The goal of this Thesis is to give a presentation of some key issues regarding the non-perturbative renormalization of the periodic scalar field theories. As an example of the non-perturbative methods, we use the differential renormalization group approach, particularly the Wegner-Houghton and the Polchinski renormalization group equations, in order to investigate the renormalization of a one-component periodic scalar field theory. The Wegner-Houghton equation provides a resummation of the loop-expansion, and the Polchinski equation is based on the resummation of the perturbation series. Therefore, these equations are exact in the sense that they contain all quantum corrections. In the framework of these renormalization group equations, field theories with periodic self interaction can be considered without violating the essential symmetry of the model: the periodicity. Both methods - the Wegner-Houghton and the Polchinski approaches - are inspired by Wilson's blocking construction in momentum space: the Wegner-Houghton method uses a sharp momentum cut-off and thus cannot be applied directly to non-constant fields (contradicts with the "derivative expansion"); the Polchinski method is based on a smooth cut-off and thus gives rise naturally to a "derivative expansion" for varying fields. However, the shape of the cut-off function (the "scheme") is not fixed a priori within Polchinski's ansatz. In this thesis, we compare the Wegner--Houghton and the Polchinski equation; we demonstrate the consistency of both methods for near-constant fields in the linearized level and obtain constraints on the regulator function that enters into Polchinski's equation. Analytic and numerical results are presented which illustrate the renormalization group flow for both methods. We also briefly discuss the relation of the momentum-space methods to real-space renormalization group approaches. For the two-dimensional Coulomb gas (which is investigated by a real-space renormalization group method using the dilute-gas approximation), we provide a systematic method for obtaining higher-order corrections to the dilute gas result.

The goal of this Thesis is to give a presentation of some key issues regarding the non-perturbative renormalization of the periodic scalar field theories. As an example of the non-perturbative methods, we use the differential renormalization group approach, particularly the Wegner-Houghton and the Polchinski renormalization group equations, in order to investigate the renormalization of a one-component periodic scalar field theory. The Wegner-Houghton equation provides a resummation of the loop-expansion, and the Polchinski equation is based on the resummation of the perturbation series. Therefore, these equations are exact in the sense that they contain all quantum corrections. In the framework of these renormalization group equations, field theories with periodic self interaction can be considered without violating the essential symmetry of the model: the periodicity. Both methods - the Wegner-Houghton and the Polchinski approaches - are inspired by Wilson's blocking construction in momentum space: the Wegner-Houghton method uses a sharp momentum cut-off and thus cannot be applied directly to non-constant fields (contradicts with the "derivative expansion"); the Polchinski method is based on a smooth cut-off and thus gives rise naturally to a "derivative expansion" for varying fields. However, the shape of the cut-off function (the "scheme") is not fixed a priori within Polchinski's ansatz. In this thesis, we compare the Wegner--Houghton and the Polchinski equation; we demonstrate the consistency of both methods for near-constant fields in the linearized level and obtain constraints on the regulator function that enters into Polchinski's equation. Analytic and numerical results are presented which illustrate the renormalization group flow for both methods. We also briefly discuss the relation of the momentum-space methods to real-space renormalization group approaches. For the two-dimensional Coulomb gas (which is investigated by a real-space renormalization group method using the dilute-gas approximation), we provide a systematic method for obtaining higher-order corrections to the dilute gas result.