2026-10-01 · chapter Yang–Mills and the mass gapPart 7 From the side of physicsextra dimensions, Kaluza–Klein

How extra dimensions become invisible — waves on a small circle look like a tower of heavy particles

To see, in physics, is to collide particles and examine how they bounce off. To resolve a structure of size $\ell$ needs a probe with wavelength shorter than $\ell$, that is, high energy (2-05).

1 TeV gives $2.0\times10^{-19}$ m, the LHC collision energy of 13.6 TeV gives $1.5\times10^{-20}$ m, and the Planck mass $1.22\times10^{19}$ GeV gives $1.6\times10^{-35}$ m Computed. From here on we set $\hbar=c=1$ (2-04 §08). As a metaphor, a garden hose seen from afar is a line, but up close the direction going around it becomes visible. More precisely, what this chapter says is that "fluctuations along a small circle need high energy".

Consider the five-dimensional spacetime $\mathbb R^{1,3}\times S^1$ with a circle of radius $R$ attached to each point of space (the circle coordinate $y$ identified with $y+2\pi R$). A massless scalar field $\Phi(x,y)$ obeys the wave equation (2-02), and since it is periodic in $y$ it can be expanded in a Fourier series (1-11).

Substituting, $\partial_y^2$ is replaced by $-n^2/R^2$, and the equation splits coefficient by coefficient.

This is the Klein–Gordon equation of mass $m_n$ (2-02 §06). One five-dimensional field looks, in four dimensions, like a massless zero mode $\phi_0$ and an infinite sequence of fields of mass $1/R,2/R,\dots$. This is the Kaluza–Klein tower (KK tower) Known. The PDG writes 'At energies smaller than 1/R, the KK modes can be neglected, leaving the zero-mode action'. The couplings of the zero mode are diluted; for a quartic interaction $\lambda_4=\lambda_5/(2\pi RM_5)$ (PDG formula 84.6), weaker the larger the circle. $M_5$ is the fundamental mass scale of the five-dimensional theory, put in to make $\lambda_5$ dimensionless.

For a product of $\delta$ circles (a torus $T^\delta$ with radii $R_i$), each tuple of integers $\vec n$ gives a field with $m^2=\sum_i n_i^2/R_i^2$. Figure 1 is the case of two circles.

ComputedThe numbers of Table 1 and Figure 1 were computed on this machine (the figure is drawn on the spot from formula (3)). If JavaScript is disabled, see Table 1.

The row $R_2/R_1=0.1$ is the same as one circle for $ER_1\le5$, and the thin direction first appears at $E\gtrsim1/R_2$. The count grows as $E$ to the power of (the number of visible extra dimensions).

The lowest rung $1/R$ can be restated as an inequality. For a periodic function with mean 0, $f=\sum_{n\ne0}c_ne^{iny/R}$, Parseval's identity gives (Wirtinger's inequality) Known

This is the simplest example of the Poincaré inequality of 2-05 §08 and 1-10 §07, and the best constant $R^2$ is the reciprocal of the gap $1/R^2$ (the square of the lowest rung of the tower).

Measuring lengths — metric, diameter, Laplacian

To replace the circle by a general compact manifold $K$ (1-05) we need a way of measuring lengths. An inner product given on each tangent space (1-05 §06), varying smoothly from point to point, is called a metric. It is a way of measuring lengths and angles. With a metric we can measure the lengths of curves, and the length of the shortest path joining two points is their distance. The maximum of that distance over all pairs of points is the diameter $d$ (for a circle of circumference $2\pi R$, $d=\pi R$; the $d$ of this chapter is the diameter, not the spacetime dimension $d$ of other chapters). The size of the gradient of a function, $|\nabla f|$, is also fixed by the metric, and the operator $-\Delta_K$ with $\int_K|\nabla f|^2=\int_K\bar f\,(-\Delta_Kf)$ is called the Laplacian (the Laplace–Beltrami operator). On a flat torus, in coordinates $y_i$ measured by length, $-\Delta_K=-\sum_i\partial^2/\partial y_i^2$.

If spacetime is the unwarped product $\mathbb R^{1,3}\times K$ and the field is a scalar, expanding in the eigenfunctions of $-\Delta_K$ (eigenvalues $0=\lambda_0<\lambda_1\le\cdots$) gives $m_j^2=\lambda_j$. The first KK mass is the minimum of the Rayleigh quotient (1-10 §06).

For form or spinor fields, or in warped backgrounds, it becomes the eigenvalue of another operator (the Hodge Laplacian, the Dirac operator and so on). In this chapter we restate "extra dimensions are invisible" as the Poincaré inequality on the part without the zero mode, $\int|\nabla f|^2\ge\lambda_1\int|f|^2$.

When the diameter suffices — Ricci curvature

Multiplying the metric by $s^2$ (multiplying all lengths by $s$) multiplies $\lambda_1$ by $s^{-2}$, so shrinking raises the tower. That "small" may be measured by the diameter $d$ alone is a theorem when the Ricci curvature is non-negative.

Ricci curvature, as a metaphor, is the average of how much a bundle of geodesics (locally shortest paths) leaving a point around one direction gathers together as it goes. On a sphere it is positive (meridians gather at the poles), on a flat torus 0, on a saddle-shaped surface negative (they spread). More precisely, for a direction $v$ of length 1, it is the sum $\operatorname{Ric}(v,v)=\sum_iK(v,e_i)$ of the bending (sectional curvature) of the $n-1$ sections spanned by $v$ and the directions $e_i$ orthogonal to $v$, which is $n-1$ times the average. In two dimensions there is a single section, so it is the same as the Gauss curvature. The bending itself is the curvature in the sense of 1-07 §05 (the discrepancy after going round a small loop) when the bundle of tangent spaces is given the connection fixed by the metric. $\operatorname{Ric}_K\ge0$ is the condition that this sum is at least 0 at every point in every direction Known.

The tori were computed numerically and agree to 5 digits with the closed forms (rectangle $1+\rho^2$, hexagon $16/9$). The spheres are known values.

For thin tori the ratio approaches 1 but does not reach it. The curvature condition cannot be dropped. Join two spheres by a thin neck and put into (5) a function that is $+1$ on one and $-1$ on the other: the numerator comes only from the neck, so making the neck thinner leaves the diameter nearly unchanged while the Rayleigh quotient becomes as small as we like. At the base of the neck the curvature becomes strongly negative, and $\operatorname{Ric}\ge0$ fails. This is a well-known example Known.

The idea of extra dimensions goes back to Kaluza and Klein in the 1920s (the introduction of the PDG). Five-dimensional spacetime also has a metric in the sense of §03 (in spacetime, what corresponds to the inner product at each point has, like the $\eta=\operatorname{diag}(-1,1,1,1)$ of 2-04, exactly one negative sign, and it may vary from point to point). Its 15 components split into 10 for the four-dimensional metric $g_{\mu\nu}$, 4 for a vector $A_\mu$ and 1 for a scalar $\phi$. Rewriting formulas 5 and 8 of the review by Overduin–Wesson with signature $(-,+,+,+,+)$ gives^{Note 1}

The Einstein equations are the equations of gravity that tie the bending (curvature) of the metric to the energy of matter. Here we do not go into their content and use only the counting of components. If no component depends on $y$, the five-dimensional Einstein equations split into 10+4+1=15 four-dimensional equations (ibid. formula 6). Setting $\phi$ constant, among the 15 the equations for the metric and vector components become the four-dimensional Einstein equations and Maxwell's equations (ibid. formula 7) Known. But the remaining scalar equation is compatible with $\phi=$ constant only when $F_{\mu\nu}F^{\mu\nu}=0$ (pointed out by Jordan and Thiry, ibid. §3.4). The change of coordinates $y\mapsto y+\kappa\lambda(x)$ induces $A_\mu\mapsto A_\mu-\partial_\mu\lambda$, and the gauge transformation (2-03) appears as a re-choice of the coordinate in the direction of the circle.

Klein (1926) connected Kaluza's theory with wave mechanics (abstract) and made the circle small. The momentum $n/R$ of the $n$-th component in (2) becomes, under (7), a charge (ibid. formula 25, $\phi=1$).

Charge is restricted to integer multiples of $q_1$ (the title of Klein's short communication of the same year is 'The Atomicity of Electricity as a Quantum Theory Law'). As a metaphor, charge is momentum around an invisible circle; more precisely, it is the momentum $n/R$ times a constant $\kappa$ fixed by convention.

Setting $q_1=e$ ($e^2/4\pi=\alpha\approx1/137.036$) gives $R=2\ell_P/\sqrt\alpha\approx3.8\times10^{-34}$ m (about 23 times the Planck length), and the first rung of the tower is $1/R\approx5.2\times10^{17}$ GeV, about $10^{21}$ times the electron mass Computed. Overduin–Wesson, taking the radius as the Planck length, estimate the discrepancy at about 22 orders of magnitude, and write that it 'played a large role in the abandonment of five-dimensional Kaluza-Klein theory'. If the internal space has a non-abelian symmetry, non-abelian gauge fields can come out (7-07) Physics.

Spacetime in superstring theory is ten-dimensional (7-05 §06), so to make it look four-dimensional a six-dimensional internal space $K$ is needed. Candelas–Horowitz–Strominger–Witten (1985) studied vacua leaving one supersymmetry in four dimensions, and wrote in the abstract 'This condition permits only a few possibilities'. The $K$ satisfying the condition is said to be a Kähler manifold with Ricci curvature 0 (a complex manifold whose metric and complex structure mesh) whose holonomy (1-07) lies in SU(3): a Calabi–Yau manifold Physics.

That such metrics exist is a mathematical theorem. Yau announced in 1977 'We announce a proof of Calabi's conjectures on the Ricci curvature of a compact Kähler manifold' (the paper is from 1978). As a consequence, a compact Kähler manifold with vanishing first Chern class has a Ricci-flat Kähler metric Known. If it is Ricci-flat, (6) applies, so the scalar KK tower on a Calabi–Yau of diameter $d$ starts above $\pi/d$.

On the size, the introduction of the PDG writes 'The extra dimensions were supposed to be compactified at a scale close to the Planck scale, and thus not testable experimentally in the near future'. The shape affects the particle content (the model of CHSW has 'an E6 gauge group with four standard generations of fermions').

In 1998, Arkani-Hamed–Dimopoulos–Dvali (ADD) proposed a way of hiding in which the fields of the Standard Model are confined to a three-dimensional membrane (a 3-brane) and only gravity spreads into $4+\delta$ dimensions. Since they can be probed only by gravity, the extra dimensions may be large. The abstract reads 'The observed weakness of gravity on distances ≳ 1 mm is due to the existence of n ≥ 2 new compact spatial dimensions large compared to the weak scale' Physics. The weakness of gravity is the same dilution as the $\lambda_4$ of §02, and the fundamental scale $M_D$ and the reduced Planck mass $M_P\approx2.4\times10^{18}$ GeV are related by (PDG formula 84.10)

$\delta=1$ is the size of the solar system, 'clearly ruled out' (PDG), and $\delta=2$ gives 0.5 mm. The next year, 1999, Randall–Sundrum (RS) used a slice of five-dimensional anti-de Sitter space (PDG formula 84.16).

Anti-de Sitter space is a spacetime of constant negative curvature (7-08). The metric (10) is the slice $0\le y\le\pi R$ of that spacetime with curvature radius $1/k$. Scales at the end $y=\pi R$ are redshifted by a factor $e^{-k\pi R}$. The PDG writes that $k\sim M_P$ and $R\simeq11/k$ give the TeV; at $kR=11$, $ke^{-k\pi R}\approx2.4$ TeV, and changing $kR$ from 10 to 12 moves it from 55 TeV to 0.1 TeV Computed. Making a large ratio with an exponential resembles dimensional transmutation (7-01). More precisely, in RS the exponent comes from geometry ($kR$), in dimensional transmutation from the running of the coupling ($1/g^2$). The view that connects warped extra dimensions with strongly coupled four-dimensional theories is AdS/CFT (7-08), at the level of a conjecture Physics.

Detour: extra dimensions without a gap

Deviations from Newton's law are by custom expressed in Yukawa form (PDG formula 84.12).

The exchange of a field of mass $m$ gives a Yukawa potential $e^{-mr}/r$, and one term is added for each KK mode. In a scalar (Newtonian) model, summing the whole tower of $T^\delta$ (radius $R$) gives

$2\delta$ is the number of fields on the first rung; for $\delta=1$ the sum is $\coth(r/2R)$. For $r\ll R$, $V\propto r^{-1-\delta}$: the law of a space of $3+\delta$ dimensions. For the graviton the coefficient changes. The KK modes of the graviton are massive spin-2 fields, and the contribution of their polarisations is said to multiply the coefficient of the first rung by 4/3; the PDG takes, for a two-dimensional torus, $\alpha=16/3$ (4/3 of the 4 in (12)) and $\lambda=R$ Physics. (12) is not the formula for gravity itself but a model for seeing how summing the tower produces a Yukawa-type deviation.

ComputedThe numbers of Figure 2 and Table 4 were computed on this machine (sums over $|\vec k|\le420$; $\delta=1$ agrees with $\coth$ to 12 digits, and $r\to0$ with the integral approximation to 5 digits). If JavaScript is disabled, see Table 4.

That the deviation vanishes as $e^{-r/R}$ for $r\gg R$ is the quantitative content of "invisible".

One of the inverse-square-law experiments (Tan et al. 2016) confirms $|\alpha|\le1$ down to $\lambda=59$ µm (abstract). The observed gauge fields are four-dimensional down to about $5\times10^{-20}$ m, and gravity follows the inverse-square law down to some tens of µm. Extra dimensions have not been found, but neither has it been shown that there are none.

The KK gap is not the mass gap. The lowest rung of the tower is a gap that comes only from the internal space being compact; free fields have it too, and the zero mode has no gap. It is the same computation as the gap of a free field in a box (2-02 §06, 2-05 §06). What the prize problem asks for is a gap arising from interaction on $\mathbb R^4$, which has no compact directions (4-11), and Jaffe–Witten write 'no present ideas point the direction to establish the existence of a mass gap that is uniform in the volume'.

Do not mix the two kinds of extra dimensions. The lower bounds of §07 concern real extra dimensions. The extra dimensions of holographic models (7-08, 7-09) (the radial direction of anti-de Sitter space, $S^5$, the auxiliary circle of Witten's model) are tools of a dual description or of regularisation, and are not claimed to be real dimensions.

Within the models the tower is visible. In models that read the gap from geometry, the KK tower of the extra dimensions comes at the same height as the glueballs (in the Maldacena–Núñez model of $\mathcal N=1$ supersymmetric Yang–Mills, if the gravity approximation is valid, 'comparable to the masses of Kaluza Klein states'; the authors also write that quantising the string on this background might allow the KK states to be separated) Physics. Removing the tower requires shrinking the extra dimensions, and then the coupling becomes weak (in four dimensions, because of asymptotic freedom), which is the opposite side from the region where the computation works (Csáki et al., of Witten's three-dimensional model: 'This is the opposite of the limit that is required for the supergravity description to be valid'). The order of limits is treated in 7-10.

What this chapter can say

Sources and reproduction

Next chapter to read: 7-07 Where gauge fields come from — extracting gauge groups from branes and geometry.

← 7-05 String theory and extra dimensionsContents7-07 Where gauge fields come from →