The constraint operator is represented in terms of the Bosonic fields for the charge degrees of freedom: ϑ$_{e}$ ( x ) = ( $^{ϑ}$↑ ( x ) + $^{ϑ}$↓ ( x )) , ϕ$_{e}$ ( x ) = 1 $_{2}$( $^{ϕ}$↑ ( x ) + $^{ϕ}$↓ ( x )) and P$_{e}$ ( x ) = ∂$_{x}$ϕ$_{e}$ ( x ). Similarly, the spin degrees of freedom are given by: ϑ$_{s}$ ( x ) = 1 √$_{2}$ ( $^{ϑ}$↑ ( x ) − $^{ϑ}$↓ ( x )), ϕ$_{s}$ ( x ) = 1 √$_{2}$ ( $^{ϕ}$↑ ( x ) − $^{ϕ}$↓ ( x )) and P$_{s}$ ( x ) = ∂$_{x}$ϕ$_{s}$ ( x ). In order to find the commutation relation, we will use the new ground state which is restricted by the constraint. The commutator [ ϑ$_{σ}$ ( x ) , P$_{σ}$ ′ ( x ′ )] will be defined by the value of the expectation value with respect the ground state [31, 33]. This expectation value is the central result of this paper and is derived in the next section. The expectation value of the commutator is a function of h ( x, d ) given by: h ( x, d ) = 1 for | x | ≤ d $_{2}$, ( U = ∞ ) and h ( x, d ) = 0 for | x | > d $_{2}$, ( U = 0). This result allows us to define the Dirac commutator [ ϑ$_{e}$ ( x ) , P$_{e}$ ( x ′ )]$_{Dirac}$ = iδ ( x − x ′ )[1 − h ( x, d ) 1 $_{2}$]. The Bosonized Hamiltonian for this case is given by : v is the Fermi velocity for the non-interacting fermions. We assume perfect coupling between the wire and the leads. The Bosonic fields obey continuous boundary conditions: ϑ$_{σ}$ ( x = d 2 − ϵ ) = ϑ$_{σ}$ ( x = d $_{2}$+ ϵ ), ϑ$_{σ}$ ( x = − d 2 − ϵ ) = ϑ$_{σ}$ ( x = − d $_{2}$+ ϵ ) and ϕ$_{σ}$ ( x = d 2 − ϵ ) = ϕ$_{σ}$ ( x = d $_{2}$+ ϵ ), ϕ$_{σ}$ ( x = − d 2 − ϵ ) = ϕ$_{σ}$ ( x = − d 2 + ϵ ) , where ϵ represents the overlapping wire-leads region. Therefore, we can replace the Hamiltonian H$_{wire}$ + H$_{leads}$ by a free Bosonic model. 8 H$_{reservoir}$ = ∑ α = ↑ , ↓ ∫ L 2 − L 2 dx [ µ$_{L}$ ( x ) n$_{L,σ}$ ( x ) + µ$_{R}$ ( x ) n$_{R,σ}$ ( x )] (14) H$_{wire}$ + H$_{leads}$ = ∑ σ = ↑ , ↓ ∫ L 2 − L 2 dx ¯ hv 2 [( ∂$_{x}$ϕ$_{σ}$ ( x )) 2 + ( ∂$_{x}$ϑ$_{σ}$ ( x )) $^{2}$] H$_{Leads}$ = ∑ σ = ↑ , ↓$^{[}$ ∫ − d 2 − L 2 dx ¯ hv 2 [( ∂$_{x}$ϕ$_{σ}$ ( x )) 2 + ( ∂$_{x}$ϑ$_{σ}$ ( x )) $^{2}$] + ∫ L 2 d 2 dx ¯ hv 2 [( ∂$_{x}$ϕ$_{σ}$ ( x )) 2 + ( ∂$_{x}$ϑ$_{σ}$ ( x )) $^{2}$] (13) H$_{wire}$ = ∑ σ = ↑ , ↓ ∫ d 2 − d 2 dx ¯ hv 2 [( ∂$_{x}$ϕ$_{σ}$ ( x )) 2 + ( ∂$_{x}$ϑ$_{σ}$ ( x )) $^{2}$] (12) < F | [ ϑ$_{e}$ ( x ) , P$_{e}$ ( x $^{′$_{)]}$}$| F > = iδ ( x − x ′$_{)[1}$ − h ( x, d ) 1 2 ] (11) $^{Ψ}$↑( x $^{)Ψ}$↓( x ) | F > = 2 e − $_{i}$√$_{4}$$_{πϕ}$$_{e}$$_{(}$$_{x}$$_{)}$$_{cos}$$_{[2}$$_{K}$ $_{F}$x + √$_{πϑ}$$_{e}$$_{(}$$_{x}$$_{)] +}$ cos $_{[}$√$_{2}$$_{πϑ}$$_{s}$$_{(}$$_{x}$$_{)]}$ | F > = 0; | x | ≤ d 2 (10)