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= Compactification in Kaluza-Klein =

{{Audience_Sidebar
| difficulty = Advanced
| reading_time = 9 minutes
| prerequisites = [[Kaluza-Klein_Unification]]; basic differential geometry (circles, manifolds, periodic identification).
| if_too_advanced_see = [[Why_Does_Physics_Need_Extra_Dimensions]]
| if_you_want_the_math_see = This page
}}

{{Notation
| signature = Mostly-plus (−,+,+,+,+) in 5D.
| units = Geometrized; ℏ = c = 1. x5 has period 2πL. Capital Latin indices M, N = 0..4; Greek μ, ν = 0..3.
}}

'''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 <math>x^5</math> is compactified on a circle of radius <math>L</math>:

:<math>x^5 \sim x^5 + 2\pi L</math>

This page covers the compactification procedure in [[Kaluza-Klein_Unification|Kaluza-Klein theory]] and in the framework's [[5D_Action_Principle|5D action]].

== The compactification idea ==

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

:<math>(x^0,\,x^1,\,x^2,\,x^3,\,x^5) \sim (x^0,\,x^1,\,x^2,\,x^3,\,x^5 + 2\pi L)</math>

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

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

== Mode expansion ==

Any field <math>\varphi(x^\mu, x^5)</math> on <math>M^4 \times S^1</math> is periodic in <math>x^5</math>:

:<math>\varphi(x^\mu, x^5) = \varphi(x^\mu, x^5 + 2\pi L)</math>

It can therefore be Fourier-decomposed:

:<math>\varphi(x^\mu, x^5) = \sum_{n=-\infty}^{\infty} \varphi_n(x^\mu)\,e^{i n x^5 / L}</math>

Each Fourier mode <math>\varphi_n(x^\mu)</math> is a 4D field. Substituting into the 5D wave equation:

:<math>\Box_5\,\varphi = \partial_\mu\partial^\mu\varphi + \partial_5^2\,\varphi = 0</math>

gives, for each Fourier mode:

:<math>\bigl(\Box_4 - n^2/L^2\bigr)\,\varphi_n(x^\mu) = 0</math>

Each mode appears as a '''massive 4D field''' with mass <math>m_n = |n|/L</math>. This is the '''Kaluza-Klein tower''':

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

For <math>L \sim</math> Planck length, the first KK mode has mass <math>\sim</math> 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 <math>n = 0</math> mode''' — the [[Cylinder_Condition|cylinder condition]] taken as a limit. The 5D field is treated as <math>x^5</math>-independent:

:<math>\varphi(x^\mu, x^5) \approx \varphi_0(x^\mu)</math>

Substituting into the 5D action and integrating over <math>x^5</math>:

:<math>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)]</math>

— the 5D action becomes <math>2\pi L</math> times a 4D action. Quantities are then rescaled: <math>G = \tilde{G}/(2\pi L)</math>, 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 [[Psi_Field|ψ field]] in the [[5D_Action_Principle|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 Td''' — d copies of S1; gives U(1)d gauge symmetry.
* '''Sphere Sd''' — 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|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 [[5D_Action_Principle|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.

[[Category:Psionics]]
[[Category:Kaluza-Klein]]
[[Category:Physics]]

Compactification in Kaluza-Klein - Revision history

JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to ; lint-clean per tools/wiki_latex_lint.py

JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to $; lint-clean per tools/wiki_latex_lint.py$