Manuscript XII-A@2 · XII-A-v2.tex · section · 955d4b22

Named composites and their formal sector

Named composites and their formal sector · lines 1051–1141

XII-A@2#eq:B, XII-A@2#eq:T, XII-A@2#eq:sumzero, XII-A@2#eq:compositesupports

Set

a^{α μ, ab} = \frac{1}{2} η^{α μ} η^{ab} - η^{α (a} η^{b) μ}, Λ^{μ, ab} (x, z) = σ_{ε} (z) a^{α μ, ab} \partial_{α}^{z} E_{0}^{R} (x - z) .

Define

X_{1}^{μ} [γ] (x) = \int Λ^{μ, ab} (x, z) γ_{ab} (z) d z

and

B_{f} = - \int f X_{1}^{μ} [γ] \partial_{μ} ϕ, b_{f}^{ab} (x, z) = \partial_{μ}^{x} [f (x) Λ^{μ, ab} (x, z)] .

Thus $B_{f} = \int b_{f}^{ab} (x, z) γ_{ab} (z) ϕ (x) d x d z$ . The kernel of $s_{0} B_{f}$ is obtained by applying $R_{z}^{T}$ to its metric slot.

For the scalar stress, let $τ^{ab} (y; y_{1}, y_{2})$ be the symmetric bilinear local kernel of the frozen quadratic stress $T_{0}^{ab}$ :

T_{0}^{ab} [ϕ] (y) = ⟨ τ^{ab} (y; \cdot, \cdot), ϕ \otimes ϕ ⟩ .

Equivalently $τ$ is a fixed finite sum of products of derivatives of $δ (y - y_{1}) δ (y - y_{2})$ with constant coefficients. This convention includes no polarization factor beyond the one displayed.

The coordinate response to the metric Green operator is

Q^{μ}_{ab} (x, y) = \int Λ^{μ, c d} (x, z) (G_{gr}^{R})_{c d, ab} (z, y) d z .

For compact external sets the integration is over a compact causal region, because $z \in J^{-} (x) \cap J^{+} (y)$ and $t_{z} \geq 0$ . The retarded wavefront relations imply $WF (Q) \subset {(x, p; y, - p)}$ .

The frozen tree and its symmetric kernel are

T (f, g) t_{f, g} Θ_{f, g} Ξ^{μ} (x, y, y_{1}, y_{2}) = \frac{1}{2} \int f (x) Q^{μ}_{ab} (x, y) g (y) T_{0}^{ab} [ϕ] (y) \partial_{μ} ϕ (x) d x d y, = Sym_{3} Θ_{f, g}, = - \frac{1}{2} \partial_{μ}^{x} π_{y *} [(f \otimes g \otimes 1 \otimes 1) Ξ^{μ}], = Q^{μ}_{ab} (x, y) τ^{ab} (y; y_{1}, y_{2}) .

The factor $\frac{1}{2}$ is retained after integration by parts.

The product defining $Ξ$ exists: a canceled external covector would force $p = 0$ , hence the $Q$ covector to vanish; the other external covectors then force the $τ$ covector to vanish. The input relations and pushforward give

WF (b_{f}), WF (R_{z}^{T} b_{f}) WF (t_{f, g}) \subset Z_{2} := {p + q = 0} ∖ 0, \subset Z_{3} := {p + r_{1} + r_{2} = 0} ∖ 0.

A tuple in one closed causal cone cannot have zero sum unless it is zero. These kernels are therefore microcausal.

For tests supported in $K_{f}, K_{g}$ , valid compact support bounds are

K_{B} = K_{f} \times (J^{-} (K_{f}) \cap {t \geq 0}), K_{T} = π \in S_{3} ⋃ π (K_{f} \times K_{g} \times K_{g}) .

Support and cone bounds for a graded-symmetric quadratic kernel are understood after the corresponding finite permutations.

Displayed formulas in this passage

XII-A@2#eq:B · line 1069
$B_{f} = - \int f X_{1}^{μ} [γ] \partial_{μ} ϕ, b_{f}^{ab} (x, z) = \partial_{μ}^{x} [f (x) Λ^{μ, ab} (x, z)] .$
XII-A@2#eq:T · line 1102
$T (f, g) t_{f, g} Θ_{f, g} Ξ^{μ} (x, y, y_{1}, y_{2}) = \frac{1}{2} \int f (x) Q^{μ}_{ab} (x, y) g (y) T_{0}^{ab} [ϕ] (y) \partial_{μ} ϕ (x) d x d y, = Sym_{3} Θ_{f, g}, = - \frac{1}{2} \partial_{μ}^{x} π_{y *} [(f \otimes g \otimes 1 \otimes 1) Ξ^{μ}], = Q^{μ}_{ab} (x, y) τ^{ab} (y; y_{1}, y_{2}) .$
XII-A@2#eq:sumzero · line 1121
$WF (b_{f}), WF (R_{z}^{T} b_{f}) WF (t_{f, g}) \subset Z_{2} := {p + q = 0} ∖ 0, \subset Z_{3} := {p + r_{1} + r_{2} = 0} ∖ 0.$
XII-A@2#eq:compositesupports · line 1134
$K_{B} = K_{f} \times (J^{-} (K_{f}) \cap {t \geq 0}), K_{T} = π \in S_{3} ⋃ π (K_{f} \times K_{g} \times K_{g}) .$
unlabeled · line 1055
$a^{α μ, ab} = \frac{1}{2} η^{α μ} η^{ab} - η^{α (a} η^{b) μ}, Λ^{μ, ab} (x, z) = σ_{ε} (z) a^{α μ, ab} \partial_{α}^{z} E_{0}^{R} (x - z) .$
unlabeled · line 1065
$X_{1}^{μ} [γ] (x) = \int Λ^{μ, ab} (x, z) γ_{ab} (z) d z$
unlabeled · line 1081
$T_{0}^{ab} [ϕ] (y) = ⟨ τ^{ab} (y; \cdot, \cdot), ϕ \otimes ϕ ⟩ .$
unlabeled · line 1091
$Q^{μ}_{ab} (x, y) = \int Λ^{μ, c d} (x, z) (G_{gr}^{R})_{c d, ab} (z, y) d z .$

Named composites and their formal sector

Displayed formulas in this passage

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