conductance will be half in comparison with a non -interacting wire. 

The plan of this paper is as follows: For pedagogical reasons, we present in chapter 2 the method of Dirac’s constraints used in quantum mechanics. This method will be used in the remaining chapters. In chapter 3, we present the model for an interacting wire ( U = ∞ ) of length d coupled to the non-interacting leads and reservoir ( U = 0). In chapter 4, we derive the anomalous Dirac commutator for the quantum wire model. This is done by extending the Bosonization method to the constraints . In chapter 5, we identify the electric current operator and in chapter 6, we compute the static current at a finite temperature. In chapter 7, we consider the effect of the Zeeman interaction, and in chapter 8, we present the conclusion. 

## 2. Dirac’s method in quantum mechanics 

In this section, we will present Dirac’s method in quantum mechanics. This method will be extended in the next sections to the strongly interacting quantum wires. 

We consider the Hamiltonian H with the constraint operator Q$_{1}$ . The constraint op- erator is defined by the condition that when it acts on the true ground state | F > it satisfies the equation Q$_{1}$ | F > = 0. Formally this condition is introduced by demanding that the ground state wave function | F > of the Hamiltonian H has to obey, in addi- tion the constraint equation, Q$_{1}$ | F > = 0 . Since the constraint must be obeyed at any time, we must have dQ$_{1}$ ( t ) dt | F > = 0. From the Heisenberg equation of motion, we ob- tain the condition i ¯ h dQ$_{1}$ ( t ) dt | F > = [ Q$_{1}$ ( t ) ,H ] | F > = 0. Therefore, we find that the con- straint is satisfied at any time, only if the constraint operator Q$_{1}$ commutes with the Hamiltonian [ Q$_{1}$ ( t ) ,H ] or satisfies the equation [ Q$_{1}$ ( t ) ,H ] ∝ Q$_{1}$ ( t ). If this is the case, Q$_{1}$ will be the only constraint. Here we will assume that this not the case! There- fore, the only way to satisfy the condition dQ$_{1}$ ( t ) dt | F > = 0 is to introduce a second con- straint Q$_{2}$ ∝ [ Q$_{1}$,H ] (or Q$_{2}$ ∝ [ Q$_{1}$,H ] − constant · Q$_{1}$ ). For simplicity we assume that no more constraints are needed. As a result, we obtain a problem with two constraints Q$_{1}$ | F > = 0, and Q$_{2}$ | F > = 0 which do not commute [ Q$_{1}$,Q$_{2}$ ] ̸ = 0. To enforce the constraints on the wave function, we use the method of Lagrange multipliers λ$_{1}$ and λ$_{2}$ . This allows us to replace the Hamiltonian H by H$_{T}$ = H + λ$_{1}$Q$_{1}$ + λ$_{2}$Q$_{2}$ . We will determine the Lagrange multipliers using the Heisenberg equations of motion. 

