Manuscript XIV-C@1 · XIV-C.tex · section · a34d13df

Individual line orders

J1: copied branches, line orders, and collision inventory › Individual line orders · lines 681–766

XIV-C@1#eq:cross-orders

A displayed divergence is transferred to its incoming numerical line. Derivatives on unused fields are not transferred to an unrelated line.

In the following table a superscript on a field slot is its derivative order, and an edge superscript is the derivative order on that internal retarded line. The scalar kinetic terms give the displayed maxima; the mass terms have the corresponding lower orders.

Branch	Observable field slots	Internal lines and orders	Maximum total
$B_{1}$	$Γ^{0} (u), Γ^{0} (v), φ^{1} (v)$	$(x - u)^{1}, (u - v)^{2}$	$4$
$B_{2}$	$Γ^{0} (u), Γ^{0} (u), φ^{1} (u)$	$(x - u)^{1}$	$2$
$B_{3}$	$Γ^{0} (u), Γ^{0} (v), φ^{1} (v)$	$(x - u)^{1}, (x - v)^{2}$	$4$
$B_{4}$	$Γ^{0} (u), Γ^{0} (v), φ^{1} (x)$	$(x - u)^{1}, (x - v)^{2}$	$4$
$B_{5}$	$Γ^{0} (u), Γ^{0} (v), φ^{1} (x)$	$(x - u)^{1}, (u - v)^{2}$	$4$
$B_{6}$	$Γ^{0} (u), Γ^{0} (u), φ^{1} (x)$	$(x - u)^{1}$	$2$
$B_{7}$	$Γ^{0} (u), Γ^{0} (v), φ^{2} (x)$	$(x - u)^{1}, (x - v)^{1}$	$4$
$B_{8}$	$Γ^{∣ α ∣} (v), Γ^{∣ β ∣} (v), φ^{1} (x)$	$(x - u)^{1}, (u - v)^{0}$	$4$
$B_{9}$	$Γ^{∣ α ∣} (v), Γ^{∣ β ∣} (v), φ^{1} (u)$	$(x - u)^{1}, (u - v)^{0}$	$4$
$B_{10}$	$φ^{e_{s}} (v), φ^{e_{s}} (v), φ^{1} (x)$	$(x - u)^{1}, (u - v)^{0}$	$4$
$B_{11}$	$φ^{e_{s}} (v), φ^{e_{s}} (v), φ^{1} (u)$	$(x - u)^{1}, (u - v)^{0}$	$4$

For the gravitational source terms, $∣ α ∣ + ∣ β ∣ \leq 2$ , including their fixed layer coefficients. For a stress monomial $e_{s} = 1$ in the kinetic case and $e_{s} = 0$ in the mass case.

The interaction slots at $y$ have orders

Γ^{0} (y), φ^{e_{v}} (y), φ^{e_{v}} (y), e_{v} \in {0, 1} .

Therefore each cross-line order is fixed by its endpoints:

d_{γ γ} = d_{Γ, obs}, d_{ϕϕ} = e_{v} + d_{φ, obs} .

In $B_{7}$ the scalar cross-line can have order three. In $B_{8}$ and $B_{9}$ the metric cross-line carries the selected source derivative. For $B_{10}$ and $B_{11}$ , each of the two scalar cross-lines separately uses (eq:cross-orders).

These assignments name line orders, not a tensor truncation. The complete graph has maximum total derivative count six after inclusion of the interaction.

Displayed formulas in this passage

XIV-C@1#eq:cross-orders · line 752
$d_{γ γ} = d_{Γ, obs}, d_{ϕϕ} = e_{v} + d_{φ, obs} .$
unlabeled · line 746
$Γ^{0} (y), φ^{e_{v}} (y), φ^{e_{v}} (y), e_{v} \in {0, 1} .$