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In the limit of a very weak interaction V$_{0}$ , typical ˜ φ$_{r}$ ( τ ) are small and one can neglect them in the expression for M r,r ′ , Eq. (4.51). Then, eigenvalues λ$_{k}$ and eigenfunc- tions v k r of this operator do not depend on the number of the slices and can easily be obtained in the form 

where c$_{k}$ are normalization coefficients. 

In this limit, one can solve Eq. (4.50) by Fourier trans- forming both the sides in space and time. This allows one to write the Fourier transformed solution as 

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Substituting Eq. (4.55) into Eqs. (4.44, 4.47) and taking the limit γ → 0 and Δ → 0 we come to the RPA result, Eq. (3.14). The integral over u and summation over σ are trivial in this calculation. 

The presence of the regularizing parameter γ results in the vanishing of the contribution of the state with k = k $^{′}$, ω = 0. However, the error is small because the sum extends over many states with different k that inevitably exist due to the bath. 

The differential with respect to time equation (4.50) can be written in the integral form with the help of the bare Green function G 0 r,r ′ ; r$_{1}$,r ′ 1 ( τ, τ$_{1}$ ) introduced as the solution of the equation 

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The function G 0 r,r ′ ; r$_{1}$,r ′ 1 ( τ, τ$_{1}$ ) is a 2 × 2 matrix and is uniquely defined by the boundary conditions 

The corresponding homogeneous equation does not have solutions due to the presence of the parameter γ . The bare Green function 0 r,r ′ ; r$_{1}$,r ′ ( τ, τ$_{1}$ ) can easily be 

G 1 found by Fourier transformation and can be written ex- plicitly as 

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where the Pauli matrices Λ Eq. (2.42). 

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2. Integral form of the equation 

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Eq. (4.58) is written for the entire system including the bath and the momenta k , k ′ correspond to this enlarged system. Accordingly, the total number of sites N total d enters the normalization factor in Eq. (4.58). The state with ω = 0, ε$_{k}$ = ε$_{k}$ ′ is included in the sum in Eq. (4.58). 

One can perform summation over the bosonic frequen- cies ω = 2 πT n in Eq. (4.58) using the formula 

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where the symbol “ sgn ” stands for the sign and a is a number. This gives the following expression for the Green function G 0 r,r ′ ; r$_{1}$,r ′ 1 ( τ, τ$_{1}$ ) for arbitrary coordinates and times 

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where 

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In Eq. (4.60) the operators ˆ ε$_{r}$$_{1}$ and ˆ ε$_{r}$ ′ 1 act on the corre- sponding variables entering the δ -functions on the right. The function G 0 r,r ′ ; r$_{1}$,r ′ 1 ( τ, τ$_{1}$ ), Eq. (4.60), is explicitly real and free of singularities. 0 

Using the Green function G r,r ′ ; r$_{1}$,r ′ 1 ( τ, τ$_{1}$ ) , Eq. (4.60), we reduce Eq. (4.50) to the form 

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Eq. (4.62) can serve as the basis of many numerical approaches. One can, e.g., solve this equation recur- sively neglecting in the zeroth approximation the func- tion ˜ Φ$_{r}$$_{1}$$_{,r}$ ′ 1 ( τ$_{1}$ ) in the L.H.S. and then calculating all or- ders of ˜ Φ$_{r}$$_{1}$$_{,r}$ ′ 1 ( τ$_{1}$ ) or one can just invert the operator act- ing on A$_{a}$$_{;}$$_{r,r}$ ′ ( z ) in the L.H.S. of the equation. Due to the regularization, the zero eigenvalues of the operator in the L.H.S. are discarded. Therefore all these operations are well defined and one should not encounter any singu- larity. In particular, one can invert the operator acting on A$_{a}$$_{;}$$_{r,r}$ ′ in the L.H.S. of Eq. (4.62). 

As a result, after integration over u the exponent in the function Z$_{b}$ [ ˜ φ ] remains real and Z$_{b}$ [ ˜ φ ] positive. This demonstrates that this bosonization method of compu- tations is free of the sign problem. Substituting the so- lution obtained by inversion of the L.H.S. operator into 

