Compactification (physics)

Compactification makes some spatial dimensions compact, so modes along them acquire discrete momenta. At energies below the first nonzero compact-mode mass, a dimensional reduction to the zero modes gives a lower-dimensional effective field theory. The compact volume fixes the gravitational kinetic coefficient in an unwarped product background.

In a brane-world model some fields, typically ordinary matter and gauge fields, are localized on a lower-dimensional brane, while gravity propagates through the higher-dimensional bulk. Compact volume or a warped metric can then separate the observed gravitational scale from local matter scales. A model must specify field localization, boundary conditions and stabilization rather than identifying a brane with an arbitrary compactification.

Kaluza-Klein theory obtains lower-dimensional fields from a higher-dimensional metric and other fields on compact dimensions. In the displayed fixed-radius circle ansatz, the off-diagonal metric supplies a vector. The coordinate change $y\mapsto y+\lambda (x)$ is compensated by $A\mapsto A-d\lambda$, producing an Abelian gauge symmetry. Allowing the radius to vary adds a scalar modulus. Kaluza-Klein modes also include nonzero internal momenta, giving a tower of massive fields.

// Target: quantum-field-theory.bigb

For the Poisson equation ${\nabla }_4^2\varphi =-4\pi Rq{\delta }^3(\mathbf{x}){\delta }_{2\pi R}(z)$ on ${\mathbb{R}}^3\times S^1$, expand the periodic delta as $(2\pi R{)}^{-1}{\sum }_n{e}^{inz/R}$. The $n$th Kaluza-Klein mode has radial potential $q{e}^{-|n|r/R}/(2\pi r)$, since $({\nabla }_3^2-m^2)e^{-mr}/r=-4\pi {\delta }^3(\mathbf{x})$. Summing the geometric series gives the displayed Green function. At equal compact coordinate it reduces to $q\coth (r/(2R))/(2\pi r)$. Thus the interaction energy changes from $q_1{q}_2/(2\pi r)$ at large separation to $q_1{q}_{2}R/(\pi r^2)$ at small separation, with normalization fixed by the stated Poisson equation.

For a massless field on Minkowski space times a circle of circumference $R$, the compact delta distribution expands as $R^{-1}{\sum }_n{e}^{2\pi in\Delta w/R}$. The noncompact retarded Green function in each Fourier sector is that of a four-dimensional field with mass $m_n$. Equivalently, the time-frequency denominator is ${\mathbf{k}}^2+m_n^2-(\omega +i0{)}^2$, putting both poles below the real axis. At equal compact positions the sum has one zero mode and a doubly counted nonzero mass tower.

At zero frequency, sum the Kaluza-Klein tower of three-spatial-dimensional Yukawa kernels: $[4\pi Rr{]}^{-1}(1+2{\sum }_{n\ge 1}{e}^{-2\pi nr/R})$. The geometric series gives the displayed function. It behaves as $1/(4{\pi }^2{r}^2)$ for $r\ll R$, the four-spatial-dimensional Coulomb kernel, and as $1/(4\pi Rr)$ for $r\gg R$, the three-spatial-dimensional kernel with a zero-mode normalization factor.

A real five-dimensional Maxwell field on a circle of radius $r$ has a Fourier series with coefficients $A_M^{(n)}$, $A_M^{(-n)}=(A_M^{(n)}{)}^{\ast }$. Its mixed field-strength component is $F_{\mu 5}^{(n)}={\partial }_{\mu }{A}_5^{(n)}-in{A}_{\mu }^{(n)}/r$. The nonzero modes transform as $\delta A_{\mu }^{(n)}={\partial }_{\mu }{\Lambda }^{(n)}$, $\delta A_5^{(n)}=in{\Lambda }^{(n)}/r$. Thus $A_5^{(n)}$ is a Stueckelberg mechanism field and the gauge-invariant massive vector is $A_{\mu }^{(n)}-r{\partial }_{\mu }{A}_5^{(n)}/(in)$. Its mass is $|n|/r$ and it has three physical polarizations. The zero modes instead consist of a massless four-dimensional Maxwell field and a real massless scalar field; the latter cannot be removed by a periodic infinitesimal gauge transformation. A positive Fourier level and its negative level are conjugate, not independent real fields.

A gravitational magnetic monopole for a compact circle, whose fibre is nontrivially twisted over a surrounding two-sphere. With flat spectator directions its nonsingular local geometry is a Taub–NUT space. Reducing along the M-circle yields the D6-brane. The magnetic dual of the IIB circle vector is instead a five-brane in nine noncompact dimensions.

// Target: geometry-and-topology.bigb

A radion is a lower-dimensional scalar field describing the size or separation modulus of compact extra dimensions. In a two-brane Randall–Sundrum model, its expectation determines the inter-brane separation and the warp-generated hierarchy. A background size alone does not supply a radion potential or a stabilization mechanism.

// Target: supersymmetry.bigb

Radius stabilization fixes the size of a compactification dynamically rather than choosing it as an undetermined parameter. If the four-dimensional low-energy effective action contains a radion $R$ with positive kinetic coefficient $Z(R)$ and scalar potential $V(R)$, a stable radius $R_0$ requires $V^{\prime }(R_0)=0$ and $m_R^2=V^{\prime \prime }(R_0)/Z(R_0)>0$. This matters for the hierarchy problem: the volume or warp factor explains a hierarchy only if a consistent dynamics selects and maintains the required geometry.

The Goldberger–Wise stabilization mechanism generates a radion potential in the Randall–Sundrum model using a weakly backreacting bulk scalar field. On $0\le y\le L$, its static profile obeys ${\chi }^{\prime \prime }-4k{\chi }^{\prime }-m_{\chi }^2\chi =0$, hence $\chi =A{e}^{(4+ϵ)ky}+B{e}^{-ϵky}$, where $ϵ=\sqrt{4+m_{\chi }^2/k^2}-2$. Boundary potentials approximately fix $\chi (0)=v_0$, $\chi (L)=v_L$. For $ϵ\ll 1$ and $kL\gg 1$, the leading radius-dependent potential is proportional to $e^{-4kL}(v_L-v_0{e}^{-ϵkL}{)}^2$. Its leading minimum satisfies $kL\simeq {ϵ}^{-1}\log (v_0/v_L)$. Thus a modest ratio of boundary values can generate a large warp exponent. Corrections shift this minimum; the approximation requires small scalar field backreaction, and the four-dimensional cosmological constant still needs adjustment.

A warped five-dimensional geometry redshifts masses of fields localized on the distant brane: $m_{\mathrm{IR}}=e^{-kL}{m}_0$. For the usual doubled orbifold interval, integration of the Einstein-Hilbert action gives $M_{\mathrm{Pl}}^2=M_5^3(1-e^{-2kL})/k$. Exponential redshift can generate a large separation of observable scales with a moderately sized dimensionless distance $kL$. Stabilizing the radion remains a dynamical requirement.

// Target: physics.bigb

For a spatially flat, tuned one-brane background without energy transfer to the bulk, the Israel junction conditions produce the displayed modified Friedmann equation. Positive brane tension $\lambda$ gives ordinary expansion when $\rho \ll \lambda$, while the quadratic density correction enhances expansion at high energy. Big Bang nucleosynthesis, early acoustic evolution and primordial spectra can constrain this change. Spatial curvature, a residual cosmological constant and bulk energy exchange require their additional terms rather than this special formula.

Projected bulk Weyl tensor can contribute a radiation-scaling term to the brane Friedmann equation, even without ordinary brane photons. For an anti-de Sitter black-hole bulk the integration constant is related to its mass. This dark radiation on a cosmological brane changes the expansion rate tested by Big Bang nucleosynthesis and Cosmic microwave background anisotropy; it should not automatically be identified with a freely specifiable thermal particle species.

In a one-brane warped background, the proper extra dimension can be infinite while the four-dimensional graviton zero mode has finite norm. The Einstein-Hilbert action weighs its four-dimensional curvature term by the warp factor integral shown above, finite for $k>0$. Thus the proper length is not the effective gravitational volume. This differs from an unwarped noncompact direction, whose volume integral diverges. It also differs from the two-brane hierarchy mechanism, where finite brane separation redshifts masses on the distant brane. The noncompact construction is established in the original single-brane warped-gravity paper.

An inertial observer of a fixed-$y$ flat brane in metric $d{y}^2+e^{-2y/L}{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$ is accelerated relative to the higher-dimensional bulk. Normalization and ${\Gamma }_{\mu \nu }^y=e^{-2y/L}{\eta }_{\mu \nu }/L$ give $a^y=-1/L$. This is its invariant bulk proper acceleration, not a nonzero intrinsic four-dimensional acceleration on the flat induced brane metric.

Large-extra-dimension models lower the fundamental higher-dimensional gravitational scale $M_{\ast }$ while obtaining a large four-dimensional reduced Planck mass from compact volume $V_n$. Ordinary matter can be localized on a brane while gravity propagates in the bulk. The Planck mass from compactification volume relation explains weak effective four-dimensional gravity but leaves volume stabilization and the origin of a large internal volume as separate problems.

// Target: physics.bigb

For a completely regular Hausdorff space, the Stone-Čech compactification is the weak-star closure of the evaluation functionals in the dual unit ball of $C_b(X)$. The Banach-Alaoglu theorem gives compactness; complete regularity makes evaluation a topological embedding. Every bounded real continuous function extends by $\tilde{f}(p)=p(f)$, yielding an isometric isomorphism $C_b(X)\cong C(\beta X)$. For a continuous map into a compact Hausdorff space $K$, the adjoint of pullback sends this closure into the closed evaluation copy of $K$, yielding the unique continuous extension. Uniqueness follows from density. It is a universal compactification, not a homeomorphism between $X$ and the whole dual space.

For a Tychonoff space $X$, its Stone-Čech compactification $\beta X$ has the following universal property: every continuous map $f:X\to K$ into a compact Hausdorff space extends to a continuous map $\overline{f}:\beta X\to K$. The original space is embedded as a dense subset of $\beta X$. The hypothesis on $X$ matters because a general space need not embed in any compact Hausdorff space.

The extension is unique: two continuous maps into a Hausdorff space that agree on a dense subset agree everywhere. If two Hausdorff compactifications have this extension property, applying it to the inclusion of $X$ into each one gives comparison maps in both directions. Their composites fix the dense subset $X$, so uniqueness makes both composites identity maps. The comparison maps are therefore inverse homeomorphisms fixing $X$. Thus the universal property characterizes the Stone-Čech compactification up to equivalence, rather than depending on a particular construction.

In the Stone-Čech compactification of a discrete space, the closure of each subset of $X$ is clopen, because its indicator extends to a continuous zero-one-valued function. For disjoint closed sets $A,B\subseteq \beta X$, choose $u=0$ on $A$ and $u=1$ on $B$ by Urysohn's lemma. Then $U=\overline{\{x\in X:u(x)<1/2\}}$ and $V=\beta X\setminus U$ are disjoint clopen sets containing $A$ and $B$, respectively. Density proves these containments.

For a discrete space $X$ and $D\subseteq X$, the continuous extension of $1_D$ to $\beta X$ remains idempotent by density, so it takes values in $\{0,1\}$. Its one-set is the clopen closure of $D$. If $O$ is any open set in $\beta X$, density gives $\overline{O}=\overline{O\cap X}$, so this closure is clopen as well. Thus $\beta X$ is an extremally disconnected space. For infinite $X$, this does not imply that every subset of $\beta X$ has open closure.

For an unwarped product metric with fixed compact volume $V_n$, integrating the higher-dimensional Einstein-Hilbert action gives $M_{\mathrm{Pl}}^2=M_{\ast }^{n+2}{V}_n$ when the action coefficient is $M_D^{D-2}/2$. The reduced Planck mass convention must be distinguished from the unreduced one, and stabilizing the internal metric is a separate dynamical issue.

A Kaluza-Klein mode is a coefficient in a field expansion in eigenfunctions on compact extra dimensions. On a circle of circumference $L$, momentum is $2\pi \ell /L$, and a higher-dimensional massless field has lower-dimensional squared mass $(2\pi \ell /L{)}^2$. Nonzero modes give exponentially suppressed static-potential corrections at distances much larger than $L$.

For a field periodic on a circle of radius $R$, a Fourier series gives modes with quantized internal momentum $n/R$. Each mode is a lower-dimensional field with the displayed squared mass. The entire sequence is the tower; retaining only modes below a chosen energy gives an effective field theory. The tower is distinct from any individual Kaluza-Klein mode.

The compactification scale is the first nonzero internal-mode mass. On a circle it equals $1/R$ for a higher-dimensional massless field. A zero-mode dimensional reduction applies at energies well below it. Warping, nontrivial geometry and boundary conditions can change the relation between the first mass and a geometric radius.

A pointlike pulse can contain frequencies above the compactification gap even when its travel time and distance greatly exceed the circle size. Its retarded circle-compactified massless propagator can resolve winding images and massive-mode oscillations. Suppressing those modes requires low-frequency or long-wavelength sources/measurements, or suitable smearing; mere coordinate separation is not that restriction. The static Green function across a circle compactification gives a precise long-distance zero-mode limit without this ambiguity.

For a periodic scalar on a circle of circumference $2\pi R$, its Fourier series has modes ${\psi }_n(x)e^{inz/R}$. In the fixed-radius metric $d{s}_5^2={\gamma }_{ij}d{x}^{i}d{x}^j+(dz+A_{i}d{x}^i{)}^2$, the five-dimensional Klein-Gordon equation becomes $(D_i{D}^i-M^2-n^2/R^2){\psi }_n=0$ with $D_i={\nabla }_i-in{A}_i/R$. Thus the geometric gauge charge is quantized in units $1/R$ in units $\hslash =1$, or the conserved compact momentum in units $\hslash /R$. A rescaling to a canonically normalized electromagnetic potential changes the corresponding numerical unit of physical electric charge.

On three noncompact spatial dimensions times a circle of circumference $L$, the method of images gives ${\sum }_{j\in \mathbb{Z}}[r^2+j^2{L}^2{]}^{-1}=\pi \coth (\pi r/L)/(Lr)$ at equal compact coordinates. This interpolates from $1/r^2$ at short distances to $\pi /(Lr)$ at long distances. A gravitational circle-radius scalar can change the long-range coefficient if it remains unstabilized.

Dimensional reduction retains fields independent of chosen internal coordinates. In a flat-torus zero-mode reduction, a metric supplies a metric, internal-component vectors and internal metric scalars; a massless p-form gauge field supplies lower-rank fields with internal indices. Fluxes, twists or projections can alter the spectrum and preserved supercharges.

The zero-mode dimensional reduction on a flat $(D-4)$-torus splits a massless p-form gauge field by the number $r$ of spacetime indices. Each sector has $\left(\genfrac{}{}{0ex}{}{D-4}{p-r}\right)$ copies and $\left(\genfrac{}{}{0ex}{}{2}{r}\right)$ local polarizations per copy. Comparing coefficients in $(1+t{)}^{D-4}(1+t{)}^2$ gives ${\sum }_r\left(\genfrac{}{}{0ex}{}{D-4}{p-r}\right)\left(\genfrac{}{}{0ex}{}{2}{r}\right)=\left(\genfrac{}{}{0ex}{}{D-2}{p}\right)$, proving polarization matching. Four-dimensional three- and four-form potentials contribute no local propagating states.

// Target: supersymmetry.bigb