Manuscript X-I-ST@1 · X-I-ST.tex · section · 9745fd40

Explicit coefficient specification

A finite reference action through third order › Explicit coefficient specification · lines 815–890

X-I-ST@1#eq:Iinv, X-I-ST@1#eq:referencegauge, X-I-ST@1#eq:Ij

For any formal metric $p$ , put

J_{p} = - det p, Γ^{ρ}_{μν} [p] = \frac{1}{2} p^{ρ λ} (\partial_{μ} p_{λ ν} + \partial_{ν} p_{λ μ} - \partial_{λ} p_{μν}),

R_{μν} [p] = \partial_{ρ} Γ^{ρ}_{μν} - \partial_{ν} Γ^{ρ}_{μ ρ} + Γ^{ρ}_{ρ λ} Γ^{λ}_{μν} - Γ^{ρ}_{ν λ} Γ^{λ}_{μ ρ} .

Write

E [p] = J_{p} p^{μν} R_{μν} [p], L_{p} [ϕ] = \frac{1}{2} J_{p} (p^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}) .

The generalized invariant action is

I_{inv} = - \frac{1}{2 κ ^{2}} \int (E [q] - E [q ˉ]) d^{4} x + \int L_{q} [ϕ] d^{4} x .

The sole negative-power coefficient is a divergence:

- \frac{1}{2 κ} (\partial_{μ} \partial_{ν} r^{μν} - □ tr r) .

It is removed as a variational class before passing to a nonnegative formal coefficient ring.

Let $\nabla ˉ$ be the connection of $q ˉ$ , and set

C_{q ˉ}^{μ} [r] = \nabla ˉ_{ν} (q ˉ^{ν α} q ˉ^{μ β} r_{α β} - \frac{1}{2} q ˉ^{ν μ} q ˉ^{α β} r_{α β}),

Ψ_{q ˉ} = \int J_{q ˉ} c ˉ_{μ} (C_{q ˉ}^{μ} [r] - q ˉ^{μν} b_{ν}) d^{4} x, I = I_{inv} + s Ψ_{q ˉ} .

The auxiliary field remains independent.

For $j = 0, 1, 2, 3$ , define the local variational coefficients

I_{j} = \int [- \frac{1}{2} [κ^{j + 2}] (E [q] - E [q ˉ]) + [κ^{j}] L_{q} [ϕ] + [κ^{j}] (s Ψ_{q ˉ})_{dens}] d^{4} x .

Coefficient extraction means differentiation at zero divided by the relevant factorial. Thus (eq:Ij) is a finite prescription, not an unnamed cubic or quartic vertex.

Displayed formulas in this passage

X-I-ST@1#eq:Iinv · line 842
$I_{inv} = - \frac{1}{2 κ ^{2}} \int (E [q] - E [q ˉ]) d^{4} x + \int L_{q} [ϕ] d^{4} x .$
X-I-ST@1#eq:Ij · line 876
$I_{j} = \int [- \frac{1}{2} [κ^{j + 2}] (E [q] - E [q ˉ]) + [κ^{j}] L_{q} [ϕ] + [κ^{j}] (s Ψ_{q ˉ})_{dens}] d^{4} x .$
unlabeled · line 818
$J_{p} = - det p, Γ^{ρ}_{μν} [p] = \frac{1}{2} p^{ρ λ} (\partial_{μ} p_{λ ν} + \partial_{ν} p_{λ μ} - \partial_{λ} p_{μν}),$
unlabeled · line 826
$R_{μν} [p] = \partial_{ρ} Γ^{ρ}_{μν} - \partial_{ν} Γ^{ρ}_{μ ρ} + Γ^{ρ}_{ρ λ} Γ^{λ}_{μν} - Γ^{ρ}_{ν λ} Γ^{λ}_{μ ρ} .$
unlabeled · line 834
$E [p] = J_{p} p^{μν} R_{μν} [p], L_{p} [ϕ] = \frac{1}{2} J_{p} (p^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}) .$
unlabeled · line 850
$- \frac{1}{2 κ} (\partial_{μ} \partial_{ν} r^{μν} - □ tr r) .$
unlabeled · line 858
$C_{q ˉ}^{μ} [r] = \nabla ˉ_{ν} (q ˉ^{ν α} q ˉ^{μ β} r_{α β} - \frac{1}{2} q ˉ^{ν μ} q ˉ^{α β} r_{α β}),$
X-I-ST@1#eq:referencegauge · line 865
$Ψ_{q ˉ} = \int J_{q ˉ} c ˉ_{μ} (C_{q ˉ}^{μ} [r] - q ˉ^{μν} b_{ν}) d^{4} x, I = I_{inv} + s Ψ_{q ˉ} .$

Explicit coefficient specification

Displayed formulas in this passage

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