KaluzaKlein

Kaluza-Klein theory is a framework for unifying gravity with other forces by introducing extra spatial dimensions beyond the familiar four. It originated with Theodor Kaluza in 1921, who proposed a five-dimensional version of general relativity that, upon compactification of the fifth dimension, yielded four-dimensional gravity plus electromagnetism. Oskar Klein later provided a quantum interpretation, suggesting the extra dimension is compactified on a small circle, making the fifth-dimensional momentum discrete and manifesting as electric charge in four dimensions.

In this scheme, the five-dimensional metric includes components that correspond to the electromagnetic vector potential, and the dynamics reduce to Einstein-Maxwell theory at low energies when the extra dimension is small. The approach links geometry and gauge fields, using the fifth dimension to encode electromagnetic interactions within the gravitational framework.

In modern contexts, one extends the idea to more than one extra dimension, leading to Kaluza-Klein towers: fields in higher dimensions decompose into a tower of four-dimensional fields with increasing mass, determined by the size and shape of the compact space. These KK modes are a generic feature of compactified higher-dimensional theories and provide a mechanism for introducing new heavy states in the low-energy effective theory.

Kaluza-Klein ideas have influenced higher-dimensional theories and string theory, where extra dimensions are central. They motivate searches for KK excitations of gravitons or gauge fields in particle colliders and precision gravity experiments. To date, no conclusive experimental evidence supports extra dimensions at accessible energies.

See also: extra dimensions, string theory, compactification, braneworld, Kaluza-Klein mass spectrum.