Field Redefinitions and Redundant Operators

Two effective actions can assign different coefficients to individual operators and still give the same on-shell predictions through a stated order. The reason is that local perturbative field redefinitions move terms among the action, sources, boundary contributions, and higher orders. The invariant content is an observable calculated consistently in one nonredundant basis, not the presence of a particular equation-of-motion operator in a chosen set of variables.

Required background. Einstein Equations provides the lower-order metric equation used in the gravity example; Gravitational Actions and Boundary Terms provides the surface completion that cannot be discarded during a metric redefinition; Variational and Symplectic Geometry provides the variational identity and reduced-phase-space criterion. Operator sizes are counted as on Scales, Operators, and Power Counting.

Helpful background. Formulations of General Relativity supplies the constraint and boundary-data distinctions needed to decide whether a metric redefinition preserves the same solution space.

Perturbative changes of field coordinates

Let $S_0[\Phi]$ be the leading action and make a local change of variables

$$\Phi^A\longmapsto\Phi^A+\epsilon F^A[\Phi],$$

where $\epsilon$ records the EFT order. Taylor expansion gives

$$S_0[\Phi+\epsilon F] =S_0[\Phi] +\epsilon\int_M\mathbf E_A^{(0)}F^A +\epsilon\int_{\partial M}\boldsymbol\Theta_0(\Phi,F) +O(\epsilon^2).$$

Consequently an operator proportional to the lower-order equation of motion $\mathbf E_A^{(0)}$ can be removed at order $\epsilon$ by the inverse field map. Its removal is not deletion: the map shifts other Wilson coefficients, changes source and composite-operator couplings, modifies boundary terms, and generates definite $O(\epsilon^2)$ contributions. The equivalence theorem applies when the map is local and perturbatively invertible and when external states are transformed consistently. Criado and Pérez-Victoria develop the higher-order expansion and its matching consequences in Criado and Pérez-Victoria 2019, §§2–3, pp. 3–12.

Three elementary operations are usually combined when building a basis.

Operation	Redundancy removed	Qualification
Integration by parts	Operators differing by a total divergence	The boundary integral and allowed data must be retained.
Algebraic identity	Dimension- or representation-dependent tensor combinations	The identity may fail in another dimension or with torsion/nonmetricity.
Field redefinition	Operators proportional to lower-order equations of motion	Sources, Jacobian, coefficients, and the next retained order must transform too.

An equation of motion is therefore a basis tool, not a relation that may be inserted everywhere inside a higher-order calculation. Using the leading equation twice can change terms precisely at the next order one is trying to predict.

A scalar example with visible coefficient shifts

Take

$$\mathcal L =-\frac12(\partial\phi)^2 -\frac12m^2\phi^2 -\frac{\lambda}{4!}\phi^4 +\frac{c}{\Lambda^2}\phi^3\Box\phi.$$

The leading Euler–Lagrange expression is

$$E_0(\phi)=\Box\phi-m^2\phi-\frac{\lambda}{6}\phi^3.$$

Hence

$$\phi^3\Box\phi =\phi^3E_0 +m^2\phi^4 +\frac{\lambda}{6}\phi^6.$$

Write the old field in terms of a new one as

$$\phi=\chi-\frac{c}{\Lambda^2}\chi^3.$$

The variation of $S_0$ produces $-c\chi^3E_0/\Lambda^2$, canceling the redundant part of the displayed operator. Through $O(\Lambda^{-2})$ the transformed Lagrangian contains

$$\mathcal L' =-\frac12(\partial\chi)^2 -\frac12m^2\chi^2 -\frac{\lambda}{4!}\chi^4 +\frac{cm^2}{\Lambda^2}\chi^4 +\frac{c\lambda}{6\Lambda^2}\chi^6.$$

The derivative operator has disappeared, but its physical effects have moved into the quartic and sextic coefficients. A tree-level four-point amplitude computed from the original derivative vertex agrees with the amplitude from the shifted quartic coupling once external legs obey the same mass-shell condition. Off-shell Green functions need not agree, and a source coupled linearly to $\phi$ becomes $J\chi-cJ\chi^3/\Lambda^2$ rather than remaining $J\chi$.

This example also fixes the sign convention. Reversing whether the old field is written in terms of the new or the new in terms of the old reverses the displayed intermediate sign; the fully transformed observable does not care.

Curvature-squared gravity in four dimensions

Consider, for the moment, pure metric gravity in four dimensions,

$$S=\int_M d^4x\sqrt{-g}\left[ \frac{M_{\mathrm{Pl}}^2}{2}R +c_1R^2+c_2R_{ab}R^{ab}+c_3R_{abcd}R^{abcd} \right].$$

The Euler density

$$E_4=R_{abcd}R^{abcd}-4R_{ab}R^{ab}+R^2$$

has a topological bulk integral on a closed four-manifold. With the appropriate boundary completion it removes one curvature-squared bulk combination from the local equations. Next make the perturbative metric redefinition

$$g_{ab}\longmapsto g_{ab} +\frac{a}{M_{\mathrm{Pl}}^2}R_{ab} +\frac{b}{M_{\mathrm{Pl}}^2}Rg_{ab}.$$

The first-order change of the Einstein–Hilbert action is proportional to $G^{ab}\delta g_{ab}$ plus a surface term. In four dimensions its bulk part spans $R_{ab}R^{ab}$ and $R^2$, so $a$ and $b$ can remove the remaining two curvature-squared combinations at this order. Thus pure-vacuum on-shell graviton amplitudes receive no independent local four-derivative correction; the first nonredundant parity-even local terms can be chosen at six derivatives. The curvature identity and redefinition argument are summarized in Burgess 2004, §3.1.1, pp. 23–25, equations (3.2)–(3.3).

Every qualifier in that statement matters.

• Matter: the same metric map transforms $S_m[g,\psi]$ and induces operators such as $R_{ab}T^{ab}$ and $RT$. Curvature-squared effects have not vanished from arbitrary matter observables.
• Boundary: $E_4$ needs its boundary completion, while the metric map changes both the Einstein and higher-curvature surface variations.
• Dimension: $E_4$ is not topological in dimensions above four. A scalar-dependent coefficient also makes it dynamical in four dimensions.
• Background: a cosmological constant changes the lower-order equation and therefore the coefficient shifts. Nontrivial topology and asymptotic data can retain global information.
• Order: removing a four-derivative term generates definite six-derivative terms. They must be kept when the calculation claims that accuracy.

The dedicated Gauss–Bonnet page later proves the dimensional and boundary statement rather than extrapolating this EFT shortcut.

Sources, observables, and the functional measure

For an on-shell scattering amplitude, local invertible redefinitions preserve the pole locations and residues associated with the same physical states after wave-function normalization. For a classical waveform or cosmological observable, the analogous rule is to transform the detector or matter metric, initial data, source multipoles, and observable functional. Merely substituting the leading equations into a field equation can fail because it does not specify this complete map.

In the path integral,

$$\mathcal D\Phi=\mathcal D\Phi'\, \det\!\left(\frac{\delta\Phi}{\delta\Phi'}\right).$$

For ordinary local perturbative transformations the Jacobian can often be represented by local terms and handled in a consistent regularization scheme. It is not permissible to assume this for a nonlocal, singular, anomalous, or nonperturbative map. Matching calculations are especially sensitive: changing the low-energy basis also changes the map between ultraviolet parameters and Wilson coefficients. The source and matching qualifications are analyzed in Criado and Pérez-Victoria 2019, §§4–5, pp. 13–28.

An equivalence checklist

Before declaring two operator bases physically equivalent, verify all of the following:

1. The field map is local and invertible order by order throughout the branch and domain of interest.
2. Both actions are truncated at the same order, and all coefficient shifts and induced higher-order terms at that order are retained.
3. Boundary, corner, initial, asymptotic, and regularity data map into each other without changing the solution space.
4. Matter actions, external sources, composite operators, units, and the final observable are transformed.
5. The functional Jacobian and possible anomaly are included in a quantum calculation.
6. A shared on-shell observable agrees, while an off-shell quantity is not mistaken for invariant evidence.

This checklist distinguishes an equivalence class of descriptions from a family of genuinely different gravitational theories.

Common pitfalls

Set the equations of motion to zero inside the action. Equation-of-motion operators are removed by an explicit field redefinition. Direct substitution misses coefficient, source, boundary, and higher-order changes.

Discard every total derivative. A total derivative has no bulk variation under suitable compactly supported variations, but it can change charges or the variational problem when a boundary is present.

Transfer four-dimensional vacuum redundancy to all settings. Matter, boundaries, a cosmological constant, scalar-dependent couplings, or dimensions above four change the curvature-squared conclusion.

Exercises

For the scalar Lagrangian above, perform the redefinition $\phi=\chi-c\chi^3/\Lambda^2$ explicitly through $O(\Lambda^{-2})$. Determine the induced quartic and sextic couplings, and explain how a source term $J\phi$ must change for the two descriptions to produce the same on-shell observables.

Solution

Taylor expansion of the leading action gives

$$S_0[\chi-c\chi^3/\Lambda^2] =S_0[\chi] -\frac{c}{\Lambda^2}\int d^4x\,\chi^3E_0(\chi) +O(\Lambda^{-4}),$$

up to the boundary term fixed by the chosen data. At the same accuracy the higher-order operator can be evaluated at $\phi=\chi$:

$$\frac{c}{\Lambda^2}\chi^3\Box\chi =\frac{c}{\Lambda^2}\left[ \chi^3E_0+m^2\chi^4+\frac{\lambda}{6}\chi^6 \right].$$

The two $\chi^3E_0$ terms cancel. The remaining Lagrangian shifts are

$$\Delta\mathcal L =\frac{cm^2}{\Lambda^2}\chi^4 +\frac{c\lambda}{6\Lambda^2}\chi^6.$$

If the quartic and sextic potentials are parameterized as $-\lambda_{\mathrm{eff}}\chi^4/4!$ and $-g_6\chi^6/(6!\Lambda^2)$, then

$$\lambda_{\mathrm{eff}} =\lambda-\frac{24cm^2}{\Lambda^2}, \qquad g_6=-120c\lambda.$$

Finally,

$$J\phi=J\chi-\frac{c}{\Lambda^2}J\chi^3.$$

Keeping only $J\chi$ would compare different source functionals and could make off-shell correlators or response coefficients appear inequivalent. With the source, external-state normalization, and boundary data transformed, the on-shell amplitudes agree through $O(\Lambda^{-2})$; any difference is of the next omitted order.

References

• Burgess, C. P. (2004), “Quantum Gravity in Everyday Life: General Relativity as an Effective Field Theory,” Living Reviews in Relativity 7, article 5. DOI; Open PDF.
• Criado, Juan Carlos, and Manuel Pérez-Victoria (2019), “Field Redefinitions in Effective Theories at Higher Orders,” JHEP 03, 038. DOI; Open PDF.