From i ¯ h dQ$_{1}$ ( t ) dt | F > = ([ Q$_{1}$, H ] + λ$_{2}$ [ Q$_{1}$, Q$_{2}$ ]) | F > = 0, we obtain λ$_{2}$ = − ([ Q$_{1}$, Q$_{2}$ ]) − $^{1}$[ Q$_{1}$, H ], and from i ¯ h dQ$_{2}$ ( t ) dt | F > = ([ Q$_{2}$, H ]+ λ$_{1}$ [ Q$_{1}$, Q$_{2}$ ]) | F > = 0, we obtain λ$_{2}$ = − ([ Q$_{1}$, Q$_{2}$ ]) − $^{1}$[ Q$_{1}$, H ]. We substitute the explicit form of the Lagrange multipliers into the Hamiltonian H$_{T}$ , and find the following Heisenberg equation of motion for any operator A : The conclusion from this calculation is that the effect of the constraints have changed the regular commutator [ A, H ] to the Dirac bracket commutator [ A, H ]$_{D}$, which is given by [ A, H ]$_{D}$ ≡ [ A, H ] − [ A, Q$_{1}$ ]([ Q$_{1}$, Q$_{2}$ ]) − $^{1}$[ Q$_{2}$, H ] − [ A, Q$_{2}$ ]([ Q$_{2}$, Q$_{1}$ ]) − $^{1}$[ Q$_{1}$, H ]. Using this result, we define the Dirac bracket for two operators A and B in the following way: This methodology will be applied in the next sections to the problem of strongly inter- acting electrons. We consider a quantum wire of length d which is perfectly coupled to the leads of length L − d shown in figure 1. The leads are in thermal equilibrium with the two electronic reservoirs which have chemical potentials µ$_{R}$ and µ$_{L}$ . The reservoirs are described by three dimensional non-interacting electrons. Physically, once the electrons reach the reservoir they do not return to the wire . At a finite temperature T , the infinite leads can be replaced by finite leads L ≈ L$_{T}$ ( L$_{T}$ is the thermal length for which coherency in the leads is preserved). At finite temperatures we have the model: H = H$_{wire}$ + H$_{leads}$ + H$_{reservoir}$ U ( x ) is the space dependent Hubbard interaction, n$_{σ}$ ( x ) = Ψ † $_{σ}$( x )Ψ$_{σ}$( x ) represents the electronic density for the spin polarization σ = ↑ , ↓ and E$_{F}$ is the Fermi energy. 5 H$_{wire}$ + H$_{leads}$ = ∫ L 2 − L 2 dx [ ∑ σ = ↑ , ↓ − ¯ h 2 2 m $_{Ψ}$† $_{σ}$( x ) ∂ 2 ∂x $_{2}$Ψ$_{σ}$ ( x ) + U ( x ) $^{n$_{σ}$$_{=}$}$↑ ( x ) $^{n$_{σ}$$_{=}$}$↓ ( x ) − E$_{F}$ ∑ σ = ↑ , ↓ $_{Ψ}$† $_{σ}$( x )Ψ$_{σ}$( x )] (3) [ A, B ]$_{D}$ ≡ [ A, B ] − [ A, Q$_{1}$ ]([ Q$_{1}$, Q$_{2}$ $_{])}$− $^{1}$[ Q$_{2}$, B ] − [ A, Q$_{2}$ ]([ Q$_{2}$, Q$_{1}$ $_{])}$− $^{1}$[ Q$_{1}$, B ] (2) i ¯ h dA dt = [ A, H$_{T}$ ] = [ A, H ] − [ A, Q$_{1}$ ]([ Q$_{1}$, Q$_{2}$ $_{])}$− $^{1}$[ Q$_{2}$, H ] − [ A, Q$_{2}$ ]([ Q$_{2}$, Q$_{1}$ $_{])}$− $^{1}$[ Q$_{1}$, H ] ≡ [ A, H ]$_{D}$ (1) 3. The model for a short wire with exclusion of double occupancy coupled to non-interacting leads and reservoir