r - qc / 0 30 30 97 v 3 2 1 M ay 2 00 3 Oskar Klein , the sixth dimension and the strength of a magnetic pole

This work extends to six dimensions the idea first proposed by Klein regarding a closed space in the context of a fifth dimension and its link to quantum theory. The main result is a formula that expresses the value of the characteristic length of the sixth dimension in terms of the strength of a magnetic monopole g. Possible consequences of the idea are discussed. It is well known that the world lines of charged particles in the presence of gravitational and electromagnetic fields can be viewed as geodesics in the 5D space-time first proposed by T. Kaluza [1]. Kaluza’s treatment was later improved by Klein in 1926 [2], in a quantum theoretical approach, giving rise to the so-called Kaluza-Klein theory [4]. This formalism also makes possible to propose, based on plausible grounds, a simple formula that gives the value of the characteristic length of the fifth dimension in terms of fundamental constants of nature. The original Kaluza-Klein theories have, since the 1 1920’s been the subject of further analysis that include generalizations to a larger number of dimensions [5], cosmological implications [6] and magnetic monopoles [7], most of them treated a-la Dirac [8]. This last subject deserves more attention. Recently [9], the authors have observed that a simple 6D generalization of Kaluza’s 5D metric leads naturally to the geodesics of particles possessing (hypothetical) fundamental magnetic charges. Further, this 6D space-time reproduces Maxwell’s equations in the presence of magnetic monopoles and allows the establishment of a wave equation for the vector potential, in the presence of point sources without using any singularity arguments [10]. In view of these facts, we have reanalyzed Klein’s formalism in order to establish a link between the eigenvalue of the sixth dimensional quantum operator and the discrete nature of the hypothetical point charges. We also examine other physical features of the sixth dimension, such as its characteristic length. We start this section observing that the geodesic equation dx dt + Γαβλ dx dt dx dt = 0 (1) contains the equation of motion of an electric-magnetic charged particle in the presence of a static electromagnetic field given that: x = 