In the opposite limit, α ≫ Δ , the fluctuations of the difference of the fields at different slices are strongly sup- pressed and one can replace the difference in Eq. (4.9) by the derivative. Then, we can use Eq. (4.5) for estimates to obtain while the average squared difference of ˜ φ at neighboring slices equals Eqs. (4.12, 4.13) show that in the limit α ≫ Δ the field ˜ φ$_{r}$ ( τ ) changes slowly from slice to slice. Now we are prepared to discuss the origin of the sign problem and how it can be avoided using the bosoniza- tion. Although the HS field is now discontinuous, Eq. (4.8) is a standard continuous time representation for the par- tition function. This means that one can use Eqs. (2.21- 2.25) replacing everywhere the function φ$_{r}$ ( τ ) by ˜ φ$_{r}$ ( τ ). For example, the equation for the Green function takes the form Instead of using Eq. (4.6) we write the partition func- tion Z as Δ → 0 ∫ W [ φ ] D φ where ˜ φ$_{r}$ ( τ ) = φ$_{r,l}$ for ( l − 1) Δ ≤ τ < l Δ and the func- tional Z$_{f}$ [ ˜ φ ] equals The weight ˜ W [ ˜ φ ] is chosen in the form We assume, as for the continuous limit, that ˜ φ$_{r}$ ( τ ) = ˜ φ$_{r}$ ( τ + β ) . Of course, the representations (4.6) and (4.7- 4.9) are equivalent in the continuous limit. In Eq. (4.8) one integrates the exponent over each slice instead of taking it in the middle of the slices as it is done in the second of Eqs. (4.6). The function ˜ φ$_{r}$ ( τ ) is discontinuous and its typical form is represented in Fig. 6. The fluctuations of this function are essentially dependent on the ratio α/ Δ. In the limit α ≪ Δ, the typical average value 〈 [ ˜ φ$_{r}$ ( τ )] $^{2}$〉 ˜ W calculated with ˜ W$_{α}$ , Eq. (4.9), equals In Eq. (4.16), the multipliers are ordered in l , such that the multiplier with l = 1 is on the right. The same representation is usually used in the standard MC simulations [Actually, one replaces each multi- plier in the second line of Eq. (4.16) by the product exp ( − (ˆ ε$_{r}$ − µ $^{′}$) Δ) exp ( σφ$_{r,l}$ Δ) , which is clearly justi- fied in the limit Δ ≪ β ]. Eq. (4.16) was obtained in Refs. 22,25 as a result of discretizing the time. At the same time, we see from Eqs. (4.14, 4.15) that only the HS field ˜ φ$_{r}$ ( τ ) is discrete now, while the derivative ∂/∂τ is continuous as before. This allows us to bosonize the fermion model even in this discrete time representation. However, before making the proper transformations, let us discuss the origin of negative signs of Z$_{f}$ [ ˜ φ ]. whereas the correlation at neighboring slices is which shows that 〈 [ ˜ φ$_{r}$ ( τ ) − ˜ φ$_{r}$ ( τ +Δ)] $^{2}$〉 ˜ W and 〈 [ ˜ φ$_{r}$ ( τ )] $^{2}$〉 ˜ W are of the same order. In the same way as in Ref. 22 we can reduce the func- tional Z$_{f}$ [ ˜ φ ] to the form Substituting the solution of Eq. (4.14) into Eq. (2.21) one comes to the fermionic determinant for Z$_{f}$ [ ˜ φ ], 〈 [ ˜ φ$_{r}$ ( τ ) ] 2 〉 ˜ W = V$_{0}$/ (2 α ) , (4.12) 〈 ( ˜ φ$_{r}$ ( τ ) − ˜ φ ( τ + Δ) ) 2 〉 ˜ W = V$_{0}$ Δ /α 2 ≪ 〈 [ ˜ φ$_{r}$ ( τ ) ] 2 〉 ˜ W (4.13) Z = lim Δ → 0 ∫ Z$_{f}$ [ ˜ φ ] ˜ W [ ˜ φ ] D ˜ φ ˜ W [ ˜ φ ] D ˜ φ (4.7) ( − ∂ ∂τ − ˆ h [ u ˜ φ ] ) G ( uφ ) r,r ′ ; σ ( τ, τ $^{′}$) = δ$_{r,r}$ ′ δ ( τ − τ $^{′}$) , (4.14) ˆ h [ u ˜ φ ] = ˆ ε$_{r}$ − µ ′ − σu ˜ φ$_{r}$ ( τ ) . Z$_{f}$ [ ˜ φ ] = Tr$_{¯}$ c , c [ exp ( − β ˆ H$_{0}$ ) (4.8) × T$_{τ}$ exp ( − ∫ β 0 ∑ r,σ σ ˜ φ$_{rσ}$ ( τ ) ¯ c$_{rσ}$ ( τ ) c$_{rσ}$ ( τ ) dτ )] , ˜ W$_{α}$ [ ˜ φ ] = exp { − 1 2 V$_{0}$ ∑ r ∫ β 0 ( ˜ φ 2 r ( τ ) (4.9) + ( α/ Δ) 2 [ ˜ φ$_{r}$ ( τ + Δ) − ˜ φ$_{r}$ ( τ ) ] 2 ) dτ } Z$_{f}$ [ ˜ φ ] = det r,σ,τ [ − ∂ ∂τ − ˆ h [ ˜ φ ] ] . (4.15) Z$_{f}$ [ ˜ φ ] = det r,σ [ 1 + T$_{τ}$ exp ( − ∫ β 0 ˆ h [ ˜ φ ] dτ ) ] = det r,σ [ 1 + N ∏ l =1 exp ( − (ˆ ε$_{r}$ − µ ′ − σφ$_{r,l}$ ) Δ) ] (4.16) 〈 [ ˜ φ$_{r}$ ( τ ) ] 2 〉 ˜ W = V$_{0}$/ Δ , (4.10) 〈 ˜ φ$_{r}$ ( τ ) ˜ φ$_{r}$ ( τ + Δ) 〉 ˜ W = V$_{0}$α $^{2}$/ Δ 3 ≪ 〈 [ ˜ φ$_{r}$ ( τ ) ] 2 〉 ˜ W , (4.11) FIG. 6: Typical form of the function ˜ φ r ( τ ) . 17