Cosets and Nonlinear Realizations

Once an exact continuous internal symmetry is broken from $G$ to $H$, the Goldstone fields can be treated as local coordinates on $G/H$. They do not usually form a linear representation of all of $G$. Instead, a transformation by $g\in G$ moves a chosen coset representative out of its chosen section, and a field-dependent compensator in $H$ brings it back. The Maurer–Cartan form then separates into a coset vielbein and a composite $H$ connection; contractions invariant under $H$ are automatically invariant under the nonlinear action of $G$.

This page constructs those local building blocks for exact ordinary internal symmetries in a relativistic phase. It does not decide whether the broken phase exists, count its propagating modes, determine effective-theory coefficients, or develop spacetime-symmetry cosets and inverse-Higgs constraints.

Required background. Vacuum Orbits and Unbroken Subgroups supplies the homogeneous space $G/H$ and its tangent space $\mathfrak g/\mathfrak h$. Lie Groups, Lie Algebras, and Exponential and Adjoint Maps supplies Lie brackets, exponentials, adjoint actions, and changes of generator basis.

Helpful background. Differential Forms, Integration, Orientation, and Stokes Theorem supplies the one-form and wedge-product language used for the Maurer–Cartan identity.

Coset representatives and compensators

Let $G$ be a finite-dimensional compact Lie group, or more generally suppose that the required reductive structure exists, and let $H\subset G$ be the closed unbroken subgroup. Choose Hermitian generators $T_i$ for $\mathfrak h$ and representatives $X_a$ of the broken directions. We assume an $\operatorname{Ad}(H)$-stable split

$$\mathfrak g=\mathfrak h\oplus\mathfrak m, \qquad hX_a h^{-1}=R(h)_a{}^bX_b.$$

When $H$ is compact—as it is for a closed subgroup of compact $G$—averaging an inner product over $H$ supplies such a stable complement. For a more general non-reductive homogeneous space, the simple component transformation laws below require refinement.

On a neighborhood of the identity coset, choose the local section

$$\xi(\pi) =\exp\!\left( \frac{i\pi^aX_a}{f} \right),$$

where $f$ is a convenient normalization scale. The exponential is a coordinate choice, not a claim that one chart covers all of $G/H$.

For $g\in G$, and wherever the transformed point remains in the chosen chart, there is a unique local factorization

$$g\,\xi(\pi) =\xi(\pi')\,h(g,\pi), \qquad h(g,\pi)\in H.$$

Equivalently, $\xi(\pi')=g\xi(\pi)h^{-1}$. This equation defines both the generally nonlinear transformation $\pi\mapsto\pi'$ and the compensator $h(g,\pi)$. Near $\pi=0$, a broken transformation $g=e^{i\epsilon^aX_a}$ begins as

$$\pi'^a=\pi^a+f\epsilon^a +O(\epsilon\pi),$$

whereas an unbroken transformation acts linearly through $R(h)$. A matter field $\psi$ that carries a representation $D$ of $H$ is assigned the nonlinear $G$ action

$$\psi'(x) =D\!\left(h(g,\pi(x))\right)\psi(x).$$

The compensators obey the composition rule

$$\begin{aligned} h(g_2g_1,\pi) &=h(g_2,g_1\mathbin{\cdot}\pi) \\ &\qquad\times h(g_1,\pi), \end{aligned}$$

so these field transformations realize the group law. This local factorization and the induced $H$ action are developed in Weinberg 1995, § 19.6, pp. 211–217; the general Goldstone-coordinate viewpoint is summarized in Schwartz 2014, § 28.2.4, pp. 573–574.

A different local section replaces $\xi(\pi)$ by $\xi(\pi)k(\pi)$ with $k(\pi)\in H$, together with a coordinate redefinition. It changes the compensator and the intermediate components below, but not the $G$-invariant theory. The compensator is therefore not an additional propagating field.

The Maurer–Cartan vielbein and composite connection

Pull back the left-invariant Maurer–Cartan form along the section and use Hermitian generators:

$$\begin{aligned} \Omega(\pi) &\equiv-i\xi^{-1}\mathrm d\xi \\ &=e^aX_a+\omega^iT_i. \end{aligned}$$

Here $e=e^aX_a$ is the broken component and $\omega=\omega^iT_i$ is the unbroken component. Under a constant $g\in G$, write $h=h(g,\pi)$ and use $\xi'=g\xi h^{-1}$. Direct differentiation gives

$$\Omega' =h\Omega h^{-1} -i h\,\mathrm d h^{-1}.$$

Because the complement $\mathfrak m$ is $\operatorname{Ad}(H)$ stable, the two components transform separately:

$$\begin{aligned} e'^aX_a &=h(e^aX_a)h^{-1}, \\ \omega'^iT_i &=h(\omega^iT_i)h^{-1} -i h\,\mathrm d h^{-1}. \end{aligned}$$

Thus $e$ transforms homogeneously, like a coframe on the coset, while $\omega$ has the inhomogeneous term of an $H$ connection. After $\pi=\pi(x)$, $e_\mu$ is the pullback of this target-space coframe; it is not a gravitational spacetime vielbein. The connection $\omega$ is a composite function of the Goldstone coordinates, not a new microscopic gauge field.

Writing

$$\Pi=\frac{\pi^aX_a}{f},$$

the Baker–Campbell–Hausdorff expansion provides a sign check:

$$\Omega =\mathrm d\Pi -\frac{i}{2}[\Pi,\mathrm d\Pi] +O(\Pi^2\mathrm d\Pi).$$

In particular, $e^a=\mathrm d\pi^a/f+\cdots$. The full form also satisfies the Maurer–Cartan identity

$$\mathrm d\Omega+i\Omega\wedge\Omega=0.$$

This is a geometric identity, not an equation of motion. The transformation laws and their use in invariant interactions are derived in Weinberg 1995, § 19.6, pp. 215–220.

Invariant local building blocks

On spacetime, write

$$\begin{aligned} e^a&=e_\mu^a\,\mathrm dx^\mu, \\ \omega^i&=\omega_\mu^i\,\mathrm dx^\mu. \end{aligned}$$

If $t_i$ are the Hermitian $H$ generators in the representation of $\psi$, define

$$\nabla_\mu\psi =\left( \partial_\mu+i\omega_\mu^i t_i \right)\psi.$$

The inhomogeneous term in $\omega'$ cancels the derivative of the field-dependent compensator, giving

$$(\nabla_\mu\psi)' =D(h)\nabla_\mu\psi.$$

The sign is tied to the definition $\Omega=-i\xi^{-1}\mathrm d\xi$ and the convention $\psi'=D(h)\psi$.

Let $g_{ab}$ be an $H$-invariant positive target-space metric on the broken representation:

$$g_{ab}R(h)^a{}_cR(h)^b{}_d =g_{cd}.$$

The leading Lorentz-invariant Goldstone term is then

$$\mathcal L_2 =\frac{f^2}{2}\, g_{ab}e_\mu^ae^{\mu b}.$$

If $\mathfrak m$ splits into inequivalent $H$ representations, there can be several independent invariant tensors and therefore several coefficients already at this order. Symmetry determines the allowed contractions, not their numerical values.

More generally, any local $H$-singlet constructed from $e_\mu$, $\psi$, $\nabla_\mu\psi$, and their covariant derivatives is invariant under the nonlinear action of $G$. An exact internal symmetry forbids a nonconstant invariant potential on a transitive coset, so undifferentiated Goldstone coordinates cannot acquire an ordinary mass term. Integrations by parts, equations of motion, and dimension-specific identities can still make an apparently distinct local basis redundant.

This construction captures strictly invariant local terms. Wess–Zumino terms, which may shift by a total derivative, require global and cohomological information beyond one chart. Power counting, loop renormalization, matching, and model-dependent coefficients belong to Renormalization and Effective Field Theory.

A non-Abelian local check

For $SO(3)\to SO(2)$, let $H$ be generated by $J_3$ and take $X_1=J_1$, $X_2=J_2$, with

$$[J_i,J_j]=i\epsilon_{ijk}J_k.$$

The local coset is a patch of $S^2$. Substituting

$$\Pi=\frac{\pi^1J_1+\pi^2J_2}{f}$$

into the expansion of $\Omega$ gives

$$\begin{aligned} e^1 &=\frac{\mathrm d\pi^1}{f} +O\!\left( \frac{\lVert\pi\rVert^2\mathrm d\pi}{f^3} \right), \\ e^2 &=\frac{\mathrm d\pi^2}{f} +O\!\left( \frac{\lVert\pi\rVert^2\mathrm d\pi}{f^3} \right), \\ \omega^3 &=\frac{ \pi^1\mathrm d\pi^2 -\pi^2\mathrm d\pi^1 }{2f^2} \\ &\qquad +O\!\left(\frac{\pi^3\mathrm d\pi}{f^4}\right). \end{aligned}$$

The pair $(e^1,e^2)$ rotates homogeneously under the unbroken $SO(2)$, while $\omega^3$ supplies its composite connection. The calculation is local: another chart is needed near points where these exponential coordinates cease to be regular.

The complex scalar and its shift field

For the exact global complex-scalar symmetry

$$G=U(1), \qquad H=\{1\},$$

choose the charge-one generator and

$$\xi(\pi)=e^{i\pi/f}.$$

There is no nontrivial compensator or unbroken connection. A phase rotation by $\alpha$ gives

$$\pi\longmapsto\pi+f\alpha,$$

with the global identification $\pi\sim\pi+2\pi f$, and

$$\Omega=e=\frac{\mathrm d\pi}{f}.$$

The leading invariant is therefore

$$\frac{f^2}{2}e_\mu e^\mu =\frac12\partial_\mu\pi\,\partial^\mu\pi.$$

In the weakly coupled scalar model, write locally

$$\phi =\frac{v+\sigma}{\sqrt2} e^{i\pi/f}.$$

Its kinetic term becomes

$$\begin{aligned} \partial_\mu\phi^\dagger\partial^\mu\phi &=\frac12(\partial\sigma)^2 \\ &\quad +\frac{(v+\sigma)^2}{2f^2} (\partial\pi)^2. \end{aligned}$$

At tree level and at energies well below the radial mass, setting the leading solution $\sigma=0$ and choosing $f=v$ reproduces the canonical Goldstone kinetic term. Eliminating $\sigma$ more accurately generates higher-derivative interactions. In the exact interacting theory, $f$ is a physical low-energy normalization and need not equal a chosen order-parameter expectation value. The angular variable, shift action, and derivative interactions are worked out in Schwartz 2014, § 28.2.1, pp. 564–566. This Abelian example checks the sign and normalization but cannot test a non-Abelian compensator or connection.

Two finite-group statements must be kept distinct. If the exact symmetry is $U(1)$ but the unbroken subgroup is $\mathbb Z_N$, then $U(1)/\mathbb Z_N$ is still one-dimensional and has a Goldstone coordinate. If a permanent deformation instead reduces the exact phase-rotation group itself to $\mathbb Z_N$, there is no continuous coset tangent and no exact shift symmetry. The lifted angular mode belongs to Explicit Breaking and Pseudo-Goldstone Modes. A temporary selector removed after the infinite-volume limit does not alter the final exact group and is not retained in the coset action.

A reliable construction sequence

1. Identify the exact physical global group $G$ and the actual stabilizer $H$ of the selected phase.
2. Check that $H$ is closed and choose a local section of $G\to G/H$; for the simple component rules, choose an $\operatorname{Ad}(H)$-stable complement.
3. Factor $g\xi(\pi)=\xi(\pi')h(g,\pi)$ to determine the nonlinear action and compensator.
4. Compute $-i\xi^{-1}\mathrm d\xi$ and separate its broken vielbein from its unbroken connection.
5. Assign non-Goldstone fields to representations of $H$ and form $H$-invariant contractions of the covariant building blocks.
6. Only then reduce the operator basis and impose power counting, matching, and loop organization in the effective-theory treatment.

This sequence is local in field space. Global patching, defects, Wess–Zumino terms, anomalies, and quantized coefficients need additional information. For broken spacetime symmetries, even the list of independent coordinates can change through demonstrated inverse-Higgs-type redundancies; the internal construction here should not be applied mechanically. See Goldstone Counting, Low-Dimensional Obstructions, and Spacetime Exceptions.

Common pitfalls

“$G/H$ must be a quotient group.” It is a homogeneous space of cosets. It becomes a group only when $H$ is normal in $G$.

“The compensator is a new gauge field.” The compensator is the field-dependent $H$ transformation that returns a representative to the chosen section. The connection $\omega$ is built from $\pi$; neither introduces an independent microscopic gauge degree of freedom.

“The exponential representative is global.” It supplies a useful chart near the identity coset. Compact or topologically nontrivial cosets generally require identifications or additional patches.

“The Maurer–Cartan connection transforms homogeneously.” The vielbein does; the connection has an inhomogeneous derivative term. That term is precisely what makes derivatives of $H$-multiplet matter fields covariant.

“One coset coordinate always means one propagating mode.” That equality holds for the relativistic internal setting under the counting theorem’s assumptions. At finite density, type-B pairing can reduce the number of modes; spacetime generators can be redundant.

“An unbroken finite subgroup removes every Goldstone field.” If the exact group remains continuous, $G/H$ can still have a continuous tangent even when $H$ is finite. No continuous coset exists only when the exact symmetry available for the construction is itself finite.

Calculate a coset

These questions are for self-study and are not graded.

1. Starting from $\xi'=g\xi h^{-1}$ with constant $g$, derive the transformation of $\Omega=-i\xi^{-1}\mathrm d\xi$.
2. Use $\Omega=\mathrm d\Pi-\tfrac{i}{2}[\Pi,\mathrm d\Pi]+\cdots$ to recover the leading $\omega^3$ for $SO(3)/SO(2)$.
3. Explain why both $U(1)/\{1\}$ and $U(1)/\mathbb Z_N$ have one local Goldstone coordinate, while a theory whose exact group is only $\mathbb Z_N$ does not.

Check

1. Differentiate $\xi'=g\xi h^{-1}$ and multiply by $\xi'^{-1}=h\xi^{-1}g^{-1}$:

   $$\xi'^{-1}\mathrm d\xi' =h(\xi^{-1}\mathrm d\xi)h^{-1} +h\,\mathrm d h^{-1}.$$

   Multiplication by $-i$ gives

   $$\Omega' =h\Omega h^{-1} -i h\,\mathrm d h^{-1}.$$

2. With $\Pi=(\pi^1J_1+\pi^2J_2)/f$ and $[J_1,J_2]=iJ_3$,

   $$[\Pi,\mathrm d\Pi] =\frac{i}{f^2} \left( \pi^1\mathrm d\pi^2 -\pi^2\mathrm d\pi^1 \right)J_3.$$

   Therefore the unbroken component is

   $$\omega^3 =\frac{ \pi^1\mathrm d\pi^2 -\pi^2\mathrm d\pi^1 }{2f^2} +\cdots.$$

3. Dividing a one-dimensional Lie group by either the trivial subgroup or a finite subgroup leaves a one-dimensional homogeneous space locally. By contrast, a finite exact group has a zero-dimensional Lie algebra and no infinitesimal broken direction from which to construct a Goldstone coordinate.

What follows

• Explicit Breaking and Pseudo-Goldstone Modes adds symmetry-breaking operators and follows the shift mode away from zero mass.
• Finite-Density Goldstone Counting develops the nonrelativistic relation between broken coordinates and propagating modes.
• Renormalization and Effective Field Theory supplies power counting, operator-basis reduction, matching, and loops.
• Chiral Effective Theory and Nonlinear Symmetry applies these building blocks to a developed non-Abelian EFT.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
• Weinberg, Steven. The Quantum Theory of Fields, Volume II: Modern Applications. Cambridge: Cambridge University Press, 1995. DOI.