Manuscript XIII-B@1 · XIII-B.tex · section · b8a5c0f5

The composite entry

B5: the free differential, displayed without a defect › The composite entry · lines 1197–1294

XIII-B@1#eq:dotk, XIII-B@1#eq:hats, XIII-B@1#eq:weighted-divergences, XIII-B@1#eq:first-ghost-terms, XIII-B@1#eq:second-ghost-terms

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We give a compact explicit form of XIII-A's entry derivative. Set

z = R c, X = X_{1} [γ], p = p_{1} [γ; ϕ], Z = X_{1} [z], p_{z} = p_{1} [z; ϕ], k = r_{D} [γ] + r_{g} [ϕ] .

Let $X_{2}, p_{2}$ be the inherited second completion at $(γ, k; ϕ)$ . The composite is

O = \int f (p_{2} - X^{μ} \partial_{μ} p + (X^{ν} \partial_{ν} X^{μ} - X_{2}^{μ}) \partial_{μ} ϕ + \frac{1}{2} X^{μ} X^{ν} \partial_{μ} \partial_{ν} ϕ) d^{4} x .

Write $k ˙ = r_{D}^{'}_{γ} [z]$ . Since the actual source is $J_{D} = σ_{ε} J_{3} + 2 C_{2} σ_{ε}^{''} - 2 M_{2} σ_{ε}^{'}$ in the time-only ramp convention, its inherited derivative is

k ˙ = P E_{0}^{R} (σ_{ε} J_{3}^{'}_{γ} [z] + 2 C_{2}^{'}_{γ} [z] σ_{ε}^{''} - 2 M_{2}^{'}_{γ} [z] σ_{ε}^{'}) .

The tensors $C_{2}, M_{2}$ are exactly the finite quadratic source coefficients supplied by XII-D. This equation differentiates the existing response; it introduces no quantum argument $J_{D}$ .

Define

b ˙_{2} a_{2} ˙^{μν} a = \frac{1}{4} (tr_{η} γ) (tr_{η} z) - \frac{1}{2} γ_{α β} z^{α β}, = z^{μ}_{α} γ^{α ν} + γ^{μ}_{α} z^{α ν} - \frac{1}{2} ((tr_{η} z) γ^{μν} + (tr_{η} γ) z^{μν}) + b ˙_{2} η^{μν}, = a_{1} [k ˙] + a ˙_{2}, b = b_{1} [k ˙] + b ˙_{2} .

For any tensor coefficient $a$ and scalar or coordinate component $v$ , put

D_{σ}_{ε} (a, v) D_{g} (a, v) = σ_{ε} a^{μν} \partial_{μ} \partial_{ν} v + σ_{ε} (\partial_{μ} a^{μν}) \partial_{ν} v + σ_{ε}^{'} a^{0 ν} \partial_{ν} v, = g a^{μν} \partial_{μ} \partial_{ν} v + g (\partial_{μ} a^{μν}) \partial_{ν} v + (\partial_{μ} g) a^{μν} \partial_{ν} v .

Thus every external derivative of the weights is visible.

On the free shell, $D z = 0$ and, with $v = c^{μ} \partial_{μ} ϕ$ ,

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

The coordinate layer and scalar switch terms survive for unrestricted homogeneous ghosts.

Finally put $Y = s_{0} X_{2}$ and $p_{2, z} = s_{0} p_{2}$ . Their complete divergence-form derivatives are

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

Together with (eq:dotk), these formulas retain the source $σ_{ε}^{'}, σ_{ε}^{''}$ terms as well as all coordinate and scalar switching terms.

Displayed formulas in this passage

XIII-B@1#eq:dotk · line 1219
$k ˙ = P E_{0}^{R} (σ_{ε} J_{3}^{'}_{γ} [z] + 2 C_{2}^{'}_{γ} [z] σ_{ε}^{''} - 2 M_{2}^{'}_{γ} [z] σ_{ε}^{'}) .$
XIII-B@1#eq:hats · line 1231
$b ˙_{2} a_{2} ˙^{μν} a = \frac{1}{4} (tr_{η} γ) (tr_{η} z) - \frac{1}{2} γ_{α β} z^{α β}, = z^{μ}_{α} γ^{α ν} + γ^{μ}_{α} z^{α ν} - \frac{1}{2} ((tr_{η} z) γ^{μν} + (tr_{η} γ) z^{μν}) + b ˙_{2} η^{μν}, = a_{1} [k ˙] + a ˙_{2}, b = b_{1} [k ˙] + b ˙_{2} .$
XIII-B@1#eq:weighted-divergences · line 1248
$D_{σ}_{ε} (a, v) D_{g} (a, v) = σ_{ε} a^{μν} \partial_{μ} \partial_{ν} v + σ_{ε} (\partial_{μ} a^{μν}) \partial_{ν} v + σ_{ε}^{'} a^{0 ν} \partial_{ν} v, = g a^{μν} \partial_{μ} \partial_{ν} v + g (\partial_{μ} a^{μν}) \partial_{ν} v + (\partial_{μ} g) a^{μν} \partial_{ν} v .$
XIII-B@1#eq:first-ghost-terms · line 1262
$Z^{μ} p_{z} = - E_{0}^{R} (σ_{ε}^{'} a_{1}^{0 μ} [z]), = \frac{1}{2} g v - E_{m}^{R} (\frac{1}{2} (□ g) v + (\partial^{μ} g) \partial_{μ} v + (\partial_{μ} g) a_{1}^{μν} [z] \partial_{ν} ϕ) .$
XIII-B@1#eq:second-ghost-terms · line 1278
$Y^{μ} = - E_{0}^{R} (p_{2, z} = - E_{m}^{R} (D_{σ}_{ε} (a_{1} [z], X^{μ}) + D_{σ}_{ε} (a_{1} [γ], Z^{μ}) + σ_{ε} \partial_{α} a^{α μ} + σ_{ε}^{'} a^{0 μ}), D_{g} (a_{1} [z], p) + m^{2} g b_{1} [z] p + D_{g} (a_{1} [γ], p_{z}) + m^{2} g b_{1} [γ] p_{z} + D_{g} (a, ϕ) + m^{2} g b ϕ) .$
unlabeled · line 1201
$z = R c, X = X_{1} [γ], p = p_{1} [γ; ϕ], Z = X_{1} [z], p_{z} = p_{1} [z; ϕ], k = r_{D} [γ] + r_{g} [ϕ] .$
unlabeled · line 1208
$O = \int f (p_{2} - X^{μ} \partial_{μ} p + (X^{ν} \partial_{ν} X^{μ} - X_{2}^{μ}) \partial_{μ} ϕ + \frac{1}{2} X^{μ} X^{ν} \partial_{μ} \partial_{ν} ϕ) d^{4} x .$

The composite entry

Displayed formulas in this passage

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