By Peter Kopietz

The writer offers intimately a brand new non-perturbative method of the fermionic many-body challenge, enhancing the bosonization method and generalizing it to dimensions *d*1 through practical integration and Hubbard--Stratonovich variations. partly I he basically illustrates the approximations and obstacles inherent in higher-dimensional bosonization and derives the perfect relation with diagrammatic perturbation thought. He indicates how the non-linear phrases within the strength dispersion may be systematically integrated into bosonization in arbitrary *d*, in order that in *d*1 the curvature of the Fermi floor should be taken under consideration. half II supplies functions to difficulties of actual curiosity. The publication addresses researchers and graduate scholars in theoretical condensed subject physics.

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**Example text**

As will be shown in Chap. 5, in this book we shall be able to treat the full quantum dynamics of the HubbardStratonovich field non-perturbatively – we shall neither rely on saddle point approximations, nor on the naive perturbative calculation of fluctuation corrections around saddle points! 46 3. Hubbard-Stratonovich transformations 4. Bosonization of the Hamiltonian and the density-density correlation function We use our functional integral formalism to bosonize the Hamiltonian of an interacting Fermi system with two-body density-density interactions.

Note also that −1 the appearance of f˜q is only an intermediate step in our calculation. The final expressions for physical correlation functions can be written entirely in terms of f˜q , and remain finite even if this matrix is not positive definite. Such a rather loose use of mathematics is quite common in statistical field theory, although for mathematicians it is certainly not acceptable. Formally, −1 the appearance of f˜q at intermediate steps can be avoided with the help of the two-field Hubbard-Stratonovich transformation discussed in Sect.

54 4. Bosonization of the Hamiltonian and . . The generalized closed loop theorem implies that the Gaussian approximation is justified in a parameter regime where the approximations (A1) and (A2) discussed in Sect. 1 are accurate. We now calculate the vertices U1 and U2 . 19) k α is the number of occupied states in sector KΛ,λ in the non-interacting limit. Thus, α Skin,1 {φα } = i φα . 20) 0 N0 α The second-order vertex is given by U2 (q1 α, q2 α′ ) = −δq1 +q2 ,0 α′ 1 2β 2 ′ Θα (k)Θα (k + q 2 )G0 (k)G0 (k + q2 ) k α + Θ (k)Θ (k + q 1 )G0 (k)G0 (k + q1 ) .