C. The negative sign problem for fermions In the static case, when φ$_{r,l}$ does not depend on l , the product in Eq. (4.16) is trivially calculated because the multipliers commute with each other. Then one comes to the standard expression for the partition function of fermions in a static external po tential, which is clearly positive. If the function φ$_{r,l}$ varies sufficiently slowly in time, the adiabatic approximation can be used and the function Z$_{f}$ [ ˜ φ ] takes the form 22 spins “up” and “down” [ N total d is the number of the eigen- functions corresponding to the total number of the sites in the system]. The matrices P$_{±}$ are real but not neces- sarily symmetric. For fields ˜ φ ( τ ) slowly varying in time, the eigenfunc- tions v ( k$_{l}$ ) r,l on the neighboring slices are almost equal to each other and the matrices π$_{l}$ are close to the unity ma- trix. In this case, the matrices π$_{l}$ can be calculated using the standard perturbation theory for wave functions where δφ$_{r,l}$ = φ$_{r,l}$$_{+1}$ − φ$_{r,l}$ is small. Then, the off-diagonal elements of the matrices π$_{l}$ take in the main order in δφ the following form where λ ( k ) l is the eigenenergy of k -state of the Hamilto- nian ˆ h [ ˜ φ ] obtained on each slice l . This expression is, again, positive. Actually, the negative sign of Z$_{f}$ [ ˜ φ ] may arise only when the function ˜ φ$_{r,l}$ strongly fluctuates in time. Let us see how it happens. For a given ˜ φ ( τ ), we con- sider a matrix element P$_{nm}$ of the product in Eq. (4.16) for one of the spin directions  λ l − λ l In the limit Δ ≪ α, the characteristic values of the quantities | δφ$_{r,l}$ | follow from Eqs. (4.12, 4.13) while the average values of φ$_{r,l}$ and δφ$_{r,l}$ vanish. It is clear from Eq. (4.24) that | δφ$_{r,l}$ | ≪ | φ$_{r,l}$ | and this justi- fies using the perturbation theory for computation of the matrix elements π mn l , Eq. (4.24). As we assume that α − 1 ≫ { t$_{r,r}$ ′ , V$_{0}$ } , the wave func- tions φ$_{r,l}$ are almost localized on the sites r$_{n}$ and the states n are characterized by the positions of centers of the localization. Then, we estimate the denominators in Eq. (4.24) as Completely localized wave functions cannot give a contri- bution to non-diagonal matrix elements ( δφ$_{r,l}$ ) mn in Eq. (4.24) and one should take into account their overlap at finite t$_{r,r}$ ′ leading to finite values of the wave functions on sites in the neighborhood of the center of the localization r$_{0}$ . These values can be estimated [considering the tun- neling term t$_{r,r}$ ′ as a perturbation] as t$_{r,r}$$_{0}$ | φ$_{r,l}$ − φ$_{r}$$_{0}$$_{,l}$ | − 1 and one comes to typical absolute values of randomly fluctuating off-diagonal elements of the matrix π$_{l}$ The estimate (4.27) is written under the assumption that all the states are non-degenerate, which is most proba- ble for randomly chosen fields φ$_{r,l}$ . The matrix elements π mn l , Eq. (4.27), are small in the limit Δ → 0 but the matrix P , Eq. (4.20), contains a product of N = β/ Δ matrices π$_{l}$ and can essentially be different from a diago- nal one. Since the matrix elements π$_{l}$ fluctuate randomly where N total d × N total d matrices P$_{±}$ are matrices with real matrix elements ( P$_{±}$ )$_{mn}$, Eq. (4.18), corresponding to are elements of orthogonal matrices π$_{l}$ , and write the function Z$_{f}$ [ ˜ φ ] as where we represent the matrix element P$_{nm}$ in the form where v ( m ) r,l is the m -th eigenfunction of the Hamiltonian ˆ h$_{l}$ ( ˜ φ ) at the l -th slice [we assume that the eigenfunctions v ( m ) r,l are real and the product in Eq. (4.18) is ordered in time from the right to the left]. Using the completeness of the sets of the wave func- tions v ( k$_{l}$ ) r,l for each slice l, 18 v ( m ) r,l +1 = v ( m ) r,l + ∑ k = m ( δφ$_{r,l}$ ) km λ ( m ) l − λ ( k ) l v ( k ) r,l , (4.23) Z$_{f}$ [ ˜ φ ] ≈ ∏ k [ 1 + exp ( − Δ N ∑ l =1 λ ( k ) l )] (4.17) π mn l =    ( δφ$_{r,l}$ ) nm λ ( m ) l − λ ( n ) l , m = n − 1 2 ∑ k = n [ ( δφ$_{r,l}$ ) nk λ ( k ) l − λ ( n ) l ] 2 , m = n . (4.24) | φ$_{r,l}$ | ∼ ( V$_{0}$/α ) 1 / 2 , | δφ$_{r,l}$ | ∼ ( V$_{0}$ Δ /α $^{2}$) 1 / 2 , (4.25) P$_{nm}$ = ∑ r v ( n ) r, 1 [ N ∏ l =1 exp ( − ˆ h$_{l}$ Δ ) ] v ( m ) r, 1 (4.18) ∣ λ ( m ) l − λ ( n ) l ∣ = | φ$_{r}$$_{m}$$_{,l}$ − φ$_{r}$$_{n}$$_{,l}$ | ∼ ( V$_{0}$/α ) 1 / 2 (4.26) ∑ k v ( k ) r,l v ( k ) r ′ ,l = δ$_{rr}$ ′ , (4.19) P$_{nm}$ = ∑ { k$_{l}$ } δ$_{n,k}$$_{N}$ δ$_{k}$$_{1}$$_{m}$ exp ( − Δ N ∑ l =1 λ ( k$_{l}$ ) l ) N ∏ l =1 π k$_{l}$$_{+1}$k$_{l}$ l , (4.20) π k$_{l}$$_{+1}$k$_{l}$ l = ∑ r v ( k$_{l}$$_{+1}$ ) r,l +1 v ( k$_{l}$ ) r,l , (4.21) Z$_{f}$ [ ˜ φ ] = det [1 + P$_{+}$ ] det [1 + P$_{−}$ ] , (4.22) | π mn l | ∼ ( t $^{2}$Δ /V$_{0}$ ) 1 / 2 , m = n (4.27)