KKReduktionen

KKReduktionen, or Kaluza-Klein reduction, is a method in higher-dimensional field theories for deriving a lower-dimensional effective theory by compactifying the extra spatial dimensions on a small, compact manifold. The concept originates from the work of Theodor Kaluza and Oskar Klein in the 1920s and is a standard tool in string theory, supergravity, and related frameworks.

In a typical setup, spacetime is taken to be a product M^D = M^d × K, where d is the observed spacetime dimension (often 4) and K is a compact internal space. Fields on M^D are expanded in harmonics on K; the zero modes produce massless fields in d dimensions, while higher harmonics appear as an infinite tower of massive Kaluza-Klein modes with masses set by the inverse size of K.

The higher-dimensional metric decomposes into a d-dimensional metric, gauge fields from mixed components, and scalar fields that describe the size and shape of K (moduli). The resulting d-dimensional action contains gravity, gauge interactions, and a spectrum of scalars and KK modes, with couplings determined by the geometry of K and the compactification scale. In practical model building, one usually truncates to the lightest modes to obtain a finite effective field theory valid below the compactification scale. Warping and background fluxes can modify the spectrum and couplings.

Limitations include moduli stabilization, control over the infinite KK tower, and the need to ensure a consistent quantum theory. KKReduktionen remains central to connecting higher-dimensional theories with observable four-dimensional physics.

See also: Kaluza-Klein theory, extra dimensions, compactification, string theory, effective field theory.