content_image/348432.jpg content_image/348431.jpg
H$_{reservoir}$ is the reservoir Hamiltonian which is a function of the voltage difference µ$_{R}$ ( x ) − µ$_{L}$ ( x ) e = V and the right and left electronic densities n$_{R,σ}$ ( x ) ≡ ρ$_{R,σ}$ ( x )+ < F | n$_{R,σ}$ ( x ) | F > , n$_{L,σ}$ ( x ) ≡ ρ$_{L,σ}$ ( x )+ < F | n$_{L,σ}$ ( x ) | F > . The Bosonic representation of the densities is: ρ$_{R,σ}$ ( x ) = 1 $_{2}$√ $_{π}$( ∂$_{x}$ϑ$_{σ}$ ( x ) − ∂$_{x}$ϕ$_{σ}$ ( x )) and ρ$_{L,σ}$ ( x ) = 1 $_{2}$√ $_{π}$( ∂$_{x}$ϑ$_{σ}$ ( x ) + ∂$_{x}$ϕ$_{σ}$ ( x )) [32, 33]. Due to the fact that the commutator for the electronic densities is space depen- dent, the average density is space dependent < F | n$_{R,σ}$ ( x ) + n$_{L,σ}$ ( x ) | F > = n$_{e}$ [1 − h(x,d) 2 ] where n$_{e}$ = 1 − δ a is the electronic density in the leads. The space dependent electrostatic potential caused by the space dependent charge density is given by δv ( x ) = µ$_{R}$ ( x )+ µ$_{L}$ ( x ) ( − e ) [34]. This potential will introduce current fluctuations. 

## 4. Dirac’s method for the exclusion of double occupancy-an application to the finite wire-leads system-A Bosonization formulation for the constraints 

We will extend the Quantum Mechanical results obtained in the previous section to the wire Hamiltonian H$_{wire}$ and we will introduce a new formulation for the constraints using the method of Bosonization. For the wire Hamiltonian H$_{wire}$ , we will enforce the exclusion of double occupancy. The operator for exclusion of double occupancy [8] leads to the constraint condition for the ground state | F > which replaces the non-interacting Fermi sea | F (0) > . The exclusion of double occupancy is enforced by the two electrons operator $^{Ψ}$↑( x $^{)Ψ}$↓( x ) | F > = 0 for | x | ≤ d 2 $^{. Except for half filling, Ψ}$↑( x $^{)Ψ}$↓( x ) is the only primary constraint [1]. In order to find all the secondary constraints [1], we have to commute the $^{constraint operator Ψ}$↑( x $^{)Ψ}$↓( x ) with the Hamiltonian. If this commutator is not equal to the constraint field, new constraints are generated. Using the Bosonic representation Ψ$_{σ}$( x ) $^{given in equation (5), we compute the representation of the pair operator Ψ}$↑( x $^{)Ψ}$↓( x ). The pair operator is represented in terms of the Bosonic fields, ϑ$_{e}$ ( x ) , ϕ$_{e}$ ( x ), ϑ$_{s}$ ( x ) and ϕ$_{s}$ ( x ): 

<img src='content_image/348432.jpg'>

where Q$_{1}$ ( x ) is the Bosonic constraint : 

<img src='content_image/348431.jpg'>

In order to understand the effect of the constraints, we will present a simplified description. For this purpose we will ignore the 2 K$_{F}$ oscillations, and we will approximate Q$_{1}$ ( x ) by Q$_{1}$ ( x ) ≈ cos $_{[}$√$_{2}$$_{πϑ}$$_{s}$$_{(}$$_{x}$$_{)].}$ From the equation cos $_{[}$√$_{2}$$_{πϑ}$$_{s}$$_{(}$$_{x}$$_{)}$ | F > ≈ 0, we learn that the 

