Proposition: Complete source as a jet polynomial

For each $A\in \{C_2,M_2,J_3\}$ there are unique coefficients ${P^{ab,\mu \nu }}_{A,ij}(y)$, indexed by $0\le i,j\le 9$ and multi-indices $a,b$, such that

for all smooth symmetric $X,Y$, read as an identity of polynomials in the independent jet coordinates of $X$ and $Y$. Moreover:

1. Quadratic homogeneity. Every monomial of the unpolarized ${\mathfrak{s}}_A(\gamma )$ has degree exactly $2$ in the jet of $\gamma$; hence ${\mathfrak{s}}_A(\lambda \gamma )={\lambda }^2{\mathfrak{s}}_A(\gamma )$ for $\lambda \in \mathbb{R}$.

2. Bilinearity and diagonal. ${\mathfrak{s}}_A[X,Y]$ is bilinear, and ${\mathfrak{s}}_A[X,X]={\mathfrak{s}}_A(X)$.

3. Slot symmetry. ${P^{ab,\mu \nu }}_{A,ij}={P^{ba,\mu \nu }}_{A,ji}$; also ${P^{ab,\mu \nu }}_{A,ij}={P^{ab,\nu \mu }}_{A,ij}$.

4. Distribution of derivatives (exact, not only a maximum).

   • $A=C_2$: $P_{C_2}$ is a linear form in $\partial \partial {\sigma }_{\epsilon }(y)$, and ${P^{ab}}_{C_2,ij}\ne 0$ only for $|a|=|b|=0$.

   • $A=M_2$: $P_{M_2}$ is a linear form in $\partial {\sigma }_{\epsilon }(y)$, and ${P^{ab}}_{M_2,ij}\ne 0$ only for $|a|+|b|=1$.

   • $A=J_3$: $P_{J_3}$ is a constant multiple of ${\sigma }_{\epsilon }(y)$, and ${P^{ab}}_{J_3,ij}\ne 0$ only for $|a|+|b|=2$.

   In every monomial of ${\mathcal{J}}_{\mathrm{D}}$ the number of derivatives on ${\sigma }_{\epsilon }$ plus the number on the two slots equals $2$.

5. Rational coefficients. Each ${P^{ab,\mu \nu }}_{A,ij}(y)$ is a linear combination, with rational constant coefficients, of the components of the ${\sigma }_{\epsilon }$-jet of the order stated in (4). There is no other $y$-dependence; in particular neither $h$ nor ${\chi }_{\mathrm{I}}$ occurs.

6. Agreement with XVI-K. ${\mathfrak{s}}_{C_2}[X,Y]$ is the polarized source ${\mathfrak{s}}^{\mu \nu }(X,Y;S)$, $S_{\alpha \beta }={\partial }_{\alpha }{\partial }_{\beta }{\sigma }_{\epsilon }$, of XVI-K, Definition def:a (XVI-J, eq:Q), identically.

Proof. Count, for each monomial of each printed atom, its degree in $\gamma$ and its number of derivatives (on $\gamma$ and on ${\sigma }_{\epsilon }$), as in the proof of XVI-K, Proposition prop:hess. $u_n$, $b_n$ have degree $n$ and no derivative. $C_0$ is constant; $C_1$ and $C_2$ of (eq:L-C) are sums of products of $b$'s, $u$'s and $\eta$ of total degree $1$ and $2$, with no derivative. ${\Gamma }_n$ has degree $n$ and exactly one derivative. In $R_n$, $\partial {\Gamma }_n$ has degree $n$ and two derivatives (Leibniz' rule distributes them over the factors), and ${\Gamma }_a{\Gamma }_b$ has degree $a+b=n$ and two derivatives; so $R_n$, $r_n$ have degree $n$ and two derivatives. Hence $V_1$ has degree $1$, $V_2$ degree $2$ (the term $\gamma r_1$ included), both with two derivatives, and by (eq:L-J3) every monomial of $J_3$ has degree $2$ and exactly two derivatives on $\gamma$. By (eq:L-M2), every monomial of $M_2$ has degree $2$ ($C_0{\Gamma }_2$: $0+2$; $C_1{\Gamma }_1$: $1+1$) and exactly one derivative on $\gamma$. $C_2$ has degree $2$ and none. Multiplying by ${\sigma }_{\epsilon }$, ${\partial }_{\rho }{\sigma }_{\epsilon }$, ${\partial }_{\alpha }{\partial }_{\beta }{\sigma }_{\epsilon }$ respectively gives (1) and, for the unpolarized source, the statements of (4) and (5): each summand is linear in the ${\sigma }_{\epsilon }$-jet of order $0$, $1$, $2$, with no derivative of ${\sigma }_{\epsilon }$ of another order, and with $2$, $1$, $0$ derivatives on $\gamma$. All printed numerical coefficients are rational ($\pm 1,\pm 2,\pm \frac{1}{2},\frac{1}{4},\frac{1}{8}$) and $\eta$ has entries $0,\pm 1$, which gives rationality. No atom contains $h$ or ${\chi }_{\mathrm{I}}$.

A polynomial $F$ homogeneous of degree $2$ in independent jet coordinates ${\gamma }_I$ can be written $F={\sum }_{I\le J}{c}_{IJ}{\gamma }_I{\gamma }_J$; its polarization $\frac{1}{2}(F(X+Y)-F(X)-F(Y))$ equals ${\sum }_{I\le J}\frac{1}{2}{c}_{IJ}(X_I{Y}_J+X_J{Y}_I)$. This is bilinear, symmetric under $X\leftrightarrow Y$, and returns $F(X)$ on $Y=X$; this proves (2) and (3) (the $(\mu ,\nu )$ symmetry is the symmetry of $J_3$, $C_2$, $M_2$ in $\mu \nu$, immediate from (eq:L-J3)–(eq:L-M2) because $R_1$, $R_2$ are symmetric; the latter is checked by the script). A jet coordinate ${\gamma }_I$ with $I=((ij),a)$ stands for ${\partial }^a{\gamma }_{ij}$; polarization replaces the pair ${\gamma }_I{\gamma }_J$ by $X_I{Y}_J+X_J{Y}_I$, which keeps the derivative orders $|a|$, $|b|$ of the two factors. So the slot orders $(|a|,|b|)$ of (eq:jet) are the derivative orders of the two $\gamma$-factors of the unpolarized monomials, and (4) follows. The expansion (eq:jet) exists by collecting terms, and is unique because distinct jet coordinates are independent variables. (6): XVI-K, Definition def:a, polarizes $Q^{\mu \nu }=2{C_2}^{\mu \nu \alpha \beta }{\partial }_{\alpha }{\partial }_{\beta }{\sigma }_{\epsilon }$ by the same rule, with $C_2$ of the same printed formula (eq:L-C); the two definitions coincide.