The result in equation (6) allows to identify the operator P$_{σ}$ ( x ) ≡ ∂$_{x}$ϕ$_{σ}$ ( x ) as the canonical conjugated momenta operator and define the commutator [ ϑ$_{σ}$ ( x ) , P$_{σ}$ ′ ( x ′ )] = iδ$_{σ,σ}$ ′ δ ( x − x ′ ). (Once the commutators have been defined we can choose the representation representation: ϑ$_{σ}$ ( x ) | $^{ϑ}$↑$^{, ϑ}$↓ > = ϑ$_{σ}$ ( x ) | $^{ϑ}$↑$^{, ϑ}$↓ > with the conjugated momenta given by P$_{σ}$ ( x ) = − i δ δϑ$_{σ}$ ( x $_{)}$.) We consider the strongly interacting case described by the Hubbard interaction U ( x ) = ∞ . ( For finite Hubbard U interactions, the Coulomb interactions [28] drive the the Hubbard U interaction to U → ∞ . The limit U ( x ) = ∞ describes effectively the physics in the strong coupling limit ) The Hubbard interaction U ( x ) = ∞ is restricted to the region | x | ≤ d $_{2}$. As a result the the electron occupation number is restricted to the charge values n$_{e}$ ( x ) = 0 , 1. Where | F > is the electronic ground state which is different from the non-interacting ground state | F (0) > . This condition is named the exclusion of double occupancy and is implemented with the help of the constraint equation. In the complimentary region | x | > d 2 the electrons are non-interacting, and we have U ( x ) = 0. Once the constraint conditions have been introduced we can use the non-interacting Hamiltonian for the wire-leads system. Next, we Bosonize this model using the same fields as we used for the noninteracting case. In spite of the fact that the Hamiltonian for the non-interacting case is identical to the strongly interacting one, the two ground states are different. The new ground state | F > obeys the constraint equation (8) which, in the Bosonic form, is given by the representation : 7 H$_{wire}$ + H$_{leads}$ ≈ ∫ L 2 − L 2 dx [ ∑ σ = ↑ , ↓ − ¯ h 2 2 m $_{Ψ}$† $_{σ}$( x ) ∂ 2 ∂x $_{2}$Ψ$_{σ}$ ( x ) − E$_{F}$ ∑ σ = ↑ , ↓ $_{Ψ}$† $_{σ}$( x )Ψ$_{σ}$( x )] (9) $^{Ψ}$↓( x $^{)Ψ}$↑( x ) | F > = 0 for the region | x | ≤ d 2 (8) $_{Ψ}$† σ = ↑ ( x $^{)Ψ$_{σ}$$_{=}$}$↑( x $_{) + Ψ}$† σ = ↓ ( x $^{)Ψ$_{σ}$$_{=}$}$↓( x ) | F > = n$_{e}$ ( x ) | F > (7) B-The strongly interacting electrons U ( x ) = ∞