Manuscript XIII-A@1 · XIII-A.tex · section · f56e91c7

Terms remaining for homogeneous ghosts

A2: the free differential, tangency, and boundary terms › Terms remaining for homogeneous ghosts · lines 830–904

XIII-A@1#eq:ghost-coordinate, XIII-A@1#eq:ghost-scalar, XIII-A@1#eq:ghost-X2, XIII-A@1#eq:ghost-p2, XIII-A@1#eq:source-derivative-kernel

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Suppose now that the multiplet is on the free shell. Then $D z = 0$ and $\partial_{μ} a_{1}^{μν} [γ] = - b^{ν}$ . The first coordinate variation is

Z^{μ} = - E_{0}^{R} (σ_{ε}^{'} a_{1}^{0 μ} [z]) .

Thus the free ghost equation removes the $σ_{ε} D z$ term, not the coordinate layer term.

For $v = c^{μ} \partial_{μ} ϕ$ , direct differentiation gives $V_{1, 1} [z] ϕ = - \frac{1}{2} P_{m} v$ on the free scalar shell. Because $g v$ is compactly supported in the slab interior,

p_{z} = \frac{1}{2} g v - E_{m}^{R} (\frac{1}{2} (□ g) v + (\partial^{μ} g) \partial_{μ} v + (\partial_{μ} g) a_{1}^{μν} [z] \partial_{ν} ϕ) .

This formula retains the local switched term and both parts of $[P_{m}, g] v$ . It requires no compact support assumption on $c$ .

For completeness, write $Y = s_{0} X_{2}$ and $p_{2, z} = s_{0} p_{2}$ . Expanding all external weight derivatives in the completion gives

Y^{μ} = - E_{0}^{R} {p_{2, z} = - E_{m}^{R} {σ_{ε} (a_{1}^{α β} [z] \partial_{α} \partial_{β} X^{μ} + a_{1}^{α β} [γ] \partial_{α} \partial_{β} Z^{μ} - b^{β} \partial_{β} Z^{μ} + \partial_{α} a^{α μ}) + σ_{ε}^{'} (a_{1}^{0 β} [z] \partial_{β} X^{μ} + a_{1}^{0 β} [γ] \partial_{β} Z^{μ} + a^{0 μ})}, g (a_{1}^{μν} [z] \partial_{μ} \partial_{ν} p + a_{1}^{μν} [γ] \partial_{μ} \partial_{ν} p_{z} - b^{ν} \partial_{ν} p_{z} + m^{2} (b_{1} [z] p + b_{1} [γ] p_{z}) + a^{μν} \partial_{μ} \partial_{ν} ϕ + (\partial_{μ} a^{μν}) \partial_{ν} ϕ + m^{2} b ϕ) + (\partial_{μ} g) (a_{1}^{μν} [z] \partial_{ν} p + a_{1}^{μν} [γ] \partial_{ν} p_{z} + a^{μν} \partial_{ν} ϕ)} .

All source-owned ramp terms remain inside the fixed derivative $k ˙$ ; none is set to zero. This notation does not require a new expansion of the gravitational Euler source. Equivalently, if $t_{D}$ is XII-E's inherited polarized insertion kernel, its entire contribution is fixed by

A_{1} [f, g] (k ˙; ϕ) = - s_{0} C_{3, D} = - 2 ⟨ t_{D}, γ \otimes z \otimes ϕ ⟩ .

This specifies that contribution without discarding any of its smooth source coefficients.

Displayed formulas in this passage

XIII-A@1#eq:ghost-coordinate · line 835
$Z^{μ} = - E_{0}^{R} (σ_{ε}^{'} a_{1}^{0 μ} [z]) .$
XIII-A@1#eq:ghost-scalar · line 845
$p_{z} = \frac{1}{2} g v - E_{m}^{R} (\frac{1}{2} (□ g) v + (\partial^{μ} g) \partial_{μ} v + (\partial_{μ} g) a_{1}^{μν} [z] \partial_{ν} ϕ) .$
XIII-A@1#eq:ghost-X2 · line 860
$Y^{μ} = - E_{0}^{R} {p_{2, z} = - E_{m}^{R} {σ_{ε} (a_{1}^{α β} [z] \partial_{α} \partial_{β} X^{μ} + a_{1}^{α β} [γ] \partial_{α} \partial_{β} Z^{μ} - b^{β} \partial_{β} Z^{μ} + \partial_{α} a^{α μ}) + σ_{ε}^{'} (a_{1}^{0 β} [z] \partial_{β} X^{μ} + a_{1}^{0 β} [γ] \partial_{β} Z^{μ} + a^{0 μ})}, g (a_{1}^{μν} [z] \partial_{μ} \partial_{ν} p + a_{1}^{μν} [γ] \partial_{μ} \partial_{ν} p_{z} - b^{ν} \partial_{ν} p_{z} + m^{2} (b_{1} [z] p + b_{1} [γ] p_{z}) + a^{μν} \partial_{μ} \partial_{ν} ϕ + (\partial_{μ} a^{μν}) \partial_{ν} ϕ + m^{2} b ϕ) + (\partial_{μ} g) (a_{1}^{μν} [z] \partial_{ν} p + a_{1}^{μν} [γ] \partial_{ν} p_{z} + a^{μν} \partial_{ν} ϕ)} .$
XIII-A@1#eq:ghost-p2 · line 860
$Y^{μ} = - E_{0}^{R} {p_{2, z} = - E_{m}^{R} {σ_{ε} (a_{1}^{α β} [z] \partial_{α} \partial_{β} X^{μ} + a_{1}^{α β} [γ] \partial_{α} \partial_{β} Z^{μ} - b^{β} \partial_{β} Z^{μ} + \partial_{α} a^{α μ}) + σ_{ε}^{'} (a_{1}^{0 β} [z] \partial_{β} X^{μ} + a_{1}^{0 β} [γ] \partial_{β} Z^{μ} + a^{0 μ})}, g (a_{1}^{μν} [z] \partial_{μ} \partial_{ν} p + a_{1}^{μν} [γ] \partial_{μ} \partial_{ν} p_{z} - b^{ν} \partial_{ν} p_{z} + m^{2} (b_{1} [z] p + b_{1} [γ] p_{z}) + a^{μν} \partial_{μ} \partial_{ν} ϕ + (\partial_{μ} a^{μν}) \partial_{ν} ϕ + m^{2} b ϕ) + (\partial_{μ} g) (a_{1}^{μν} [z] \partial_{ν} p + a_{1}^{μν} [γ] \partial_{ν} p_{z} + a^{μν} \partial_{ν} ϕ)} .$
XIII-A@1#eq:source-derivative-kernel · line 896
$A_{1} [f, g] (k ˙; ϕ) = - s_{0} C_{3, D} = - 2 ⟨ t_{D}, γ \otimes z \otimes ϕ ⟩ .$

Terms remaining for homogeneous ghosts

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