Compactification in Kaluza-Klein

Compactification in Kaluza-Klein

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Notation on this page

Compactification is the geometric procedure that hides extra dimensions by curling them up at small scales, so that they remain mathematically present but physically invisible above a critical length. In Kaluza-Klein theory, the fifth dimension $ x^{5} $ is compactified on a circle of radius $ L $:

$ x^{5}\sim x^{5}+2\pi L $

This page covers the compactification procedure in Kaluza-Klein theory and in the framework's 5D action.

The compactification idea

In flat 5D spacetime, $ x^{5} $ ranges over all real numbers. To make the fifth dimension invisible at large scales, identify points separated by $ 2\pi L $:

$ (x^{0},\,x^{1},\,x^{2},\,x^{3},\,x^{5})\sim (x^{0},\,x^{1},\,x^{2},\,x^{3},\,x^{5}+2\pi L) $

The fifth coordinate is now an angular coordinate on a circle $ S^{1} $. Points are equivalent under shifts by $ 2\pi L $ — the topology of spacetime is $ M^{4}\times S^{1} $ (Minkowski-4 times circle).

For $ L $ sufficiently small (originally proposed Planck-scale $ \sim 10^{-33} $ cm; modern variants allow much larger), physics at scales $ \gg L $ is effectively four-dimensional.

Mode expansion

Any field $ \varphi (x^{\mu },x^{5}) $ on $ M^{4}\times S^{1} $ is periodic in $ x^{5} $:

$ \varphi (x^{\mu },x^{5})=\varphi (x^{\mu },x^{5}+2\pi L) $

It can therefore be Fourier-decomposed:

$ \varphi (x^{\mu },x^{5})=\sum _{n=-\infty }^{\infty }\varphi _{n}(x^{\mu })\,e^{inx^{5}/L} $

Each Fourier mode $ \varphi _{n}(x^{\mu }) $ is a 4D field. Substituting into the 5D wave equation:

$ \Box _{5}\,\varphi =\partial _{\mu }\partial ^{\mu }\varphi +\partial _{5}^{2}\,\varphi =0 $

gives, for each Fourier mode:

$ {\bigl (}\Box _{4}-n^{2}/L^{2}{\bigr )}\,\varphi _{n}(x^{\mu })=0 $

Each mode appears as a massive 4D field with mass $ m_{n}=|n|/L $. This is the Kaluza-Klein tower:

• $ n=0 $ mode: massless 4D field (or whatever mass the original 5D field had).
• $ n=\pm 1 $: mass $ 1/L $.
• $ n=\pm 2 $: mass $ 2/L $.
• ...

For $ L\sim $ Planck length, the first KK mode has mass $ \sim $ Planck mass — far above any accessible energy. The compactified dimension is invisible because all its excitations are inaccessible.

Zero-mode reduction

In most Kaluza-Klein derivations, one keeps only the $ n=0 $ mode — the cylinder condition taken as a limit. The 5D field is treated as $ x^{5} $-independent:

$ \varphi (x^{\mu },x^{5})\approx \varphi _{0}(x^{\mu }) $

Substituting into the 5D action and integrating over $ x^{5} $:

$ S=\int \!d^{4}x\,dx^{5}\,{\mathcal {L}}_{\text{5D}}[\varphi _{0}(x^{\mu })]=(2\pi L)\int \!d^{4}x\,{\mathcal {L}}_{\text{5D}}[\varphi _{0}(x^{\mu })] $

— the 5D action becomes $ 2\pi L $ times a 4D action. Quantities are then rescaled: $ G={\tilde {G}}/(2\pi L) $, and so on.

Compactification radius and observability

The compactification radius L is the central physical parameter. Choices:

• Planck-scale (L ~ 10⁻³³ cm): the original Kaluza-Klein assumption. KK modes at Planck mass — completely inaccessible.
• GUT-scale (L ~ 10⁻³⁰–10⁻²⁹ cm): in some grand-unified Kaluza-Klein constructions.
• Large extra dimensions (L ~ μm–mm): ADD (Arkani-Hamed-Dimopoulos-Dvali 1998) and Randall-Sundrum 1999. Predict modifications of Newton's law at sub-mm scales and TeV-scale KK gravitons; not detected, constraining L < ~ 50 μm in ADD models.
• Framework: L unspecified but small (L ≲ 10⁻¹⁸ cm): the ψ field in the framework takes the n = 0 mode plus a heavy KK tower; only the zero mode is relevant for ordinary psi phenomena.

The framework's predictions do not depend critically on L provided L is below the smallest experimentally probed scale.

Compactification of more than one dimension

Generalising to d extra dimensions, the compact manifold can be:

• Torus T^d — d copies of S¹; gives U(1)^d gauge symmetry.
• Sphere S^d — gives SO(d+1) gauge symmetry.
• Calabi-Yau 3-fold (in string theory) — gives the actual Standard Model gauge group after specific topology choices.
• Orbifolds — quotient manifolds with fixed points; produce chiral fermions.

The choice of compact manifold determines the 4D gauge symmetries and field content.

Compactification and the cylinder condition

The cylinder condition (Kaluza 1921) — that fields are x⁵-independent — is exactly the zero-mode truncation of the Kaluza-Klein tower. It can be:

• Imposed by hand (original Kaluza).
• Derived by assuming compactification with L below all probed energies, so only the zero mode is excited.
• Relaxed to allow the full KK tower (modern KK).
• Abandoned entirely (Wesson's induced-matter theory: x⁵ not compact).

Framework usage

In the framework's 5D action:

• The compactification step takes the 5D action with a ψ field and a 5D metric and integrates out x⁵.
• Result: 4D Einstein gravity + 4D Maxwell electromagnetism + 4D scalar ψ field + dilaton-like coupling.
• The ψ field's mass m, self-coupling λ, and EM-coupling α emerge as 4D parameters that descend from the 5D action.
• The KK tower for ψ is not directly relevant to psi phenomena at human scales — those involve the zero mode.

Sanity checks

• L → ∞ — the fifth dimension decompactifies; no Fourier-mode discretisation; recover 5D field theory. ✓
• L → 0 — KK modes infinitely heavy; only zero mode remains; pure 4D. ✓
• Mode-by-mode equations — substituting back recovers full 5D dynamics. ✓
• ψ = 0 (in framework) — recover standard Kaluza-Klein compactification. ✓ (Sanity_Check_Limits §2.)

See Also

• Kaluza-Klein_Unification
• Cylinder_Condition
• Dilaton
• 5D_Action_Principle
• Higher-Dimensional_Physics

References

• Klein, O. (1926a). "Quantentheorie und fünfdimensionale Relativitätstheorie." Zeitschrift für Physik 37: 895–906.
• Witten, E. (1981). "Search for a realistic Kaluza-Klein theory." Nuclear Physics B 186: 412–428.
• Overduin, J. M., Wesson, P. S. (1997). "Kaluza-Klein gravity." Physics Reports 283: 303–378. arXiv:gr-qc/9805018.
• Arkani-Hamed, N., Dimopoulos, S., Dvali, G. (1998). "The hierarchy problem and new dimensions at a millimeter." Physics Letters B 429: 263–272.