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The Hubbard interaction U ( x ) is restricted to the region | x | ≤ d 2 (with the condition d << L ). In the complimentary region | x | > d 2 the Hubbard interaction is U ( x ) = 0. 

In the remaining part of this section we will present the Bosonized version of two models which have the same Hamiltonian and differ only by the constraint conditions. The two models are: the non interacting electrons described by U = 0 and the strongly interacting electrons U = ∞ . Both problems will be investigated using the method of Bosonization. The method of Bosonization is based on the representation of the Fermion operator Ψ$_{σ}$( x ) in terms of the Bosonic fields ϑ$_{R,σ}$ = ϑ$_{σ}$ − ϕ$_{σ}$ 2 , ϑ$_{L,σ}$ = ϑ$_{σ}$ + ϕ$_{σ}$ 2 . The Bosonic representation of the electron operator is given by [31–33] : 

<img src='content_image/375050.jpg'>

where a is the lattice constant and K$_{F}$ is the Fermi momentum. Using the Bosonic fields ϑ$_{L,σ}$ ( x ) and ϑ$_{R,σ}$ ( x ), we define a new field P$_{σ}$ ( x ) ≡ ∂$_{x}$ϕ$_{σ}$ ( x ). In order to construct a quantum theory, one has to identify the canonical momentum operator which is conjugate to the Bosonic fields ϑ$_{σ}$ ( x ). Here, we have an example where both the operator ϑ$_{σ}$ ( x ) and P$_{σ}$ ( x ) ≡ ∂$_{x}$ϕ$_{σ}$ ( x ) (the candidate for conjugate momentum) are built from the same fields ( ϑ$_{R,σ}$ , ϑ$_{L,σ}$ ). Therefore, the commutator [ ϑ$_{σ}$ ( x ) , P$_{σ}$ ′ ( x ′ )] can be finite only for an infinite number of particles and vanishes otherwise. Such commutators are called anomalous [31, 33]. For such cases the commutator [ ϑ$_{σ}$ ( x ) , P$_{σ}$ ′ ( x ′ )] will be defined according to the many particle ground state | G > with the expectation value < G | [ ϑ$_{σ}$ ( x ) , P$_{σ}$ ′ ( x ′ )] | G > . 

## A-The non-interacting electrons, U ( x ) = 0 

The Hamiltonian for the non-interacting electrons is given by eq.3 with U ( x ) = 0. The Bosonized Hamiltonian is given by : 

<img src='content_image/375049.jpg'>

where n$_{R,σ}$ ( x ) and n$_{L,σ}$ ( x ) are the right and left electronic densities and µ$_{R}$ , µ$_{L}$ are the chemical potentials. The non-interacting electrons ground state is identified with the Fermi sea | F (0) > . We compute the expectation value of the operators [ ϑ$_{σ}$ ( x ) , P$_{σ}$ ′ ( x ′ )] and identify the anomalous commutator [31, 33]: 

<img src='content_image/375048.jpg'>

