Off-shell local polynomial scalar functionals

Let $\mathbb{M}={\mathbb{R}}^4$ with coordinates $x=(x^0,x^1,x^2,x^3)$, metric $\eta =\mathrm{diag}(-1,1,1,1)$ and time orientation given by increasing $x^0$. A field configuration is any smooth real function $\varphi \in C^{\mathrm{\infty }}(\mathbb{M};\mathbb{R})$; it is off shell, meaning that it is not required to satisfy a field equation. For an integer $s\ge 0$, its finite jet at $x$ is the finite tuple $j^s\varphi (x)=({\mathrm{\partial }}^{\alpha }\varphi (x){)}_{\mathrm{\mid }\alpha \mathrm{\mid }\le s}$, with multi-index derivatives in these coordinates.

An off-shell local polynomial scalar functional is a finite sum $F(\varphi )=c+{\sum }_{a=1}^N{\int }_{\mathbb{M}}{f}_a(x)P_a(j^{s_a}\varphi (x))d^{4}x,$ where $c\in \mathbb{C}$, $f_a\in C_c^{\mathrm{\infty }}(\mathbb{M};\mathbb{C})$, each $P_a$ is a polynomial in the finitely many jet coordinates, and each $s_a$ is finite. Its support is contained in the compact set ${\bigcup }_{a=1}^N\mathrm{supp}{f}_a$. The collection of these functionals is denoted ${\mathcal{F}}_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}}(\mathbb{M})$. The involution is $F^{\ast }(\varphi )=\overline{F(\varphi )}$. No physical units or field equation are imposed.

Precisely, $\mathrm{supp}F$ is the set of $x\in \mathbb{M}$ such that every neighborhood $U$ of $x$ contains the support of some $h\in C_c^{\mathrm{\infty }}(\mathbb{M};\mathbb{R})$ for which $F(\varphi +h)\mathrm{\ne }F(\varphi )$ for some smooth configuration $\varphi$. It is closed and is contained in the displayed compact union, so it is compact. This definition is independent of the chosen finite presentation of $F$.

For $r\ge 1$, define the $r$th functional derivative by $F^{(r)}(\varphi )(h_1,\dots ,h_r)={\frac{{\mathrm{\partial }}^r}{\mathrm{\partial }{t}_1\cdots \mathrm{\partial }{t}_r}F\text{\hspace{0.17em}⁣}(\varphi +{\sum }_{j=1}^r{t}_j{h}_j)\mid }_{t_1=\cdots =t_r=0},h_j\in C_c^{\mathrm{\infty }}(\mathbb{M};\mathbb{R})\mathrm{.}$ The finite polynomial expansion makes this a symmetric multilinear form. Repeated integration by parts in each $h_j$ writes it as a compactly supported distribution kernel on the thin diagonal $\{(x,\dots ,x):x\in \mathbb{M}\}\subset {\mathbb{M}}^r$. Integrals use the fixed Minkowski density $d^{4}x$; all kernels are paired bilinearly with compactly supported smooth tests.

Scalar distributions on smooth manifolds

Wavefront and positive-frequency Hadamard data