Equations de Klein Gordon sur les variétés

This work consists of studying a development of a study of dimensional reduction and this to construct a renormalizable quantum field theory. The reduction will be from a 4-dimensional space-time (D=1+3) to a variant with a smaller number of spatial dimensions (D=1+d,d<3) at sufficiently small distances. We will prove an important theorem that links the study of Klein Gordon's equation on space (with variable geometry) to the resolution of a Schrödinger equation with an effective potential generated by geometric variation. This result is based on the Fourier method (so-called varaible separation) in Klein Gordon's equation and on the fact that two-dimensional spaces are flat. We will show that in the case of the space dimension (d=2) the conformal factor of the metric between the effective potential in the Schrödinger equation due to the corresponding modifications of the variables. As an example, we will consider a space-time with a variable spatial geometry including a transition to a dimensional reduction. This example which we are going to study contains a combination between two bidimensional cylindrical regions of distinct radii connected by a transition region.Ce travail consiste à étudier un développement d'une étude de la réduction dimensionnelle et cela pour construire une théorie de champ quantique renormalisable. La réduction sera à partir d'un espace-temps à 4 dimensions (D=1+3) à une variante avec un plus petit nombre de dimensions spatiales (D=1+d,d