Equations de Klein Gordon sur les variétés

Abstract

This work consists of studying a development of a study of di- mensional reduction and this to construct a renormalizable quantum eld 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 su¢ ciently 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 e¤ective 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 at. We will show that in the case of the space dimension (d = 2) the conformal factor of the metric between the e¤ective potential in the Schrödinger equation due to the corresponding modi cations of the variables. As an example, we will consider a space-time with a variable spa- tial 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.