Page 243 - Principles and Applications of NanoMEMS Physics

B. BOSONIZATION 233

φ
where ϕ η () x and () x are constructed to exhibit periodicity L in x, and
η
a > 0 is an artifact to regularize the divergent momentum sums ( →q ∞ ),
with its reciprocal a1 interpreted as the maximum momentum difference
+
for the c k± q c -combinations occurring in the fermionic functions φ . The
k
bosonic fields obey the commutation relations,
[ ϕ η (),x ϕ ' η ( )] [ ϕ η + (),x ϕ + ' η ( )] 0' =x , (B.29)
' =
x

[ ϕ η (),ϕ η + ' ( )] = δ ηη ' ¦ 1 e − q i ( [ x − x ' )+ a ]
x '
x
q > n q , (B.30)
0
ª − i 2 π ( − xx ' − ia ) º
= δ ηη ' ln « − e1 L »
¬ ¼
∞
→
⎯ L ⎯→ δ ηη ' ln ª 2π ( − xx ' − ia ) ∞ º »
⎯
i
«
L
¬
¼
where use was made of the identity ( −1ln y ) −= ¦ y n / n . In terms of these
n =1
bosonic fields the normal-ordered electron density becomes a function of
φ
∂ x η () x , as follows,
+
+
ρ η ()≡ ψ η () () / πψ η x + + 2 = 1 ¦ e −iqx ¦ + + c k + −q ,η c kη +
x
x
+
L q k +
+
= 1 ¦ i n q (e −iqx b qη − e iqx b qη )+ ¦ + + c k + −q ,η c kη + . (B.31)
L q > 0 k +
∂ φ () x 1
= x η + N ˆ for ( →a ) 0
2π L η
B.4 Bosonization Identity and Its Application to Hamiltonian
with Linear Dispersion
The ultimate purpose of the preliminaries presented thus far, has been to
enable familiarization with the mathematical language and techniques
required to effect the transformation of Hamiltonians expressed in terms of
fermionic field operators, into Hamiltonians expressed in terms of the
bosonic field operators. This transformation is enabled by the bosonization
identity,