in this appendix are an extension of those obtained in appendix A to the case of finite δ . Assuming that δ is small we expand the field ˜ φ$_{r}$ ′ ( τ + δ ) entering Eqs. (4.33-4.36) in this parameter and repre- sent the eigenvalue problem of the operator ˜ H (+) r,r ′ ( τ ), Eq. (4.36), in the form We make a further transformation of Eq. (B7) using Eq. (2.26) and a corresponding equation written in the absence of the HS field φ$_{rσ}$ . Subtracting these equations from each other we obtain Substituting Eq. (B8) into Eq. (B7) we reduce the func- tion Z [ φ ] to the following form where the hermitian operator H r,r ′ ( z ) is defined in Eq. (2.40) and the matrix ˆ δ$_{r,r}$ ′ ( z ) can be written as Eq. (C1) is a direct generalization of Eq. (2.48) to the case of the split times. We assume that the eigenvalue ˜ E is close to zero. The regularization with the finite γ leads to level repulsion. All eigenvalues E K calculated for δ = 0 remain finite and real. In the limit γ → 0 the state with k = k ′ drops out from the sum and we obtain finally using the orthogonality of the functions w k r Considering finite δ drastically changes the situation even if δ is infinitesimally small. This is a consequence of the fact that the matrix ˆ δ$_{r,r}$ ′ ( z ) is not hermitian and therefore the operator ˜ H (+) r,r ′ ( τ ) acting on S$_{r,r}$ ′ ( z ) in the L.H.S. of Eq. (C1) is not hermitian, too. This means that the eigenvalues ˜ E can now be complex. In order to calculate the eigenvalue ˜ E we use, as in ap- pendix A the degenerate perturbation theory considering the matrices ˆ δ$_{r,r}$ ′ ( z ) and γ Λ as a perturbation. We represent the solution of Eq. (C1) in the form of Eq. (A1, A2) and multiply both sides of Eq. (C1) by S a ∗ r,r ′ ( τ ) and then by S b ∗ r,r ′ ( τ ) , summing after that over r, r $^{′}$and integrating over τ . As a result, we obtain equa- tions for the coefficients c$_{1}$ and c$_{2}$ We see that, containing the second sum, Eq. (B10) is somewhat different from Eq. (2.21). At this point we should recall that the system we consider contains a bath making the number of states large. The normal- ized eigenfunctions w k r are proportional to ( N total d ) − 1 / $^{2}$. Therefore, the second sum in the exponent in Eq. (B10) is much smaller than the first term and can be neglected provided the total number of the sites in the system N total d is large. This example of the static HS field demonstrates very well how the regularization works. with positive numbers x$_{1}$ and x$_{2}$ given by Eq. (A4). The complex parameters ˜ λ$_{1}$ and ˜ λ$_{2}$ have the form In this appendix we demonstrate that keeping in Eqs. (4.33-4.36) the parameter δ finite one can obtain imaginary part in the eigenvalues ˜ E K of the operator ˜ H (+) r,r ′ ( τ ), Eq. (4.36). The finite δ appears because the times τ and τ ′ entering the Green function G ( uφ ) r,r ′ ( τ, τ $^{′}$) should be slightly different, τ ′ = τ + δ . Although the value of δ can be infinitesimally small, the imaginary part of ˜ E K can be very important in situations when the eigenvalue ˜ E K is close to zero. The results derived where the functions v$_{r,r}$ ′ ( τ ) and ¯ v$_{r,r}$ ′ ( τ ) and the complex eigenvalues λ are introduced in Eqs. (2.46, 2.48). In the limit | λ | → 0 the functions v$_{r,r}$ ′ ( τ ) and ¯ v$_{r,r}$ ′ ( τ ) are related to each other as v$_{r,r}$ ′ ( τ ) ≃ ¯ v$_{r}$ ′ ,r ( τ ) (C5) ˜ λ$_{1}$ = σu ∑ r,r ′ ∫ β 0 ¯ v$_{r,r}$ ′ ( τ ) ∂φ$_{r}$ ′ ( τ ) ∂τ v$_{r,r}$ ′ ( τ ) dτ, (C4) ˜ λ$_{2}$ = σu ∑ r,r ′ ∫ β 0 ¯ v ∗ r,r ′ ( τ ) ∂φ$_{r}$ ( τ ) ∂τ v ∗ r,r ′ ( τ ) dτ ( ˜ E − γ ) x$_{1}$c$_{1}$ − c$_{2}$ ( λ + ˜ λ$_{1}$δ ) = 0 (C3) − λ ∗ − ˜ λ$_{2}$δ ) c$_{1}$ + ( ˜ E + γ ) x$_{2}$c$_{2}$ = 0 Z [ φ ] = Z$_{0}$ exp [ ∫ β 0 ∫ 1 0 [ ∑ r,σ σφ$_{rσ}$ ( G ( uφ ) r,r ; σ ( τ, τ + 0) − G (0) r,r ( τ, τ + 0) ) − ∑ r,r$_{1}$,r ′ $_{1}$,σ,k σφ$_{rσ}$w k $_{r}$w k k $_{r$_{1}$}$w k r ′ 1 ( G ( uφ ) r$_{1}$,r ′ $_{1}$; σ ( τ, τ + 0) − G (0) r$_{1}$,r ′ 1 ( τ, τ + 0) )]] dτ du (B10) Z [ φ ] = Z$_{0}$ exp [ ∑ r,r$_{1}$,r ′ $_{1}$,σ,k,k ′ ∫ 1 0 ∫ β 0 dudτ σφ$_{rσ}$w k $_{r}$w k ′ r w k $_{r$_{1}$}$w k ′ r ′ 1 × ( λ k − λ k ′ ) 2 u ( λ k − λ k $^{′}$) 2 + γ 2 ( G ( uφ ) r$_{1}$,r ′ 1 ( τ, τ + 0) − G (0) r$_{1}$,r ′ 1 ( τ, τ + 0) )] (B9) ˆ δ$_{r,r}$ ′ ( z ) = ( 0 σuδ ∂φ$_{r}$ ′ ( τ ) ∂τ − σuδ ∂φ$_{r}$ ( τ ) ∂τ 0 ) (C2) ˜ H (+) r,r ′ ( τ ) S$_{r,r}$ ′ ( z ) = ˜ ES$_{r,r}$ ′ ( z ) , (C1) ˜ H (+) r,r ′ ( τ ) = H r,r ′ ( z ) + ˆ δ$_{r,r}$ ′ ( z ) , − uσ ( φ$_{rσ}$ − φ$_{r}$ ′ $_{σ}$) n$_{r,r}$ ′ = ( ˆ ε$_{r}$ − uσφ$_{r}$ − ˆ ε$_{r}$ ′ + uσφ$_{r}$ ′ )( G ( uφ ) r,r ′ ; σ ( τ, τ + 0) − G (0) r,r ′ ( τ, τ + 0) ) (B8) 30 Appendix C: Splitting times in the equation for A$_{r,r}$ ′ ( z ) .