Finite-jet local operators and engineering dimension

Use natural units $c=\mathrm{\hslash }=1$ on four-dimensional Minkowski space $M={\mathbb{R}}^{1,3}$ with coordinates $x^0=ct,x^1,x^2,x^3$ and metric $\eta =\mathrm{diag}(-1,1,1,1)$. Fix a complex affine algebraic group $G$ with no nontrivial closed normal unipotent subgroup, where unipotent means that every nonzero finite-dimensional rational module has a nonzero fixed vector; this is the reductive hypothesis. Fix a finite set $I$ of field species. For each $i\in I$, fix a finite-dimensional complex $\mathbb{Z}\mathrm{/}2$-graded vector space ${\mathrm{\Phi }}_i={\mathrm{\Phi }}_i^{\bar{0}}\oplus {\mathrm{\Phi }}_i^{\bar{1}}$ with a homogeneous basis and its component parities, and an algebraic representation ${\rho }_i:G\to \mathrm{GL}({\mathrm{\Phi }}_i)$ preserving the grading. Include the declared spacetime-index action in $G$ or in its specified finite-dimensional representation data.

For the field interpretation, fix a finite Grassmann coefficient algebra $\mathcal{G}={\mathrm{\Lambda }}_{\mathbb{C}}({\zeta }_1,\dots ,{\zeta }_s)$, $s\ge 0$, with its usual grading by generator degree modulo two. A field configuration assigns to each homogeneous component of parity $\bar{p}$ a function $M\to {\mathcal{G}}_{\bar{p}}$ whose coefficient at each exterior monomial is smooth; spacetime derivatives are taken coefficientwise. Evaluation of a polynomial on an $r$-jet means the parity-preserving algebra homomorphism that sends each polynomial coordinate to the corresponding field component or derivative at that point.

Fix a derivative cutoff $r\ge 0$, a maximum polynomial field degree $q\ge 0$, and a rational engineering weight ${\mathrm{\Delta }}_i$ for each field species. A derivative ${\mathrm{\partial }}_{\mu }$ has engineering weight one, and an $s$-jet generator is a component of ${\mathrm{\partial }}_{{\mu }_1}\cdots {\mathrm{\partial }}_{{\mu }_s}{\varphi }_i(x)$ for $0\le s\le r$. If covariant derivatives are used, their finite jet generators and $G$-action are included explicitly in the same data. Let $J_r$ be the resulting finite-dimensional $\mathbb{Z}\mathrm{/}2$-graded vector space with its specified algebraic representation ${\rho }_r:G\to \mathrm{GL}(J_r)$, required to preserve the grading and the assigned engineering weights. Its polynomial algebra is the supercommutative algebra $P_r=\mathrm{Sym}(J_{r,\bar{0}}^{\ast })\otimes \mathrm{\Lambda }(J_{r,\bar{1}}^{\ast })$: even generators commute and odd generators anticommute. The induced left $G$-action on $P_r$ is $(g\cdot f)(z)=f({\rho }_r(g^{-1})z)$ for $g\in G$, $f\in P_r$, and $z\in J_r$. For a fixed weight $D\in \mathbb{Q}$, let $P_{r,q,D}$ be the complex span of monomials of total engineering weight $D$ and total polynomial degree at most $q$.

A local operator in this finite-order data is an element of $P_{r,q,D}$ evaluated at $x$ on the $r$-jet of the fields. A scalar operator invariant under the declared symmetries is an element of $V_{r,q,D}=P_{r,q,D}^G$. Engineering dimension means this assigned mass weight; it is not a statement about an observed quantity. In four dimensions a term with operator ${\mathcal{O}}_D$ has the convention $C{\mathcal{O}}_D\mathrm{/}{\mathrm{\Lambda }}^{D-4}$, where $\mathrm{\Lambda }$ has mass dimension one and $C$ is dimensionless. Reality conditions, when required for a real action, are additional conjugation data on $V_{r,q,D}$.

Reductive and linearly reductive complex algebraic groups

Burgess, §§2.1–2.2

Neubert, §2