with a gaussian distribution, we estimate the off-diagonal elements of the product of N matrices entering Eq. (4.20) as ( βt 2 r,r ′ /V$_{0}$ ) 1 / $^{2}$. This is a small number in the limit of sufficiently high temperatures where I$_{M}$ is the unity matrix, and are not necessarily real. However, as the matrix P is real, for any solution p$_{i}$ of Eq. (4.31) its complex conjugate p ∗ i is also a solution. One can see easily from Eqs. (4.30, 4.31) that As we have mentioned previously, in the adiabatic limit (4.28), when the function ˜ φ$_{r}$ ( τ ) is smooth in time, one obtains and Eq. (4.28) determines the adiabatic limit. In this limit one comes to Eq. (4.17) and the fermionic deter- minant Z$_{f}$ [ ˜ φ ] must be positive. Alternatively, one could take lower temperatures but large α ∼ β , which would lead to the same result. Let us discuss now what happens if α ≪ β and the inequality (4.28) is not fulfilled. Using Eq. (4.20) we write det P for one of the spin directions in the form and one comes to inequalities p$_{i}$ > 0 for all i . Deforming the function ˜ φ$_{r}$ ( τ ), such that it can vary fast in the inter- val [0 , β ], makes the matrix ∏ N l $_{=1}$π$_{l}$ essentially different from unity and some of the eigenvalues p$_{i}$ can become negative, which is a necessary condition for obtaining negative values of det [1 + P ]. In the limit Δ ≫ α , all the matrix elements of the ma- trices π$_{l}$ can be typically of order one and the changes of p$_{i}$ can be large after a minimum change of a configu- ration of the field φ$_{r,l}$ . In this case nothing prevents p$_{i}$ from jumping from positive to large negative values and making det [1 + P$_{±}$ ] negative. In the opposite limit α ≫ Δ, a smooth deformation in time of the field φ$_{r,l}$ results in a smooth motion of the eigenvalues p$_{i}$ . If we start with φ$_{r,l}$ slowly varying in the interval [0 , β ], all p$_{i}$ are real and positive. Deforming the function φ$_{r,l}$ results in a motion of the parameters p$_{i}$ along the real axis. They cannot cross zero because this would mean det P ↑ , ↓ = 0 , which is impossible because det π$_{l}$ = 1 for all l . However, two different values p$_{i}$ and p$_{k}$ can collide and move to the complex plane forming a pair of complex con- jugated values p and p $^{∗}$. Again, this cannot change the sign because these two eigenvalues lead to a multiplier | 1 + p | 2 in the determinant. A possibility to change the sign of the fermionic determinant Z$_{f}$ [ ˜ φ ] first arises when the values p and p ∗ collide again on the real axis at a point p$_{0}$ < 0. After this event one obtains again two real neg- ative eigenvalues p$_{i}$ and p$_{k}$ . Then, they move separately and one of them may cross the point − 1 leading to a neg- ative value of det [1 + P$_{±}$ ]. Still, this does not necessarily mean that the full fermionic determinant, Eq. (4.16), be- comes negative because the product of the contributions corresponding to spin “up” and spin “down” should be taken but the positivity of the fermionic determinant is no longer guaranteed. We see that, when deforming continuously the function ˜ φ$_{r,l}$ , a chain of several events must happen before the fermionic determinant becomes negative. Therefore, the probability of negative values of Z$_{f}$ [ ˜ φ ] may be reduced in the continuous limit, Δ ≪ α, with respect to the opposite one, Δ ≫ α , but one should not expect that it vanishes. For diagonalizable matrices each block in the matrix J is just a number and all K$_{i}$ = 1, such that n = M . The numbers p$_{i}$ are solutions of the equation The sizes K$_{i}$ of the blocks J$_{i}$ depend on the form of the matrix P but and where the block diagonal matrix J can be written as Matrix P given by Eq. (4.20) is real but not sym- metric. Such matrices cannot generally be diagonalized but one can always find a Jordan normal form for them. Assuming that P is an M × M real matrix one writes As the matrices π$_{l}$ are orthogonal, one comes to values det π$_{l}$ = 1 for all l and to positive values of det P in Eq. (4.29). Of course, this does not necessarily mean that the function Z$_{f}$ [ ˜ φ ], Eq. (4.22), is positive, too. det ( pI$_{M}$ − P ) = 0 , (4.31) n ∑ i =1 K$_{i}$ = M. J$_{i}$ =    p$_{i}$ 1 0 0 0 0 p$_{i}$ 1 0 0 · · · · · 0 0 0 p$_{i}$ 1 0 0 0 0 p$_{i}$   $^{.}$ J =    J$_{1}$ 0 0 0 0 0 J$_{2}$ 0 0 0 · · · · · 0 0 0 J$_{n}$$_{−}$$_{1}$ 0 0 0 0 0 J$_{n}$    P = QJQ − $^{1}$, (4.30) det P = exp ( − Δ N ∑ l =1 N total d ∑ k =1 λ ( k ) l ) N ∏ l =1 det π$_{l}$. (4.29) N ∏ l =1 π$_{l}$ ≈ 1 , det [1 + P ] = n ∏ i =1 (1 + p$_{i}$ ) K$_{i}$ , det P = n ∏ i =1 p K$_{i}$ i (4.32) T ≫ ( t 2 r,r ′ /V$_{0}$ ) 1 / 2 (4.28) 19