Graded kernels and evaluation

Odd fields are evaluated in arbitrary finite Grassmann algebras. An identity means equality of every Grassmann coefficient for every such evaluation. This convention applies to restriction, vanishing, polarization, and all evaluation maps below.

A polynomial has the form

where $u_n$ is a compactly supported, graded-symmetric distribution on $M^n$, with finitely many species components. Graded symmetrization is normalized:

The sign $ϵ(\pi )$ is the Koszul sign. Our kernels contain no additional $1/n!$. Consequently

All derivative and polarization identities are understood coefficientwise in the Grassmann algebra.

Let ${\tilde{V}}_{+},{\tilde{V}}_{-}$ be the closed causal covector cones, including zero. For $n\ge 1$, define

The polynomial microcausal carrier ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ consists of the above polynomials with

Degree zero consists of scalars.

The regular classical algebra ${{\mathcal{A}}^{\mathrm{cl}}}_{\mathrm{off}}$ is the subalgebra generated by compact smooth linear smearings; its kernels are finite sums of tensor products of tests. “Regular” here means finite-rank regular, not every smooth compact kernel.

Define the vector spaces

A product on ${\mathcal{E}}_H$ will be justified in Section sec:quotients; it is not assumed in this definition.