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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 ). 

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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. 

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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 ′ ). 

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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 : 

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The Dirac commutator for the canonical conjugates fields ϑ$_{σ}$ ( x ) and P$_{σ}$ ′ ( y ) will be given by: 

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