9 bath. This allows us to treat irregular expressions in a simple way because the translational invariance of the subsystem (where the interaction is finite) is broken and Green functions with coinciding momenta and frequen- cies give a vanishing contribution in the thermodynamic limit considered in this Section. Let us explain this procedure in more details. Suppose, we have an equation enforces in a very natural way the analogy between the matrix A and the density matrix in quantum mechanics. One can say that our bosonization procedure corresponds to replacing the description of the quantum mechanics in terms of the wave functions by the description in terms of the density matrix. In the classical limit, Eqs. (2.32, 2.39) play a role of the kinetic equation and A$_{r,r}$ ′ ( z ) (after Fourier transforming in r − r $^{′}$) can be considered as the distribution function for a particle moving in a fluctuating field in imaginary time. We see that the original problem of calculating the partition function of interacting fermions Z , Eq. (2.4), has been reduced after the regularization to solving Eqs. (2.39, 2.40) for a bosonic function A$_{r,r}$ ′ ( z ), sub- stituting the solution into Eq. (2.43) and calculating the functional integral over φ$_{rσ}$ ( τ ) in Eq. (2.18). For an an- alytical investigation, this integral over φ$_{rσ}$ ( τ ) can be performed before doing any approximations and this is demonstrated in the next Section. where F ( A ) is a real matrix function of a real matrix function A , and our task is to find the solution A$_{0}$ of this equation and calculate a quantity B ( A$_{0}$ ) , where B ( A ) is another matrix function. Instead of proceeding in this way, one can write B ( A$_{0}$ ) in a form of the integral ∣$^{det}$ ∣$^{da}$ over the variable a of the same structure as A . The mod- ulus of the determinant in Eq. (3.2) is included in the integrand in order to normalize the δ -function. As the next step, we represent the δ -function as where f is a real matrix variable and the symbol “ T ” means the transposition of r and r $^{′}$, f T r,r ′ = f$_{r}$ ′ ,r . C is a normalization constant. The determinant in Eq. (3.2) can also be represented in a form of an integral but now the variables of integration should be anticommuting Grassmann variables η and η $^{+}$, In this subsection, we reduce the computation of the partition function Z , Eq. (2.4), to calculations with a su- perfield model describing interacting bosonic excitations. Eq. (2.32) resembles quasiclassical equations written in Ref. 15. The solution of these equations was represented in terms of a functional integral over 48-component su- pervectors, which allowed the authors to integrate over the HS field. Unfortunately, the resulting Lagrangian and calculations with it were rather cumbersome due to the large number of the components of the supervectors. Now we use another trick, known as the Becchi-Rouet- Stora-Tuytin (BRST) transformation 28 (see also the book by Zinn-Justin $^{31}$). A similar transformation was used in the quantization of non-abelian gauge theories $^{32}$. In condensed matter physics, this trick has been used for the first time in Ref. 33. Using this transformation one can also represent the solution of an equation (not necessarily linear) in terms of a functional integral over both conventional and Grassmann anticommuting fields. Due to some special symmetries, both the types of the fields can be unified in a superfields depending not only on coordinates and times but also on additional anticom- muting variables. In order to avoid complicated formulas containing 2 × 2 matrices that appear as a result of the regularization, we restrict our consideration in this Section by the thermo- dynamic limit. In this limit the spectrum is continuous and the states with coinciding momenta and frequen- cies do not give any essential contribution anyway. This means that we put the parameter γ = 0 . At the same time, we still assume that the system is imbedded in the Writing the quadratic forms in the exponent using the variables a , f $^{T}$, η , and η + is not difficult and we do not write them in this form. A more compact form of At first glance, there should be a problem related to the presence of the modulus in Eq. (3.2) and absence of it in Eq. (3.4). Fortunately, the operator ∂/∂τ + M$_{r,r}$ ′ ( z ) is real and antisymmetric, which leads to an always positive determinant. Therefore, the modulus in Eq. (3.2) does not play any role in the case under consideration. Eq. (2.32) is linear and real and we can apply Eq. (3.2) directly. Using the transformation of Eq. (3.3) we obtain in the exponent a quadratic form in terms of the variables a and f . Then, we write the determinant with the help of Eq. (3.4) and obtain another quadratic form in the exponent containing the anticommuting variables η and η $^{+}$. After unifying the fields a , f $^{T}$, η , and η + in a super- field Ψ, the expression in the exponential can be written in a compact way. We do not present here general for- mulas that can be found in the book 31 but concentrate explicitly on Eqs. (2.31,2.32). det ( ∂F ∂a ) = ∫ exp [ − Tr ( η $_{+}$∂F ∂a η )] dη $^{+}$dη. (3.4) δ [ F ( a )] = C ∫ exp [ i Tr ( f $^{T}$F ( a ) )] df (3.3) B ( A$_{0}$ ) = ∫ B ( a ) δ [ F ( a )] ∣ ∣$^{det}$ ( ∂F ∂a )∣ ∣ ∣$^{da}$ (3.2) a of the same structure as A . The mod- F ( A ) = 0 , (3.1) III. SUPERFIELD THEORY OF INTERACTING BOSONS AND PERTURBATION THEORY A. Superfield theory of interacting bosons