leads and takes the value of one for the wire. The constraints for the wire will be enforced by replacing the Lagrange fields λ$_{i}$ ( x ) by the product h ( x, d ) λ$_{i}$ ( x ). We will apply the quantum mechanical method given in equation (9) to the Hamiltonian for the wire given in equation (18). Since the constraint must obey dQ$_{1}$ ( x,t ) dt | F > = 0, and the commutator [ Q$_{1}$ ( x ) , Q$_{2}$ ( x ′ )] can be inverted, the Lagrange multipliers can be computed. From equation (19), we obtain the Lagrange fields λ$_{i}$ ( x ). The Lagrange fields are functions of the matrix C$_{i,r}$ ( x, u ), r = 1 , 2 which are defined with the help of the the integral equation ∑ 2 r =1 ∫ duC$_{i,r}$ ( x, u )[ Q$_{r}$ ( u ) , Q$_{j}$ ( y )] = δ$_{i,j}$δ ( x − y ) , j = 1 , 2. The matrix C$_{i,r}$ ( x, u ) is given by the inverse of the commutator [ Q$_{i}$ ( x ) , Q$_{r}$ ( x ′ )]. The commutator of the constraints [ Q$_{1}$ ( x ) , Q$_{2}$ ( x ′ )] (given in equations (16 − 17)) is computed using the non-interacting Bosonic commutator [ ϑ$_{σ}$ ( x ) , P$_{σ}$ ′ ( x ′ )] = iδ$_{σ,σ}$ ′ δ ( x − x ′ ). We substitute in equation (19) the result of the commutators [ Q$_{1}$ ( x ) , Q$_{2}$ ( x ′ )] given in equa- tion (20). As a result, the equation dQ$_{i}$ ( x,t ) dt , i = 1 , 2 ( see equation (19) ) allows us to determine the Lagrange fields λ$_{i}$ ( x ), i = 1 , 2. Next, we substitute the Lagrange fields λ$_{i}$ ( x ) into the total Hamiltonian H$_{T}$ given in equation (18) and find that the Heisenberg equa- tions of motion are determined by the physical Hamiltonian H$_{wire}$ + H$_{leads}$ with the modified commutation relation [ , ]$_{Dirac}$. The Heisenberg equation of motion for any operator A ( x, t ) will be given by : The Dirac commutator for the canonical conjugates fields ϑ$_{σ}$ ( x ) and P$_{σ}$ ′ ( y ) will be given by: 11 [ ϑ$_{σ}$ ( x ) , P$_{σ}$ ′ ( y )]$_{Dirac}$ = [ ϑ$_{σ}$ ( x ) , P$_{σ}$ ′ ( y )] − ∫ L 2 − L 2 du h ( u, d ) ∫ L 2 − L 2 dv [ ϑ$_{σ}$ ( x ) , Q$_{2}$ ( u )] C$_{2}$$_{,}$$_{1}$ ( u, v )[ Q$_{1}$ ( v ) , P$_{σ}$ ′ ( y )] (22) i ¯ h dA ( x, t ) dt = [ A ( x, t ) , H$_{wire}$ + H$_{leads}$ ]$_{Dirac}$ ≡ [ A ( x, t ) , H$_{wire}$ + H$_{leads}$ ] − ∑ i =1 , 2 ∑ j =1 , $_{2}$∫ du h ( u, d ) ∫ dv [ A ( x, t ) , Q$_{i}$ ( u )] C$_{i,j}$ ( u, v )[ Q$_{j}$ ( v ) , H$_{wire}$ + H$_{leads}$ ] (21) [ Q$_{1}$ ( x ) , Q$_{2}$ ( x ′$_{)] =}$ − $_{i}$√$_{2}$$_{πδ}$$_{(}$$_{x}$ − x ′$_{)[(}$ sin $_{(}$√$_{2}$$_{πϑ}$$_{s}$$_{(}$$_{s}$$_{)))}$ 2 + ( sin (2 K$_{F}$x + √$_{πϑ}$$_{e}$$_{(}$$_{x}$$_{)))}$ $^{2}$] (20) i ¯ h dQ$_{1}$ ( x, t ) dt | F > = [ Q$_{1}$ ( x, t ) , H$_{wire}$ + H$_{leads}$ + ∫ L 2 − L 2 dx h ( x, d ) 2 ∑ i =1 λ$_{i}$ ( x ) Q$_{i}$ ( x ))] | F > = 0 (19) H$_{T}$ = H$_{wire}$ + H$_{leads}$ + ∫ L 2 − L 2 dx h ( x, d ) 2 ∑ i =1 λ$_{i}$ ( x ) Q$_{i}$ ( x )) (18)