Local versus Integrated Operator Redundancies

An equation-of-motion operator or a total derivative can be redundant only relative to a declared object and a declared set of conditions. A local variational insertion retains contact terms; a source-coupled operator retains derivatives of its source; an integrated classical action term can be reorganized by integration by parts or a field redefinition only with controlled boundaries and a perturbatively invertible redefinition; an exact quantum source-functional comparison additionally tracks the Jacobian, sources, and observables; and an on-shell matrix element applies a further quotient. These are related statements, not interchangeable definitions.

Required background. Local and Composite Operator Insertions supplies local sources, regulated composites, and full-versus-connected insertions. The Action Principle and Field Equations supplies the variational derivative and the boundary term in the classical action principle.

Helpful background. Coincident Products and Contact Terms supplies the delta-supported terms that distinguish a local equation-of-motion insertion from a pointwise zero.

Four statements that must not be conflated

Object being compared	Candidate simplification	Conditions still required
local variational or Schwinger–Dyson insertion	replace an equation-of-motion operator by zero	impossible in general: collisions with other insertions produce contacts
source-coupled local operator	integrate a divergence by parts	source derivatives, boundary flux, ordering, and contact terms remain
integrated classical action functional	remove a total derivative or an EOM-proportional term	admissible boundary data and a perturbatively invertible redefinition; quantum source-functional equality additionally requires its Jacobian and transformed sources or observables
on-shell matrix element or amplitude	discard an inverse-propagator or momentum-transfer factor	external-state construction, amputation, pole structure, and equivalence-theorem hypotheses

The word “redundant” is therefore incomplete unless it names the row. In particular, an EFT operator-basis quotient is not automatically an identity among local operator-valued distributions.

A local variational equation-of-motion insertion carries contacts

For a scalar action $S[\phi]$, define the Euler–Lagrange insertion

$$\mathcal E(x) \equiv\frac{\delta S}{\delta\phi(x)}.$$

First take a finite regulator with a flat translation-invariant measure and an admissible field-independent shift whose integration-cycle flux vanishes. With the site’s normalized Lorentzian weight $e^{iS+iJ\cdot\phi}$, define the next brackets as the regulated Schwinger–Dyson, or covariant $\mathrm T^\ast$, insertion: derivatives inside $\mathcal E$ act on the already ordered correlator. They do not denote the literal canonical product $\mathrm T\{\mathcal E(x)X\}$.

$$\left\langle \mathrm T^\ast\left\{ \mathcal E_\Lambda(x)\prod_{r=1}^n\phi(x_r) \right\} \right\rangle_\Lambda =i\sum_{r=1}^n \delta_\Lambda(x,x_r) \left\langle \mathrm T\prod_{s\ne r}\phi(x_s) \right\rangle_\Lambda.$$

The equation of motion makes $\mathcal E$ vanish on a classical solution, and the literal Heisenberg equation can hold as an operator identity. Neither fact removes the variational contact above. In a controlled continuum limit, one external field gives

$$\left\langle\mathrm T^\ast\{\mathcal E(x)\phi(y)\}\right\rangle =i\delta^{(d)}(x-y).$$

For the free action $S_0=-\tfrac12\phi\cdot P\cdot\phi$ with $P=\Box+m^2$, $\mathcal E=-P\phi$, so the same statement is

$$\left\langle\mathrm T^\ast\{\mathcal E(x)\phi(y)\}\right\rangle \equiv-P_xD_F(x-y) =i\delta^{(d)}(x-y).$$

By contrast, the literal canonical product $\langle0|\mathrm T\{(P\phi)(x)\phi(y)\}|0\rangle$ vanishes by the free Heisenberg equation. The contact appears when $P_x$ acts outside the ordered product, giving $P_xD_F=-i\delta$. Schwartz derives this time-ordering contact in Schwartz 2014, § 14.7.1, pp. 273–274.

For a nonflat regulated measure, a field-dependent shift, or a non-invariant integration cycle, the measure-divergence or boundary term in the general identity below must be retained. Passing to the displayed continuum delta also requires a controlled limit and any needed renormalization of the composite insertion.

A total derivative is local before it is integrated

Let $V^\mu(x)$ be a declared local vector insertion and let $f$ be a smooth switching function on a spacetime region $\Omega$. Weak integration by parts gives

$$\int_\Omega\mathrm d^d x\, f\,\partial_\mu V^\mu =-\int_\Omega\mathrm d^d x\, (\partial_\mu f)V^\mu +\int_{\partial\Omega}\mathrm d\Sigma_\mu\, fV^\mu.$$

The integral vanishes only when $f$ is constant on the relevant support and the boundary flux vanishes or is cancelled. For a spacetime-dependent local source $K$,

$$\int\mathrm d^d x\,K(x)\partial_\mu V^\mu(x) =-\int\mathrm d^d x\,(\partial_\mu K)V^\mu$$

under a vanishing-boundary hypothesis. The divergence has not disappeared; it has moved onto the source. Further source derivatives or time ordering can also generate contact terms.

This explains why two integrated action bases can be equivalent under constant couplings while their local insertion bases differ. Boundary observables, defects, finite regions, and nontrivial switching functions can detect the difference directly.

A scalar field redefinition produces an integrated EOM term

Consider the stable scalar model

$$S[\phi] =\int\mathrm d^d x\, \left[ \frac12(\partial\phi)^2 -\frac12m^2\phi^2 -\frac{\lambda}{4!}\phi^4 \right], \qquad m^2>0, \quad \lambda\geq0.$$

Its local Euler–Lagrange expression is

$$\mathcal E =-(\Box+m^2)\phi -\frac{\lambda}{3!}\phi^3.$$

Make the infinitesimal local change of variables

$$\phi=\chi+\epsilon a\chi^3.$$

The first-order action change is

$$S[\chi+\epsilon a\chi^3] =S[\chi] +\epsilon a\int\mathrm d^d x\, \chi^3\mathcal E[\chi] +O(\epsilon^2),$$

including the boundary conditions already required by the variational derivative. Locally,

$$\chi^3\mathcal E =3\chi^2(\partial\chi)^2 -m^2\chi^4 -\frac{\lambda}{3!}\chi^6 -\partial_\mu(\chi^3\partial^\mu\chi).$$

After integration, the last term may be removed only if its surface flux vanishes. The remaining relation is a useful algebraic preview of an EOM/IBP operator-basis reduction. It is not a local identity setting $\chi^3\mathcal E$ to zero, and inside a correlator the EOM factor still produces contacts.

This first-order separation of total derivatives from EOM-proportional terms is developed in Burgess 2021, § 2.5, pp. 45–46.

The quantum change of variables has three extra terms

At a finite regulator, write the field variables as $\phi_a$, the measure density as $\mu(\phi)$, and an infinitesimal change as $\delta\phi_a=\epsilon R_a(\phi)$. Define

$$\operatorname{div}_\mu R =\frac{1}{\mu} \frac{\partial}{\partial\phi_a} \bigl(\mu R_a\bigr), \qquad \delta_R X =R_a\frac{\partial X}{\partial\phi_a}.$$

Assume the regulated integration cycle is deformable and its boundary flux vanishes. Changing variables in a normalized expectation value gives

$$0 =\left\langle \delta_R X+X\operatorname{div}_\mu R +iX\,R_a\bigl(\mathcal E_a+J_a\bigr) \right\rangle_J.$$

Equivalently,

$$\left\langle X\,R\!\cdot\!\mathcal E\right\rangle_J =i\left\langle \delta_R X+X\operatorname{div}_\mu R \right\rangle_J -\left\langle X\,R\!\cdot\!J\right\rangle_J.$$

The three qualifications are now explicit:

1. $\delta_R X$ changes the observable or external insertions;
2. $\operatorname{div}_\mu R$ is the regulated Jacobian contribution;
3. $R\cdot J$ changes the source coupling.

For the scalar example $R_a=a\phi_a^3$ on a finite lattice, the flat-measure divergence contains $3a\sum_a\phi_a^2$. Its continuum shorthand would involve coincident kernels and must not be set to zero without a regulator-specific argument. Zinn-Justin derives the corresponding Euclidean transformation, Jacobian, source variation, and insertion term in Zinn-Justin 2021, § 7.5.3, pp. 136–137; the displayed factors of $i$ are its translation to the site’s Lorentzian convention.

If the cycle has a boundary contribution, if the transformation is not invertible on the relevant domain, or if the regulator produces an anomalous Jacobian, the displayed zero acquires an additional term. A formal substitution alone does not establish equivalence of quantum theories.

On-shell matrix elements apply a different quotient

Translation covariance gives, with $q=p'-p$,

$$\langle p'|\partial_\mu V^\mu(0)|p\rangle =iq_\mu\langle p'|V^\mu(0)|p\rangle.$$

A forward matrix element therefore annihilates this total derivative when the matrix element is regular at $q=0$. Off forward, it generally does not. Boundaries, massless singularities, or distributional momentum support can also invalidate the naive $q\to0$ step.

An EOM-proportional vertex can supply an inverse-propagator factor that vanishes on a simple external on-shell wavefunction, but the same factor can cancel a neighboring propagator and leave a contact contribution. Exact equality of source-dependent generating functionals requires transforming the action and measure together with the sources and inserted observables.

On-shell S-matrix equivalence is a different theorem. A local perturbatively invertible field redefinition can preserve amplitudes even when the off-shell source functionals differ, provided the relevant stable poles are isolated, the interpolators have finite nonzero overlap and residue, the LSZ hypotheses hold, and every induced action and regulated-Jacobian term is treated consistently order by order. Higher-order terms cannot be inferred by simply imposing the lowest-order classical EOM repeatedly. Criado and Pérez-Victoria 2019, §§ 2–3, pp. 5–11 develops these source, Jacobian, LSZ, and multiple-insertion distinctions. The full equivalence theorem belongs to the downstream EFT treatment.

What each statement permits

Claim	Permitted conclusion	Conclusion not yet permitted
$\mathcal E=0$ on a classical solution	simplify the classical solution space	erase $\mathcal E(x)$ inside ordered quantum correlators
$\int\partial_\mu V^\mu=0$ with vanishing flux	remove the integrated boundary term in that functional	declare $\partial_\mu V^\mu(x)=0$ locally
finite-regulator change of variables is valid	derive the identity including $\delta_RX$, Jacobian, source, and boundary terms	discard every EOM-proportional operator from every observable
a forward regular matrix element kills $iq_\mu$	simplify that selected matrix element	infer equality of off-forward form factors or local insertions
a downstream equivalence theorem applies	identify the stated on-shell amplitudes	infer equality of off-shell Green functions or composite definitions

The applicable quotient becomes coarser as more structure is discarded. Local insertions retain the most information; integrated functionals remove controlled boundary terms; on-shell amplitudes remove additional external-leg data under theorem-specific hypotheses.

Common mistakes about redundancy

“EOM operators vanish.” They vanish on classical solutions, and the canonical Heisenberg equation can hold as an operator identity. The variational or $\mathrm T^\ast$ insertion used in a source identity is different: it records delta-supported variations of the other fields.

“Total derivatives never matter.” Their integrals depend on switching functions and boundary flux. Their local matrix elements carry momentum transfer.

“A field redefinition only changes the Lagrangian.” It also changes sources, observables, the regulated measure, and possibly the integration domain.

“A trivial Jacobian in one scheme is a universal theorem.” Jacobian statements are regulator and transformation dependent. A nontrivial regulated measure contribution can be one manifestation of an anomaly, and its allocation among Ward-identity terms can depend on the scheme.

“On shell means all correlators agree.” On-shell amplitude equivalence is weaker than equality of off-shell Green functions, local composites, or contact terms.

Check your understanding

Check 1: recover the local contact

Set $n=1$ in the variational insertion identity. The result is $\langle\mathrm T^\ast\{\mathcal E(x)\phi(y)\}\rangle=i\delta(x-y)$. For the free action, $\mathcal E=-P\phi$ gives $-P_xD_F=i\delta$, or $P_xD_F=-i\delta$. The literal canonical product with $P\phi$ is zero; the contact comes from applying $P_x$ outside time ordering.

Check 2: integrate the scalar EOM term

Use $\partial_\mu(\chi^3\partial^\mu\chi)=3\chi^2(\partial\chi)^2+\chi^3\Box\chi$. This gives the displayed local relation and isolates the surface flux that must vanish before the integrated basis relation follows.

Check 3: identify every change-of-variables term

For an observable $X$, the infinitesimal substitution changes $X$, the measure, the action, and the source coupling. Omitting any one of $\delta_RX$, $\operatorname{div}_\mu R$, $R\cdot\mathcal E$, or $R\cdot J$ changes the identity.

Check 4: compare local and forward statements

The local divergence $\partial_\mu V^\mu$ need not vanish. Its regular forward matrix element has $q=0$ and can vanish, while an off-forward matrix element carries $iq_\mu$. State which object is being simplified before calling it redundant.

Continue to operator-basis reduction

• Integration by Parts and Equation-of-Motion Redundancy develops the source-, boundary-, ordering-, and observable-aware quotient used in EFT bases.
• Renormalization and Effective Field Theory develops renormalized operator bases, mixing, field redefinitions, equivalence theorems, and amplitudes.
• Changes of Variables and Regulated Jacobians supplies the finite-dimensional measure calculation behind the quantum identity.

References

• Burgess, C. P. Introduction to Effective Field Theory: Thinking Effectively about Hierarchies of Scale. Cambridge: Cambridge University Press, 2021. DOI.

• Criado, J. C., and M. Pérez-Victoria. “Field Redefinitions in Effective Theories at Higher Orders.” Journal of High Energy Physics 2019, article 38 (2019). DOI. Open preprint.

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. First ed. Cambridge: Cambridge University Press, 2014. DOI.

• Zinn-Justin, Jean. Quantum Field Theory and Critical Phenomena. Fifth ed. Oxford: Oxford University Press, 2021. DOI.