Manuscript Em01-F@3 · Em01-F-v3.tex · proposition · 225e59af

Proposition: Free Hessians and their inverses

One action and its normalization dictionary › Derived quadratic blocks · lines 827–880 · proof included or linked

Em01-F@3#prop:hessians, Em01-F@3#eq:quadratic, Em01-F@3#eq:inverse

At the zero-field background,

S^{(2)} = \frac{1}{2} ⟨ γ, - □_{η} P γ ⟩_{η} + \frac{1}{2} ⟨ ϕ, - (□_{η} + m^{2}) ϕ ⟩ .

The mixed free block vanishes. For the equation-normalized Green operators of $P_{m} = □_{η} + m^{2}$ , the action-Hessian inverses are

G_{γ}^{R, A} = - P E_{0}^{R, A}, G_{ϕ}^{R, A} = - E_{m}^{R, A} .

Proof. The linearized Einstein tensor is

G_{1} (h)_{μν} = - \frac{1}{2} □_{η} (P h)_{μν} + \partial_{(μ} d_{ν)} (h) - \frac{1}{2} η_{μν} \partial^{ρ} d_{ρ} (h) .

The gravitational Euler derivative in the $γ$ -variable has linear covariant output $2 G_{1} (γ)$ , by (eq:EHvariation) and $δ g = λ δ γ$ . The lowered gauge-fixing Euler derivative is

- 2 \partial_{(μ} d_{ν)} (γ) + η_{μν} \partial^{ρ} d_{ρ} (γ) .

It cancels the divergence terms, leaving $- □_{η} P γ$ .

The free scalar Euler derivative is $- P_{m} ϕ$ . Every matter interaction correction contains at least one metric field and two scalar fields, so there is no free mixed block at $(η, 0)$ . Homogeneity gives the quadratic action.

Finally $P$ commutes with $□_{η}$ and squares to the identity, so direct composition proves (eq:inverse).

Displayed formulas in this passage

Em01-F@3#eq:quadratic · line 830
$S^{(2)} = \frac{1}{2} ⟨ γ, - □_{η} P γ ⟩_{η} + \frac{1}{2} ⟨ ϕ, - (□_{η} + m^{2}) ϕ ⟩ .$
Em01-F@3#eq:inverse · line 840
$G_{γ}^{R, A} = - P E_{0}^{R, A}, G_{ϕ}^{R, A} = - E_{m}^{R, A} .$
unlabeled · line 854
$G_{1} (h)_{μν} = - \frac{1}{2} □_{η} (P h)_{μν} + \partial_{(μ} d_{ν)} (h) - \frac{1}{2} η_{μν} \partial^{ρ} d_{ρ} (h) .$
unlabeled · line 865
$- 2 \partial_{(μ} d_{ν)} (γ) + η_{μν} \partial^{ρ} d_{ρ} (γ) .$

Proposition: Free Hessians and their inverses

Displayed formulas in this passage

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