Manuscript Em06-B@1 · Em06-B-v1.tex · section · 61e38859

Density variation and Green signs

Frozen conventions and the classical object › Density variation and Green signs · lines 235–290

Em06-B@1#eq:Q, Em06-B@1#eq:action, Em06-B@1#eq:Green

Set

τ (h) a^{μν} (h) C (h)^{ν} Q_{h} v = η^{μν} h_{μν}, = \frac{1}{2} τ (h) η^{μν} - h^{μν}, b (h) = \frac{1}{2} τ (h), = \partial_{μ} h^{μν} - \frac{1}{2} \partial^{ν} τ (h), = \partial_{μ} (a^{μν} (h) \partial_{ν} v) + m^{2} b (h) v .

The switched scalar action through first order is

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 Euler equation is

P_{m} ϕ + κ Q_{χ h} ϕ = 0.

In particular, the operator is $Q_{χ h}$ , not $χ Q_{h}$ . Derivatives of the switch are retained. The metric-jet parameter $κ$ is independent of the classicality parameter $ϵ$ .

Let $G_{m}^{R}$ and $G_{m}^{A}$ be the retarded and advanced Green operators of $P_{m}$ . With $D_{m} = - Δ + m^{2}$ , our signs are

(G_{m}^{R} F) (t) (G_{m}^{A} F) (t) = \int_{- \infty}^{t} \frac{sin (( t - s ) D _{m} )}{D _{m}} F (s) d s, = \int_{t}^{\infty} \frac{sin (( s - t ) D _{m} )}{D _{m}} F (s) d s .

For compact smooth $F$ ,

P_{m} G_{m}^{R} F = P_{m} G_{m}^{A} F = F, (G_{m}^{R})^{*} = G_{m}^{A} .

We use their standard finite-propagation and Cauchy uniqueness properties BGP. The estimates needed for this model are proved in Appendix app:PDE.

Displayed formulas in this passage

Em06-B@1#eq:Q · line 238
$τ (h) a^{μν} (h) C (h)^{ν} Q_{h} v = η^{μν} h_{μν}, = \frac{1}{2} τ (h) η^{μν} - h^{μν}, b (h) = \frac{1}{2} τ (h), = \partial_{μ} h^{μν} - \frac{1}{2} \partial^{ν} τ (h), = \partial_{μ} (a^{μν} (h) \partial_{ν} v) + m^{2} b (h) v .$
Em06-B@1#eq:action · line 250
$S_{χ} [ϕ; h] = \frac{1}{2} \int_{M} (η^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}) d^{4} x + \frac{κ}{2} \int_{M} χ (a^{μν} (h) \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} b (h) ϕ^{2}) d^{4} x .$
Em06-B@1#eq:Green · line 273
$(G_{m}^{R} F) (t) (G_{m}^{A} F) (t) = \int_{- \infty}^{t} \frac{sin (( t - s ) D _{m} )}{D _{m}} F (s) d s, = \int_{t}^{\infty} \frac{sin (( s - t ) D _{m} )}{D _{m}} F (s) d s .$
unlabeled · line 263
$P_{m} ϕ + κ Q_{χ h} ϕ = 0.$
unlabeled · line 283
$P_{m} G_{m}^{R} F = P_{m} G_{m}^{A} F = F, (G_{m}^{R})^{*} = G_{m}^{A} .$