content_image/440177.jpg content_image/440176.jpg content_image/440175.jpg content_image/440167.jpg content_image/440166.jpg content_image/440174.jpg content_image/440171.jpg content_image/440169.jpg content_image/440172.jpg content_image/440168.jpg content_image/440173.jpg content_image/440165.jpg
Gibbs averaging over the states of the Hamiltonian ˆ H$_{0}$ , Eq. (2.20). 

The Green function G ( uφ ) r,r ; σ ( τ, τ $^{′}$) , Eq. (2.23), satisfies the following equation 

<img src='content_image/440177.jpg'>

where ˆ ε$_{r}$f$_{r}$ ≡ − ∑ r ′ t$_{r,r}$ ′ f$_{r}$ ′ for an arbitrary function f$_{r}$ . A conjugated equation can be written as 

<img src='content_image/440176.jpg'>

The electron Green function G ( uφ ) r,r ; σ ( τ, τ $^{′}$), Eq. (2.23), sat- isfies the fermionic boundary conditions 

<img src='content_image/440175.jpg'>

Eqs. (2.21, 2.22, 2.24, 2.25) can serve as the starting point of our bosonization scheme. In the next subsec- tion, we derive equations for the bosonic excitations in an external time dependent HS field. 

## B. Equations for bosons in the fluctuating field 

According to the results of the previous subsection one can calculate the partition function Z solving Eq. (2.24) or (2.25) for a given configuration of the field φ$_{rσ}$ ( τ ), substitute the solution into Eq. (2.21), thus obtaining the functional Z [ φ ], and then calculate Z using Eq. (2.18). Actually, solving Eqs. (2.24, 2.25) one obtains more in- formation than necessary because the Green function G ( uφ ) r,r ′ ; σ ( τ, τ $^{′}$) that should be obtained from this equa- tions depends on two times, τ and τ $^{′}$, whereas these time should be taken equal in Eq. (2.21). 

It is not difficult to derive closed equations for the Green function G ( uφ ) r,r ′ ; σ ( τ, τ + 0) entering Eq. (2.21). Sub- tracting Eq. (2.24) from Eq. (2.25) and putting τ ′ = τ +0, we obtain 

<img src='content_image/440167.jpg'>

We write here equal times in the functions φ$_{r}$ ( τ ) because we assume that they are continuous. This is not so for the Green functions G ( uφ ) r,r ′ ; σ ( τ, τ $^{′}$) because they have a jump at equal times. For discontinuous functions φ$_{r}$ ( τ ) one would have to take slightly different times τ and τ + 0, too. 

It follows from Eq. (2.26) that 

## 1. General equations. 

<img src='content_image/440166.jpg'>

which shows that the total number of particles in the system of the non-interacting fermions in the auxiliary HS field φ$_{r}$ ( τ ) does not depend on time. To avoid con- fusion, we remind the reader that we started from the grand canonical formulation for the model of the inter- acting particles, Eqs. (2.1, 2.4), and writing about the particle conservation now we do not mean that we change the formulation of our initial model. 

For subsequent manipulations, it is convenient to in- troduce a new function 

<img src='content_image/440174.jpg'>

where z = ( τ, σ, u ). 

The function A$_{r,r}$ ′ ( z ) has the property 

is periodic, 

and, hence, describes bosons. 

With this function, the functional Z [ φ ] takes a simple form 

<img src='content_image/440171.jpg'>

Eq. (2.31) is a reformulation of (2.21) in terms of A$_{r,r}$ ′ ( z ), Eq. (2.28). 

Having written Eq. (2.26) we can derive a closed equa- tion for the function A$_{r,r}$ ( z ) , Eq. (2.28). Our procedure is very similar to deriving the kinetic equation starting from equations for Green functions $^{30}$. 

We rewrite Eq. (2.26) at u = 0, and subtract it from Eq. (2.26). Using the definition of A$_{r,r}$ ′ ( z ), we come to the following equation for this function 

<img src='content_image/440169.jpg'>

where n$_{r,r}$ ′ = G (0) r,r ′ ( τ, τ + 0) is the Fermi distribution function of the ideal gas in the coordinate representation 

The solution of Eq. (2.32), as it stands, is not unique. One can easily see that there is a non-physical solution A$_{r,r}$ ′ ( z ) = n$_{r,r}$ ′ for any φ$_{rσ}$ ( τ ). Moreover, for static fields φ$_{rσ}$ , one can add to the solution of Eq. (2.32) a combination A (0) r,r ′ of the form 

<img src='content_image/440172.jpg'>

<img src='content_image/440168.jpg'>

<img src='content_image/440173.jpg'>

<img src='content_image/440165.jpg'>

