Implicaciones geométricas de la teoría de Kaluza-Klein en los modelos de compactificación: Un acercamiento a las teorías modernas de unificación en la física

Abstract

The Kaluza-Klein theory proposes the unification of gravity and the electromagnetic field using different methods but with common perspectives. On one hand, Kaluza starts from the Christoffel symbols (interpreted as components of the gravitational field) by adding a fifth dimension that is independent of those symbols to obtain a mathematical relationship between the gravitational field and the electromagnetic field; this by virtue of a mathematical analogy established by Kaluza between the gravitational and electromagnetic descriptions. On the other hand, Klein, following Kaluza's hypothesis, starts from the geometric formalism, along with particular coordinate transformation groups, to derive the field equations in which the gravitational and electromagnetic fields naturally arise; this allows him to give a wave treatment to matter (guided by de Broglie's hypothesis) in such a way that a mathematical restriction is imposed on the type of solutions the field equations must have, namely: the wave equation (Schrödinger's hypothesis). Thus, Klein gives a quantum and relativistic treatment to spacetime, by showing that the fifth dimension closes in on itself, that is, spatial loops so tiny (quantized) that they are imperceptible. The derivation methods start from the same physical ansatz (extra dimension), but from slightly different mathematical hypotheses and ansatzes. The gravitational and electromagnetic unification inherently involves the equation of these fields, that is, giving them an equivalent treatment. For this reason, a geometric interpretation of spacetime is necessary as a problem of describing measurements between observers, thus revisiting the reflections proposed by Minkowski on hyperbolic symmetries for inertial observers within the framework of special relativity that facilitate and allow for an egalitarian description of measurements and physical laws.