is given by,