TOPICS ON KALUZA-KLEIN THEORY by Tonatiuh Matos* Centro de Investigaci´ on y de Estudios Avanzados del IPN Apdo. Post. 14-740, M´ exico 07000 DF ,M´ exico and J. Antonio Nieto Escuela de F´ ısica y Matem´ aticas, Universidad Michoacana Apdo. Post. 749, Morelia 58000, Michoac´an, M´ exico Abstract A modern review of Kaluza-Klein theories is presented. We adopt the version where the whole space is a principal fiber bun- dle with the four-dimensional spacetime as base, and as typi- cal fiber a G Lie group. It is a natural generalization of gauge theories which metric is just the Kaluza-Klein metric. For the five dimensional theory we give an invariant formulation of the axisymmetric-stationary case. Some techniques for obtaining ex- act solutions and cosmology in specific dimensions are studied. Finally the method of spontaneus compactification is outlined. In 1919 Th. Kaluza presented to Albert Einstein a new idea for unifing all till that moment known interactions based on the Einstein geometrization theory. It consisted in a generalization of the general relativity theory into a five-dimensional riemannian space interpreting part of the five-dimensional metric as the four-electromagnetic potential. In a letter from Einstein to Kaluza, Einstein expressed his view of Kaluza’s idea with the comment:”Ihr Gedanke gefelt mir zun¨ achst ausserordentlich. ”(I like your idea at first sight very much).Indeed Einstein was enthusiastic with Kaluza’s idea and presented it at the Sitzungs- berichte der Preussiscchen Akademie der Wissenschaften at 8th December 1921, in a paper entitled “Zur Unit¨ atsproblem der Physic” [1]. This paper contained some inconsistences with the theory of Quanta as remarked by Einstein himself.The first important step in ascribing physical reality to the fifth dimension was taken by de Broglie and Schr¨ odinger for the treatment of quantum problems.By starting with a generalized wave equation, he discovered in the equation surprising solutions which were periodic in the fifth dimension with a period related to the Planck constant. However, the first serious attempt to assign physical meaning to the fifth dimension was made by Einstein and Bergmann [76]. These authors introduced the remarkable assumption that the space is closed in a very small circle in the direction of the fifth dimension. Through this change not only was the Kaluza theory generalized, but also a justification for the four dimensional appearence of the ”real” world was obtained. In a subsequent article, Einstein and Pauli [77] argued that the theory is still unsatisfactory with respect to the group of admissible coordinate transformations, because the fifth dimension is treated differently from the other four dimensions. However they made the following remark: ”When one tries to find a unified theory of the gravitational and electromagnetic fields,one cannot help feeling that there is some truth in Kaluza’s 5- dimensional theory”. Kaluza-Klein theory (KK) consisted basically in associate to the γ µν , (µ,ν =1...4) components of the five-dimensional metric the gravitational interaction and to γ 5µ the electromagnetic one, while γ 55 remained * Work partialy supported by CONACYT- Mexico 1