Manuscript Em06-A@1 · Em06-A-v1.tex · section · ee9923c6

Background, scalar equation, switch, and gauge domain

The classical object before the limit › Background, scalar equation, switch, and gauge domain · lines 195–236

Em06-A@1#eq:C, Em06-A@1#eq:Q, Em06-A@1#eq:action

Indices are raised and lowered with $η$ . Write

τ (h) = η^{μν} h_{μν}, a^{μν} (h) = \frac{1}{2} τ (h) η^{μν} - h^{μν}, b (h) = \frac{1}{2} τ (h),

and

C (h)^{ν} = \partial_{μ} h^{μν} - \frac{1}{2} \partial^{ν} τ (h) .

The scalar density perturbation is the differential operator

Q_{h} v = \partial_{μ} (a^{μν} (h) \partial_{ν} v) + m^{2} b (h) v .

In particular, $Q_{χ h}$ , not $χ Q_{h}$ , is the switched operator. Derivatives of the switch are retained.

For orientation, this is the coefficient in the Euler equation of the declared first-order scalar action

S_{χ} [ϕ; h] = \frac{1}{2} \int_{M} (η^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}) d^{4} x + \frac{κ}{2} \int_{M} χ (a^{μν} (h) \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} b (h) ϕ^{2}) d^{4} x .

Its equation is $P_{m} ϕ + κ Q_{χ h} ϕ = 0$ . The parameter $κ$ labels the classical metric jet; it is not the classicality parameter.

Displayed formulas in this passage

Em06-A@1#eq:C · line 204
$C (h)^{ν} = \partial_{μ} h^{μν} - \frac{1}{2} \partial^{ν} τ (h) .$
Em06-A@1#eq:Q · line 211
$Q_{h} v = \partial_{μ} (a^{μν} (h) \partial_{ν} v) + m^{2} b (h) v .$
Em06-A@1#eq:action · line 221
$S_{χ} [ϕ; h] = \frac{1}{2} \int_{M} (η^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}) d^{4} x + \frac{κ}{2} \int_{M} χ (a^{μν} (h) \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} b (h) ϕ^{2}) d^{4} x .$
unlabeled · line 198
$τ (h) = η^{μν} h_{μν}, a^{μν} (h) = \frac{1}{2} τ (h) η^{μν} - h^{μν}, b (h) = \frac{1}{2} τ (h),$