Finite-order invariant operator-basis quotient

Fix a finite jet order $r$, maximum polynomial field degree $q$, engineering weight $D$, finite field-species list, coefficient field $\mathbb{C}$, and complex reductive affine algebraic group $G$ with the finite-dimensional rational action specified on the jet generators, including the declared spacetime-index symmetries. Let $W_D$ be the finite-dimensional space of all local polynomials of weight $D$ and degree at most $q$, before imposing $G$-invariance, and let $V_D=W_D^G$ be its invariant scalar subspace. An admitted raw relation space is a specified $G$-stable linear subspace ${\tilde{R}}_D\le W_D$, given by explicit linear identities among the chosen polynomials. Its invariant part is $R_D={\tilde{R}}_D\cap V_D$. Any action, boundary, or observable interpretation of the selected identities is extra mathematical data and must be stated with the result that verifies it.

The finite-order invariant operator-basis quotient is

$${\mathcal{Q}}_D:=V_D\mathrm{/}{R}_D\mathrm{.}$$

Two invariant operators represent the same class exactly when their difference lies in $R_D$. An operator basis is any finite list of representatives in $V_D$ of a vector-space basis of ${\mathcal{Q}}_D$. The quotient records precisely the declared relation space; it does not assert that every relation relevant to an arbitrary EFT, observable, or infinite operator tower has been included.

Finite-jet local operators and engineering dimension

Finite dimensionality of fixed-order local operator spaces

Burgess, §§2.1–2.2

Neubert, §2

Criado–Pérez-Victoria, §§3 and 5.1