3. Spectral expansion. The orthogonality condition for the eigenvectors S K r,r ′ ( τ ) follows from Eq. (2.48) and can be written as In this subsection we discuss important properties of the solutions of Eqs. (2.39, 2.40) using spectral expan- sions in eigenfunctions of this equation. In order to better understand the difference between solutions of Eq. (2.32) and Eqs. (2.39, 2.40), let us consider both the equations. The operator in the L.H.S. of Eq. (2.32) is not her- mitian, which results in a double set of eigenfunctions { v K r,r ′ ( τ ) } and { ¯ v K r,r ′ ( τ ) } satisfying the equations where the symbol “+”stands for the complex conjugation and transposition “ T ”. The completeness of the system of the eigenvectors S expressed by the relation allows us to write the spectral expansion for the 2 × 2 ma- trix Green function G r,r ′ ; r$_{1}$,r ′ 1 ( τ, τ$_{1}$ ) satisfying the equa- tion with the boundary conditions v K r,r ′ ( τ ) = v K r,r ′ ( τ + β ), ¯ v K r,r ′ ( τ ) = ¯ v K r,r ′ ( τ + β ). Two other equations can be writ- ten taking the complex conjugate of Eqs. (2.46), in the form The orthogonality conditions for the functions v K r,r ′ can be written as Since all eigenvalues E K are not equal to zero, the Green function G r,r ′ ; r$_{1}$,r ′ 1 ( τ, τ$_{1}$ ), Eq. (2.52), is uniquely defined and does not have singularities. Using the properties of the eigenvectors S K r,r ′ ( τ ) explained after Eq. (2.48) we obtain a symmetry property for these functions With the Green function G r,r ′ ; r$_{1}$,r ′ 1 ( τ, τ$_{1}$ ) , Eqs. (2.51, 2.52), we write the function Z [ φ ], Eq. (2.43), in the form (the same for the complex conjugates). The eigenvalues λ K are generally complex and λ K = 0 is not excluded. For example, the function A (0) r,r ′ , Eq. (2.34), corresponds to the zero eigenvalue for static φ$_{rσ}$ . All non-zero eigenvalues λ K appear in pairs. If an eigenvalue λ K with an eigenfunction v K r,r ′ ( τ ) exists, then the eigenvalue − λ K with the eigenfunction ¯ v K r ′ ,r ( τ ) ex- ists, too. We see that nothing prevents the eigenvalues λ K from turning at some points to zero and being com- plex, which is an unpleasant feature of the theory. In contrast, the operator H r,r ′ ( τ ) is hermitian. The structure of the two-component vectors S K r,r ′ ( τ ) can be represented as Eqs. (2.52, 2.54) give an explicit unambiguous expres- sion for the function Z [ φ ]. It can be further integrated over φ$_{rσ}$ using superfields, as it is done in the next Sec- tion, or calculated numerically. In the latter case, it is more convenient to solve directly Eq. (2.39) and substi- tute the solution into Eq. (2.43). We check our regularization scheme in Appendix B on the model with a static HS field φ$_{rσ}$ . One can interpret the real function A$_{r,r}$ ′ ( z ) as a den- sity matrix, while Eqs. (2.32, 2.39) are analogues of the von Neumann equation. In this language, Eq. (2.44) is the particle conservation law. Using the density matrix in quantum mechanics implies a connection of the system to an environment. Within our approach, we use the reg- ularization based on the introduction of the bath. This and they can be found from the equation supplemented by the boundary condition S K r,r ′ ( τ ) = S K r,r ′ ( τ + β ), where the eigenvalues E K must be real. The eigenstates with eigenfunctions S K r,r ′ ( τ ) and eigen- values E K have their counterparts with the eigenfunc- tions Λ$_{1}$ S K r ′ ,r ( τ ) and eigenvalues − E $^{K}$. Z [ φ ] = Z$_{0}$ exp [ 1 2 ∑ r,r$_{1}$,r ′ $_{1}$,σ ∫ β 0 ∫ 1 0 φ$_{rσ}$ ( τ ) × ( G 12 r,r ; r$_{1}$,r ′ 1 ( τ, τ$_{1}$ ; u, σ ) + G 21 r,r ; r$_{1}$,r ′ 1 ( τ, τ$_{1}$ ; u, σ ) × un$_{r}$$_{1}$$_{,r}$ ′ $_{1}$Φ$_{r}$$_{1}$$_{,r}$ ′ $_{1}$; σ ( τ ) dτ du ] (2.54) H r,r ′ ( τ ) S K r,r ′ ( τ ) = E $^{K}$SK r,r ′ ( τ ) (2.48) S K = ( a K b K ) , ( S $^{K}$) T = ( a K b K ) ∑ r,r ′ ∫ β 0 ¯ v K r,r ′ ( τ ) v K ′ r,r ′ ( τ ) dτ = δ K,K ′ G r,r ′ ; r$_{1}$,r ′ 1 ( τ, τ$_{1}$ ) = − Λ$_{1}$ G r ′ ,r ; r ′ $_{1}$,r$_{1}$ ( τ$_{1}$, τ ) Λ$_{1}$ (2.53) G r,r ′ ; r$_{1}$,r ′ 1 ( τ, τ$_{1}$ ) = ∑ K S K r,r ′ ( τ ) ( S K r$_{1}$,r ′ 1 ( τ$_{1}$ ) ) + E K (2.52) ( ∂ ∂τ + M$_{r,r}$ ′ ) v K ∗ r,r ′ ( τ ) = λ K $^{∗}$v K ∗ r,r ′ ( τ ) , (2.47) ( − ∂ ∂τ + M$_{r,r}$ ′ ¯ v K ∗ r,r ′ ( τ ) = λ K $^{∗}$¯ v K ∗ r,r ′ ( τ ) . ( ∂ ∂τ + M$_{r,r}$ ′ ) v K r,r ′ ( τ ) = λ $^{K}$vK r,r ′ ( τ ) , (2.46) ( − ∂ ∂τ + M$_{r,r}$ ′ ¯ v K r,r ′ ( τ ) = λ $^{K}$¯ v K r,r ′ ( τ ) ∑ K S K r,r ′ ( τ ) ( S K r$_{1}$,r ′ 1 ( τ$_{1}$ ) ) + = δ$_{r,r}$$_{1}$δ$_{r}$ ′ ,r ′ $_{1}$δ ( τ − τ $^{′}$) (2.50) H r,r ′ ( τ ) G r,r ′ ; r$_{1}$,r ′ 1 ( τ, τ$_{1}$ ) = δ$_{r,r}$$_{1}$δ$_{r}$$_{1}$$_{,r}$ ′ $_{1}$δ ( τ − τ$_{1}$ ) (2.51) ∑ r,r ′ ∫ β 0 ( S K r,r ′ ( τ ) ) $^{+}$S K ′ r,r ′ ( τ ) dτ = δ K,K ′ (2.49) 8