By Peter Kopietz

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

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**Sample text**

Note that in this case |k −kα | assumes the smallest possible value. to choose the sectors as large as possible in order to avoid corrections due to the around-the-corner processes. 59)) should be well-defined. If these conditions are satisfied, we may use our formalism to calculate the single-particle Green’s function G(k α + q, i˜ ωn ) for all wave-vectors q that are small compared with the sector cutoffs. e. fqαα = fq , such as the long-range tail of the Coulomb interaction), we may identify the entire momentum space with a single sector.

Hubbard-Stratonovich transformations Our functional bosonization approach is based on two Hubbard-Stratonovich transformations, which are described in detail in this chapter. We start with the imaginary time functional integral formulation of quantum statistical mechanics. 7], so that we can be rather brief here and simply summarize the relevant representations of fermionic correlation functions as Grassmannian functional integrals. 1]. These can be viewed as a clever change of variables to collective coordinates in functional integrals, which exhibit the physically most relevant degrees of freedom.

Note that in the work by Houghton et al. 34] it is implicitly assumed that the Gaussian approximation is justified. However, in none of these works the corrections to the Gaussian approximation have been considered, so that the small parameter which actually controls the accuracy of the Gaussian approximation has not been determined. 1) truncates at the second order. In this case we have exactly ˆ 0 Vˆ ˆ 0 Vˆ ] = Tr G ˆ 0 Vˆ + 1 Tr G −Tr ln[1 − G 2 2 . 1]. A few years later T. 2], and formulated it as a theorem, which he called the closed loop theorem.