Manuscript XII-D@1 · XII-D.tex · section · 308a2080

The root kernels

D3: the completed-kernel cone calculation › The root kernels · lines 784–825

XII-D@1#eq:A-kernels, XII-D@1#eq:A-WF, XII-D@1#eq:B-kernel

Write

α_{f, g} [ϕ] (z) = \int A_{f, g} (z, x) ϕ (x) d^{4} x .

Its two components are explicitly

A_{co}^{ab} (z, x) A_{sc}^{ab} (z, x) = σ_{ε} (z) a^{λ μ, ab} \partial_{μ}^{x} [f (x) \partial_{λ}^{z} E_{0}^{A} (z, x)], = - g (z) a^{μν, ab} \partial_{μ} v_{f} (z) \partial_{ν}^{x} δ (x - z) - \frac{m ^{2}}{2} g (z) η^{ab} v_{f} (z) δ (x - z) .

In the first term of $A_{sc}$ , the coefficient depends on $z$ , so differentiation acts only on the displayed delta.

Translation invariance of the Minkowski Green kernel gives

WF (A_{f, g}) \subset Z_{2} .

This includes arbitrary diagonal Green covectors; no positive-frequency restriction is used.

Let $B_{f, g} (y, x)$ be the kernel of the compact root (eq:compact-root):

B_{f, g} (y, x) = - ϑ (y^{0}) \int E_{0}^{A} (y, z) P A_{f, g} (z, x) d^{4} z .

Displayed formulas in this passage

XII-D@1#eq:A-kernels · line 791
$A_{co}^{ab} (z, x) A_{sc}^{ab} (z, x) = σ_{ε} (z) a^{λ μ, ab} \partial_{μ}^{x} [f (x) \partial_{λ}^{z} E_{0}^{A} (z, x)], = - g (z) a^{μν, ab} \partial_{μ} v_{f} (z) \partial_{ν}^{x} δ (x - z) - \frac{m ^{2}}{2} g (z) η^{ab} v_{f} (z) δ (x - z) .$
XII-D@1#eq:A-WF · line 809
$WF (A_{f, g}) \subset Z_{2} .$
XII-D@1#eq:B-kernel · line 818
$B_{f, g} (y, x) = - ϑ (y^{0}) \int E_{0}^{A} (y, z) P A_{f, g} (z, x) d^{4} z .$
unlabeled · line 787
$α_{f, g} [ϕ] (z) = \int A_{f, g} (z, x) ϕ (x) d^{4} x .$