Redundant operators and field redefinitions

Algebraic identities, including tensor or spinor identities in a specified dimension, relate operator expressions directly. Integration by parts instead relates their contributions to an action when boundary terms vanish or are handled separately. For a scalar field that falls off appropriately, the integral of phi cubed times its d'Alembertian equals minus three times the integral of phi squared times its gradient contracted with itself. These expressions are not equal as local functions without the total derivative.

Equation-of-motion relations arise from perturbative changes of integration variables. At a fixed inverse-scale order, a local invertible field redefinition changes the leading action by its Euler–Lagrange derivative times the shift. That allows an operator proportional to the leading equations of motion to be removed while other terms are shifted. One must use the appropriate lower-order equations and keep every induced contribution at the target accuracy. Applying the full corrected equation repeatedly without an order prescription can mix orders inconsistently.

Use a free massive scalar as the leading Lagrangian and add two interactions suppressed by Lambda squared: a derivative operator with coefficient c and a mass-weighted quartic with coefficient d. Their dimensionless coefficients need not have any particular ultraviolet origin. Replace the old field by a new field plus c times its cube divided by Lambda squared. Expanding only through that order produces a term that cancels the original derivative operator and another that shifts the quartic coefficient.

The same answer follows by using the leading scalar equation of motion in the redundant operator, but the field-redefinition derivation explains what that substitution means. If a quartic interaction had already been included in the leading Lagrangian, its variation would also induce a six-field term at the same order. Omitting that induced term would change predictions. The completeness of the starting lower-order action is therefore essential when translating a reduced operator basis.

With a consistent treatment of sources, external-leg normalization, counterterms, and the functional measure, local perturbative redefinitions preserve physical S-matrix elements. Off-shell Green functions depend on the interpolating fields and can change. Thus one cannot set an equation-of-motion operator to zero inside every time-ordered correlation function without accounting for contact terms. The practical shortcut is safe only in the setting for which the operator equivalence has been established.

At loops, a regulator can require additional bookkeeping, such as evanescent operators that vanish in four dimensions but affect intermediate renormalization in a continued dimension. Coefficients quoted in different bases or schemes must be translated before comparison. Reducing redundancy does not remove genuine experimental information: it removes duplicate coordinates for that information. A basis choice should be accompanied by an order, a normalization, and a prescription for how retained coefficients and matrix elements transform together.