Manuscript XX-C@1 · XX-C.tex · section · 7a8cab37

Action Hessian and gauge normalization

Identical frozen background, operator, and rods › Action Hessian and gauge normalization · lines 305–426

XX-C@1#eq:pairings, XX-C@1#eq:H, XX-C@1#eq:curvature, XX-C@1#eq:matter, XX-C@1#eq:scalar, XX-C@1#eq:K, XX-C@1#eq:gauge, XX-C@1#eq:normalization

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Let

V = S^{2} T^{*} M \oplus R, W = T^{*} M, z = (h, φ) .

Use the nondegenerate integrated pairings

(z, w)_{0} (ξ, η)_{1} = \int_{M} (h_{μν} j^{μν} + φ ω) d V_{g ˉ}, = \int_{M} ξ_{μ} η^{μ} d V_{g ˉ}, d V_{g ˉ} = a (t)^{3} d t d^{3} x .

The invariant Euler equations are

E_{μν} = M^{2} G_{μν} - T_{μν}, E_{ϕ} = □_{g} ϕ - V^{'} (ϕ) .

The action Hessian is defined without an unspecified normalization by

H (h, φ) = (- \frac{1}{2} δ E_{μν} [h, φ], δ E_{ϕ} [h, φ]), S_{2} (z) = \frac{1}{2} (z, H z)_{0} .

All variations in (eq:H) are taken at $(g ˉ, ϕ ˉ)$ .

For completeness, the local variation rules are

Γ_{1}^{ρ}_{μν} (R_{1})_{μν} R_{1} (G_{1})_{μν} (T_{1})_{μν} (E_{ϕ})_{1} = \frac{1}{2} g ˉ^{ρ σ} (\nabla_{μ} h_{ν σ} + \nabla_{ν} h_{μ σ} - \nabla_{σ} h_{μν}), = \nabla_{ρ} Γ_{1}^{ρ}_{μν} - \nabla_{ν} Γ_{1}^{ρ}_{μ ρ}, = - h^{μν} R ˉ_{μν} + g ˉ^{μν} (R_{1})_{μν}, = (R_{1})_{μν} - \frac{1}{2} h_{μν} R ˉ - \frac{1}{2} g ˉ_{μν} R_{1}, = \partial_{μ} φ v_{ν} + v_{μ} \partial_{ν} φ - \frac{1}{2} h_{μν} (v^{ρ} v_{ρ} + 2 V) - \frac{1}{2} g ˉ_{μν} (- h^{ρ σ} v_{ρ} v_{σ} + 2 v^{ρ} \partial_{ρ} φ + 2 V^{'} φ), = □_{g ˉ} φ - V^{''} φ - h^{μν} \nabla_{μ} \nabla_{ν} ϕ ˉ - (\nabla_{μ} h^{μν} - \frac{1}{2} \nabla^{ν} h^{ρ}_{ρ}) v_{ν} .

The gauge generator and its adjoint are

K ξ = (2 \nabla_{(μ} ξ_{ν)}, v_{μ} ξ^{μ}), K^{†} (h, φ)_{ν} = - 2 \nabla^{μ} h_{μν} + v_{ν} φ .

On the on-shell background,

H^{†} = H, H K = 0, K^{†} H = 0.

Retain

F_{ν} z r = \nabla^{μ} h_{μν} - \frac{1}{2} \nabla_{ν} h^{μ}_{μ} - M^{- 2} v_{ν} φ, = \frac{M ^{2}}{2}, B = H - r F^{†} F, Q = F K .

Explicitly,

(Q ξ)_{ν} = □_{g ˉ} ξ_{ν} + R ˉ_{ν ρ} ξ^{ρ} - M^{- 2} v_{ν} v_{ρ} ξ^{ρ} .

Define four-dimensional trace reversal

T h = h - \frac{1}{2} g ˉ tr_{g ˉ} h, T^{2} = id,

and

A = diag (\frac{M ^{2}}{4} T, id), L = A^{- 1} B .

Displayed formulas in this passage

XX-C@1#eq:pairings · line 314
$(z, w)_{0} (ξ, η)_{1} = \int_{M} (h_{μν} j^{μν} + φ ω) d V_{g ˉ}, = \int_{M} ξ_{μ} η^{μ} d V_{g ˉ}, d V_{g ˉ} = a (t)^{3} d t d^{3} x .$
XX-C@1#eq:H · line 332
$H (h, φ) = (- \frac{1}{2} δ E_{μν} [h, φ], δ E_{ϕ} [h, φ]), S_{2} (z) = \frac{1}{2} (z, H z)_{0} .$
XX-C@1#eq:curvature · line 344
$Γ_{1}^{ρ}_{μν} (R_{1})_{μν} R_{1} (G_{1})_{μν} (T_{1})_{μν} (E_{ϕ})_{1} = \frac{1}{2} g ˉ^{ρ σ} (\nabla_{μ} h_{ν σ} + \nabla_{ν} h_{μ σ} - \nabla_{σ} h_{μν}), = \nabla_{ρ} Γ_{1}^{ρ}_{μν} - \nabla_{ν} Γ_{1}^{ρ}_{μ ρ}, = - h^{μν} R ˉ_{μν} + g ˉ^{μν} (R_{1})_{μν}, = (R_{1})_{μν} - \frac{1}{2} h_{μν} R ˉ - \frac{1}{2} g ˉ_{μν} R_{1}, = \partial_{μ} φ v_{ν} + v_{μ} \partial_{ν} φ - \frac{1}{2} h_{μν} (v^{ρ} v_{ρ} + 2 V) - \frac{1}{2} g ˉ_{μν} (- h^{ρ σ} v_{ρ} v_{σ} + 2 v^{ρ} \partial_{ρ} φ + 2 V^{'} φ), = □_{g ˉ} φ - V^{''} φ - h^{μν} \nabla_{μ} \nabla_{ν} ϕ ˉ - (\nabla_{μ} h^{μν} - \frac{1}{2} \nabla^{ν} h^{ρ}_{ρ}) v_{ν} .$
XX-C@1#eq:matter · line 344
$Γ_{1}^{ρ}_{μν} (R_{1})_{μν} R_{1} (G_{1})_{μν} (T_{1})_{μν} (E_{ϕ})_{1} = \frac{1}{2} g ˉ^{ρ σ} (\nabla_{μ} h_{ν σ} + \nabla_{ν} h_{μ σ} - \nabla_{σ} h_{μν}), = \nabla_{ρ} Γ_{1}^{ρ}_{μν} - \nabla_{ν} Γ_{1}^{ρ}_{μ ρ}, = - h^{μν} R ˉ_{μν} + g ˉ^{μν} (R_{1})_{μν}, = (R_{1})_{μν} - \frac{1}{2} h_{μν} R ˉ - \frac{1}{2} g ˉ_{μν} R_{1}, = \partial_{μ} φ v_{ν} + v_{μ} \partial_{ν} φ - \frac{1}{2} h_{μν} (v^{ρ} v_{ρ} + 2 V) - \frac{1}{2} g ˉ_{μν} (- h^{ρ σ} v_{ρ} v_{σ} + 2 v^{ρ} \partial_{ρ} φ + 2 V^{'} φ), = □_{g ˉ} φ - V^{''} φ - h^{μν} \nabla_{μ} \nabla_{ν} ϕ ˉ - (\nabla_{μ} h^{μν} - \frac{1}{2} \nabla^{ν} h^{ρ}_{ρ}) v_{ν} .$
XX-C@1#eq:scalar · line 344
$Γ_{1}^{ρ}_{μν} (R_{1})_{μν} R_{1} (G_{1})_{μν} (T_{1})_{μν} (E_{ϕ})_{1} = \frac{1}{2} g ˉ^{ρ σ} (\nabla_{μ} h_{ν σ} + \nabla_{ν} h_{μ σ} - \nabla_{σ} h_{μν}), = \nabla_{ρ} Γ_{1}^{ρ}_{μν} - \nabla_{ν} Γ_{1}^{ρ}_{μ ρ}, = - h^{μν} R ˉ_{μν} + g ˉ^{μν} (R_{1})_{μν}, = (R_{1})_{μν} - \frac{1}{2} h_{μν} R ˉ - \frac{1}{2} g ˉ_{μν} R_{1}, = \partial_{μ} φ v_{ν} + v_{μ} \partial_{ν} φ - \frac{1}{2} h_{μν} (v^{ρ} v_{ρ} + 2 V) - \frac{1}{2} g ˉ_{μν} (- h^{ρ σ} v_{ρ} v_{σ} + 2 v^{ρ} \partial_{ρ} φ + 2 V^{'} φ), = □_{g ˉ} φ - V^{''} φ - h^{μν} \nabla_{μ} \nabla_{ν} ϕ ˉ - (\nabla_{μ} h^{μν} - \frac{1}{2} \nabla^{ν} h^{ρ}_{ρ}) v_{ν} .$
XX-C@1#eq:K · line 382
$K ξ = (2 \nabla_{(μ} ξ_{ν)}, v_{μ} ξ^{μ}), K^{†} (h, φ)_{ν} = - 2 \nabla^{μ} h_{μν} + v_{ν} φ .$
XX-C@1#eq:gauge · line 396
$F_{ν} z r = \nabla^{μ} h_{μν} - \frac{1}{2} \nabla_{ν} h^{μ}_{μ} - M^{- 2} v_{ν} φ, = \frac{M ^{2}}{2}, B = H - r F^{†} F, Q = F K .$
XX-C@1#eq:normalization · line 421
$A = diag (\frac{M ^{2}}{4} T, id), L = A^{- 1} B .$
unlabeled · line 308
$V = S^{2} T^{*} M \oplus R, W = T^{*} M, z = (h, φ) .$
unlabeled · line 326
$E_{μν} = M^{2} G_{μν} - T_{μν}, E_{ϕ} = □_{g} ϕ - V^{'} (ϕ) .$
unlabeled · line 390
$H^{†} = H, H K = 0, K^{†} H = 0.$
unlabeled · line 409
$(Q ξ)_{ν} = □_{g ˉ} ξ_{ν} + R ˉ_{ν ρ} ξ^{ρ} - M^{- 2} v_{ν} v_{ρ} ξ^{ρ} .$
unlabeled · line 416
$T h = h - \frac{1}{2} g ˉ tr_{g ˉ} h, T^{2} = id,$

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section: Action Hessian and gauge normalization