FIG. 1: Simple electronic diagrams. where v k r are eigenfunctions of the operator ˆ h$_{r}$ [ uφ ] and C$_{k}$ are time independent coefficients. There can be less trivial solutions of the homogeneous equation corresponding to Eq. (2.32) when φ$_{rσ}$ ( τ ) is a discontinuous function of time with large jumps. Such solutions correspond to poles in the Green functions G$_{r,r}$ ′ ( τ, τ + 0) and should be treated very carefully. The sign problem in the MC simulations appears as a result of the poles in the integrals over u in Eqs. (2.31) which arise when the fields φ$_{rσ}$ ( τ ) are discontinuous in time. We discuss this question in Section IV. It is clear that the ambiguity in solving Eq. (2.32) fol- lowing from the existence of solutions of the homogeneous equation creates problems in both numerical and analyt- ical treatment. This means that a procedure fixing the proper solutions of Eq. (2.32) is necessary and we intro- duce it below. Seeking for the unique correct solution of Eq. (2.32) we are guided by the conventional diagrammatic technique for fermions which is well defined and all corresponding diagrams can be calculated at least in principle. In order to visualize how we come to solving Eq. (2.32) instead of summing the conventional diagrams we consider the simplest loop represented in Fig. 1 (a). The expression corresponding to this diagram can be written as where ε ′$_{p}$ = ε$_{p}$ − µ ′ and ε = 2 π ( n + 1 / 2) T and ω = 2 πmT are fermionic and bosonic frequencies, respec- tively. The expression in the second line of Eq. (2.35) corre- sponds to the solution of Eq. (2.32) in the lowest order in φ , which demonstrates how the bosonic modes are ob- tained from the fermionic lines. However, the first and the second lines in Eq. (2.35) are different at ω = 0 , q = 0 . The first line gives the It is clear that the leads cannot change physics of the system of the interacting electrons if the latter is suffi- ciently large. However, this model is reasonable even for which means that the interaction is finite inside the sphere of the radius R$_{0}$ containing N$_{d}$ sites and vanishes outside this sphere. In other words, we attach metallic leads to the system of the interacting electrons and as- sume that the inter-electron interaction vanishes in the leads. This goal can be achieved if we assume that the sys- tem of the interacting electrons considered here is imbed- ded in a “bath”. We introduce this bath considering a model of the interacting electrons on a d -dimensional lat- tice with the total number of sites N total d . However, we assume that the interaction V$_{r,r}$ ′ , Eq. (2.3), entering the Hamiltonian ˆ H , Eq. (2.1), vanishes outside a subsystem consisting of a considerably smaller number of sites N$_{d}$ . As an example, we can suggest the following form of the interaction V$_{r,r}$ ′ The hint at a possible regularization is given by the form of Eq. (2.35). If the momenta q were continuous rather then discrete and we could simply neglect the con- tribution of the state with q =0, the bosonization scheme would work in this order. This is not sufficient, though, because the momenta and frequencies of the two horizon- tal fermion lines in Fig. 1 (b) obtained after integration over the HS field φ coincide for any translationally invari- ant systems and one should slightly violate the transla- tional invariance in order to split the momenta. We see that our bosonization scheme is not exactly equivalent to the initial fermionic model and needs a reg- ularization in order to avoid the uncertainty. This un- certainty appears not only in the situation represented in Fig. 1 (a), where the field φ enters with ω = 0 , q =0. The same problem is encountered when calculating the contribution of, e.g., Fig. 1 (b), using the bosonization scheme because this graph contains fermionic Green func- tions at coinciding momenta and frequencies but the in- teraction lines may carry any momenta and frequencies. We will come back to this point when discussing in detail the diagrammatics of the bosonized theory. The uncertainty of the expression in the second line of Eq. (2.35) is reflected in the existence of solutions [like those given by Eq. (2.34)] of the homogeneous equation corresponding to Eq. (2.32). but the expression in the second line is not defined. Of course, one could speak about the limit q → 0 instead of just putting q = 0 but this is not justified for any finite physical system with discrete energy levels. density of states at the Fermi surface ∑ p 1 2 T cosh − 2 ( ε ′$_{p}$ 2 T ) = ∂n ∂µ (2.36) V total r,r ′ = { V$_{r,r}$ ′ , | r | , | r $^{′}$| < R$_{0}$ 0 , otherwise (2.37) T ∑ ε, p 1 iε − ε ′$_{p}$ 1 iε + iω − ε ′$_{p}$ + q = ∑ p n$_{p}$ n$_{p}$$_{+}$$_{q}$ iω − ε ′$_{p}$ + q + ε ′$_{p}$ (2.35) 2. Regularization of the bosonized model. 6