Fixed-order integration-by-parts and leading-equation field-redefinition relations

Assume full AC. Use four-dimensional natural units $c=\mathrm{\hslash }=1$ and a finite jet-polynomial field model with its declared finite rational action of a complex reductive affine algebraic group $G$. Let $W_D$ be a finite raw local-polynomial space at fixed engineering weight $D$, derivative cutoff, and field-degree cutoff; let $V_D=W_D^G$. Let ${\tilde{R}}_D$ be the $G$-stable linear span in $W_D$ of the declared polynomial identities, admitted total-divergence densities, and leading-equation densities $F^a{\mathcal{E}}_a^{(0)}$ defined below, with $R_D={\tilde{R}}_D\cap V_D$. The finite-order invariant operator quotient is ${\mathcal{Q}}_D=V_D\mathrm{/}{R}_D$.

Let $M={\mathbb{R}}^{1,3}$ with $x^0=ct$, $\eta =\mathrm{diag}(-1,1,1,1)$, and volume measure $d^{4}x$. Even field components are smooth maps $M\to {\mathrm{\Phi }}_a^{\bar{0}}$ and odd components are smooth coefficient families in a fixed Grassmann algebra valued in ${\mathrm{\Phi }}_a^{\bar{1}}$; admitted variations and currents have compact support. Let $\mathcal{I}[L]={\int }_{M}L{d}^{4}x$ on admitted local densities, with the boundary convention $\mathcal{I}({\mathrm{\partial }}_{\mu }{J}^{\mu })=0$. Let $S_0[\varphi ]$ and each $S_j[\varphi ]$ be $G$-invariant local polynomial functionals of these field configurations, so their formal Taylor expansions through order $N$ are defined. Let the variational derivative ${\mathcal{E}}_a^{(0)}[\varphi ]$ of $S_0$ be defined by

$$D{S}_0[\varphi ](h)=\mathcal{I}\text{\hspace{0.17em}⁣}(\sum _a{h}^a{\mathcal{E}}_a^{(0)}[\varphi ])$$

for every admitted compactly supported parity-compatible variation $h$. A leading-equation relation is a density $F^a{\mathcal{E}}_a^{(0)}$ with $F^a[\varphi ]$ a smooth function of $x$ and finitely many component jets of $\varphi$ at $x$, with this density in $W_D$, and satisfying the following conditions: its component parity matches that of ${\varphi }^a$; it is $G$-equivariant, meaning $F[g\cdot \varphi ]=g\cdot F[\varphi ]$; and it preserves compact support, meaning $\mathrm{supp}F[\varphi ]\subseteq \mathrm{supp}\varphi$ for compactly supported $\varphi$. At each configuration where the action identities below are applied, require $F[\varphi ]$ itself to be an admitted compactly supported variation, so the stated first-variation identity applies to this direction. An integration-by-parts relation is a density ${\mathrm{\partial }}_{\mu }{J}^{\mu }$ satisfying the stated boundary convention. Consider a finite formal action $S_{ϵ}=S_0+{\sum }_{j=1}^N{ϵ}^j{S}_j$ in the ring $\mathbb{C}[ϵ]\mathrm{/}({ϵ}^{N+1})$, where $ϵ$ is dimensionless and each ${ϵ}^k{F}^a$ has the units of ${\varphi }^a$. For every $1\le k\le N$, the local identity-leading change of variables ${\mathrm{\Psi }}_k(\varphi )=\varphi -{ϵ}^{k}F[\varphi ]$ is invertible as a formal map through order $N$ and satisfies

$$S_{ϵ}[{\mathrm{\Psi }}_k(\varphi )]=S_{ϵ}[\varphi ]-{ϵ}^k\mathcal{I}\text{\hspace{0.17em}⁣}(F^a{\mathcal{E}}_a^{(0)}[\varphi ])+O({ϵ}^{k+1})\mathrm{.}$$

Thus a coefficient $+{ϵ}^k\mathcal{I}(F^a{\mathcal{E}}_a^{(0)})$ can be removed at order $k$ without changing lower-order coefficients, provided all induced terms through the retained order are kept. The quotient ${\mathcal{Q}}_D$ identifies exactly the declared polynomial, boundary, and leading-equation relation span; it asserts no equivalence of arbitrary off-shell Green functions and does not by itself assert a quantum measure or amplitude equivalence.

Finite-jet local operators and engineering dimension

Finite-order invariant operator-basis quotient

Exactness of invariant vectors for a finite-dimensional quotient

Burgess, §2.1, equations (21)–(26), and §2.2

Neubert, §2, equations (27)–(29)

Criado–Pérez-Victoria, §§3 and 5.1