By Peter Kopietz

The writer offers intimately a brand new non-perturbative method of the fermionic many-body challenge, enhancing the bosonization approach and generalizing it to dimensions *d*1 through practical integration and Hubbard--Stratonovich alterations. partially I he sincerely illustrates the approximations and boundaries inherent in higher-dimensional bosonization and derives the right relation with diagrammatic perturbation conception. He indicates how the non-linear phrases within the power dispersion might be systematically incorporated into bosonization in arbitrary *d*, in order that in *d*1 the curvature of the Fermi floor will be taken into consideration. half II supplies functions to difficulties of actual curiosity. The booklet addresses researchers and graduate scholars in theoretical condensed subject physics.

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**Extra info for Bosonization of Interacting Fermions in Arbitrary Dimensions**

**Sample text**

7) can be represented as the functional integral average of ψk ψk† , G(k) = −β D {ψ} e−Smat {ψ} ψk ψk† D {ψ} e−Smat {ψ} . 7) The coupled electron-phonon system will be discussed in detail in Chap. 8. 10). From the Matsubara Green’s function we can obtain the real space imaginary time Green’s function via Fourier transformation, 1 ei(k·r−˜ωn τ ) G(k) . 9) D {ψ} e−Smat {ψ} ψ(r, τ )ψ † (r ′ , τ ′ ) . 10) k we can also write G(r − r′ , τ − τ ′ ) = − Two-particle Green’s functions can also be represented as functional integral averages.

18]. 19]. 5 Curved patches and reduction of the patch number If we do not require that the energy dispersion should be linearized, we are free to subdivide the Fermi surface into a small number of curved patches. In some special cases we may completely abandon the patching construction, and formally identify the entire momentum space with a single sector. Then the around-the-corner processes simply do not exist. Because in this book we shall develop a systematic method for including the non-linear terms of the energy dispersion into higher-dimensional bosonization, we shall ultimately drop the requirement that the variation of the local normal vector within a given patch must be negligible.

However, under certain conditions, which will be described in detail in Chap. 1, S˜eff {ρ˜α } can be approximated by a quadratic form. In this case bosonization enormously simplifies the many-body problem. In a sense, the collective density fields ρ˜α are the “correct coordinates” to parameterize the low-energy excitations of the system. 4 Summary and outlook In this chapter we have used well-known representations of fermionic correlation functions as Grassmannian functional integrals and Hubbard-Stratonovich transformations to eliminate the fermionic degrees of freedom in favour of bosonic ones.