| Summary: | The physical system modelled is a periodic lattice that provides a background for
interacting fermions. An effective model of the interacting fermions at zero temperature
may be studied using standard techniques of many-body quantum mechanics. All the
physical information of such a system is encoded in its Schwinger's functions that tell
us how states evolve in time and what the matrix elements of all operators are. It is
natural to do calculations on the interacting system by treating it as a perturbation
of the corresponding noninteracting system, as the eigenstates and energy levels of the
latter are known. All Schwinger functions may be conveniently cast in the path integral
formalism, and combined into a generating functional. The Schwinger functions are
shown to be sums of Feynman diagrams, where the electron propagators blow-up at the
Fermi surface. Power counting is described and used to reveal subdiagram divergences.
Renormalization via the renormalized Gallavotti-Nicolo Tree Expansion is implemented
in an attempt to circumvent these subdiagram divergences and obtain finite values for
the Schwinger functions. It is shown that if the Fermi surface has at worst quadratic
singularities, the perturbation coefficients of the renormalizing counterterms are finite
and remain so as the infrared cutoff is removed. === Science, Faculty of === Mathematics, Department of === Graduate
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