Manuscript XX-A@1 · XX-A.tex · section · 3a586110

A frozen quadratic gauge-fixing candidate

Quantum field input: the operators that are specified and those that remain missing › A frozen quadratic gauge-fixing candidate · lines 1027–1091

XX-A@1#eq:gauge-fermion, XX-A@1#eq:gauge-action, XX-A@1#eq:ghost-operator

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The field content for the proposed construction is

(h_{μν}, φ, c^{μ}, c ˉ^{μ}, B^{μ}) .

The scalar fluctuation defining the clock and slices is retained. The ghosts and auxiliary variables are gauge-fixing fields, not additional physical clocks.

Let $S_{2} (z) = \frac{1}{2} S^{(2)}_{g ˉ, ϕ ˉ} (z, z)$ , and define the linear differential

s_{0} h = L_{c} g ˉ, s_{0} φ = c^{μ} v_{μ}, s_{0} c = 0, s_{0} c ˉ = B, s_{0} B = 0.

At this linear level $s_{0}^{2} = 0$ . Freeze the gauge-fixing candidate

F_{ν} (z) = \nabla^{μ} h_{μν} - \frac{1}{2} \nabla_{ν} h^{μ}_{μ} - M^{- 2} v_{ν} φ, Ψ = \int - g ˉ c ˉ^{ν} (F_{ν} + \frac{β}{2} B_{ν}),

where $β > 0$ is fixed. Then

S_{2, gf} = S_{2} + s_{0} Ψ = S_{2} + \int - g ˉ (B^{ν} F_{ν} + \frac{β}{2} B^{ν} B_{ν} - c ˉ^{ν} (Q c)_{ν}),

with

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

The sign of the ghost term follows from the odd Leibniz rule. Equation (eq:ghost-operator) follows by applying $F$ to $K c$ and commuting background derivatives.

These formulas specify a quadratic candidate and its normalization. They do not prove that its complete reduced metric–scalar realization has the intended Green properties, nor select a state. Eliminating $B$ , changing field variables, or using another gauge requires a checked normalization and comparison, not an implicit substitution.

Hack's coupled linearized Einstein–Klein–Gordon construction is relevant framework Hack. It is not used as a premise that the particular $F$ , covariance, rods, spatial tails, or composite in this paper have been constructed. Relational quantum-observable and perturbative BV frameworks likewise do not supply this particular realization by name BFHPR,Frob,BFR.

Displayed formulas in this passage

XX-A@1#eq:gauge-fermion · line 1047
$F_{ν} (z) = \nabla^{μ} h_{μν} - \frac{1}{2} \nabla_{ν} h^{μ}_{μ} - M^{- 2} v_{ν} φ, Ψ = \int - g ˉ c ˉ^{ν} (F_{ν} + \frac{β}{2} B_{ν}),$
XX-A@1#eq:gauge-action · line 1059
$S_{2, gf} = S_{2} + s_{0} Ψ = S_{2} + \int - g ˉ (B^{ν} F_{ν} + \frac{β}{2} B^{ν} B_{ν} - c ˉ^{ν} (Q c)_{ν}),$
XX-A@1#eq:ghost-operator · line 1067
$(Q c)_{ν} = □_{g ˉ} c_{ν} + R ˉ_{ν ρ} c^{ρ} - M^{- 2} v_{ν} v_{ρ} c^{ρ} .$
unlabeled · line 1030
$(h_{μν}, φ, c^{μ}, c ˉ^{μ}, B^{μ}) .$
unlabeled · line 1039
$s_{0} h = L_{c} g ˉ, s_{0} φ = c^{μ} v_{μ}, s_{0} c = 0, s_{0} c ˉ = B, s_{0} B = 0.$