lems. A good quantitative agreement with exact results is expected for large systems implying that they include the bath. Although the arguments presented in the last subsec- tions may help to understand the general scenario of what happens in the process of the bosonization, they are not rigorous and their final validity is to be checked numeri- cally. As we have seen previously, just putting δ = 0 and keep- ing the parameter γ finite in Eqs. (4.35, 4.36) results in strictly positive weights Z$_{b}$ [ ˜ φ ] because the solutions of the homogeneous equation in Eq. (4.42) do not exist for finite γ and one comes to a real solution for A$_{a}$$_{;}$$_{r,r}$ ′ ( z ). Putting γ = 0 for finite δ we can choose the solution for A$_{a}$$_{;}$$_{r,r}$ ′ ( z ) corresponding to the initial fermionic problem and thus obtain the functional Z$_{f}$ [ ˜ φ ] that can be both positive and negative. In principle, one could obtain many other solutions of Eq. (4.42, 4.43) keeping both γ and δ finite. Using the spectral expansion, Eq. (2.52), it is clear that the singularities in the function A$_{r,r}$ ′ ( z ) can originate only from the zero eigenvalues E K = 0 of the operator in the L.H.S. of Eq. (2.32). In order to avoid this problem we have introduced a regularization that discards such solutions. Actually, the crucial step in this regularization is taking equal times in the function ˜ Φ$_{r,r}$ ′ ( τ ) (putting δ = 0 in Eq. (4.41)). On the other hand, keeping a small δ > 0 makes the eigenvalues E K complex with an imaginary part of order δ . Although this imaginary part is infinitesimally small in the limit δ → 0, it determines the imaginary part of the integral over u in Eq. (2.43) arising near the poles corresponding to the zero eigenvalues E K = 0. We calculate explicitly the imaginary part of the eigen- value E K in Appendix C assuming that the eigenvalue E K is very close to zero and show that it is finite for finite δ but vanishes in the limit δ/γ → 0. Thus, our bosonization scheme should give different re- sults depending on how we treat Eqs. (4.42, 4.43). The presence of the bath makes the partition function insen- sitive to the choice of the solution of this equation and we take the limit δ/γ → 0 leading us to the the real positive Z$_{b}$ [ ˜ φ ]. In the next subsection, we discuss the basic proper- ties of Z$_{b}$ [ ˜ φ ] and demonstrate that there is no reason to encounter the sign problem when using this functional. In this subsection we want to demonstrate that using Z$_{b}$ [ ˜ φ ] is convenient for computations and there are no singularities or bad features, like the negative sign, that would create problems. Final formulas will be brought to a form that can immediately be used for explicit compu- tations. We do not see any problems in taking the limit Δ → 0, α/ Δ → 0 in this representation and put α = 0 in the beginning of the discussion. For convenience, we display in one place the formulas that may serve as the basis of computational schemes for the case of the on-site repulsion. We consider the system in a bath, which means that the HS field ˜ φ$_{r}$ ( τ ) is not equal to zero only in a region re- We can repeat the transformations of subsection II B keeping a finite δ > 0 and introducing the regularization with the parameter γ and the bath. As a result, we come instead of Eqs. (2.39, 2.40) to the following equations and δ → +0. where a = 1 , 2, and ˜ H (+) r,r ′ is a 2 × 2 matrix where with the operator ˜ M (+) r,r ′ ( z ) , To see this, let us carry out the regularization keeping small but finite δ for τ ′ = τ + δ . Although the param- eter δ is infinitesimally small, its presence can be very important for choosing the proper solution of Eq. (2.32). As we consider discontinuous functions ˜ φ$_{r}$ ( τ ), taking the proper limit is especially important because τ and τ + 0 can belong to different slices leading to completely differ- ent values ˜ φ$_{r}$ ( τ ) and ˜ φ$_{r}$ ( τ + 0). So, we write Eq. (2.26) in the form Of course, just taking the original electron Green function G ( uφ ) r,r ′ ; σ ( τ, τ $^{′}$) from Eq. (2.24) we would satisfy Eq. (2.32) for τ ′ = τ + δ but we understand that using this solution, one encounters the sign problem. There are infinitely many other solutions of Eq. (2.32) leading to non-positive weights in the MC procedure. In this subsection, we want to emphasize that one can select the solution of Eq. (2.32) in many different ways but only one of them, Eq. (2.39), gives positive real Z$_{b}$ [ ˜ φ ] for all ˜ φ$_{r}$ ( τ ). In principle, in the limit of a large number of sites in the bath all these solutions should lead to the same result and we simply choose the most convenient one. E. Some remarks about the regularization 22 F. Properties of the bosonic action and its suitability for numerical investigations ( ∂ ∂τ − ˜ M (+) r,r ′ ( z ) G ( uφ ) r,r ′ ; σ ( τ, τ + δ ) = 0 (4.39) ˜ M (+) r,r ′ ( z ) = ˆ ε$_{r}$ − ˆ ε$_{r}$ ′ − uσ ˜ Φ (+) r,r ′ ; σ ( τ ) , (4.40) ˜ Φ (+) r,r ′ ; σ ( τ ) = ˜ φ$_{r}$ ( τ ) − ˜ φ$_{r}$ ′ ( τ + δ ) (4.41) ˜ H (+) r,r ′ ( τ ) A (+) a ; r,r ′ ( z ) = − uσn$_{r,r}$ ′ $_{,σ}$˜ Φ (+) r,r ′ ; σ ( τ ) B$_{a}$, (4.42) ˜ H (+) r,r ′ ( τ ) = Λ$_{1}$ ˜ M (+) r,r ′ ( z ) + i Λ$_{2}$ ∂ ∂τ + Λ γ. (4.43) 1. Bosonic action