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The Dirac commutator affect in a significant way the equation of motion for the particle -hole excitations. Using the Heisenberg equations of motion i ¯ h dϑ$_{e}$ ( x,t ) dt = [ ϑ$_{e}$ ( x, t ) , H$_{wire}$ ]$_{Dirac}$ and i ¯ h dϕ$_{e}$ ( x,t ) dt = [ ϕ$_{e}$ ( x, t ) , H$_{wire}$ ]$_{Dirac}$ we find: 

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The space dependent commutation rules give rise to a multi particle-hole scatter- ing state. Performing a space average, we obtain the effective velocity v$_{eff.}$ ≡ v √ < (1 − h ( x,d ) 2 )) 2 $^{>$_{space}$}$− average = v (1 − 3 d 4 $_{L}$) with the fundamental frequency ω = v$_{eff.}$ 2 π L ,. The space dependent part in equation (27) gives rise to the scattering potential V$_{sc}$ ( x ) ≡ v $^{2}$[ < (1 − h ( x,d ) 2 )) 2 $^{>$_{space}$}$− average − (1 − h ( x,d ) 2 )) $^{2}$] which generates high harmonic states. 

To conclude the anomalous commutator for a wire with exclusion of double occupancy given in equation (26) is the central result of this paper. This result was obtained restricting the infinite number of generated constraints to a finite set of constraints Q$_{1}$ ( x ) and Q$_{2}$ ( x ). We have shown that according to scaling theory [35], the higher order constraints are irrelevant and therefore can be neglected. 

## 5. The current operator for the wire 

Using the Dirac commutator, we find the equation of motion for the electronic density ρ$_{e}$ ( x ) ≡ 1 √ $_{π}$∂$_{x}$ ϑ$_{e}$ ( x ). 

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We identify from the continuity equation dρ$_{e}$ ( x,t ) dt + ∂$_{x}$J$_{e}$ ( x, t ) = 0 the conserved electronic current operator J$_{e}$ ( x ). We multiply J$_{e}$ ( x ) by the electric charge ( − e ) and find the electric current operator ˆ I$_{e}$ . 

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This equation shows that the current in the wire is density dependent. We find that the current in the wire ( | x | ≤ d $_{2}$) is given by ˆ I$_{e}$ = ev $_{2}$√ $_{π}$∂$_{x}$ ϕ$_{e}$ ( x ) is reduced by a factor of two in comparison with the non-interacting electrons ( | x | > d $_{2}$) ˆ I ( U =0) e = ev √ $_{π}$∂$_{x}$ ϕ$_{e}$ ( x ). 

