Manuscript X-A-ST@1 · X-A-ST.tex · section · d9ccbde1

Scalar equations, with every switch term

Flat Cauchy construction and coefficient equations › Scalar equations, with every switch term · lines 807–842

X-A-ST@1#eq:scalaroperators, X-A-ST@1#eq:scalarsolution

Let $P_{m} = □ + m^{2}$ . The expansion is

E_{g} [q] = P_{m} + κ V_{1, g} [h] + κ^{2} V_{2, g} [h, k],

with

V_{1, g} [h] ψ V_{2, g} [h, k] ψ = g (\partial_{μ} (a_{1}^{μν} [h] \partial_{ν} ψ) + m^{2} b_{1} [h] ψ) + (\partial_{μ} g) a_{1}^{μν} [h] \partial_{ν} ψ, = g (\partial_{μ} (B^{μν} \partial_{ν} ψ) + m^{2} (b_{1} [k] + b_{2} [h]) ψ) + (\partial_{μ} g) B^{μν} \partial_{ν} ψ .

For a fixed smooth real free input $P_{m} ϕ_{0} = 0$ , the switched scalar solution is

ρ_{g} [q; ϕ_{0}] ρ_{1, g} ρ_{2, g} = ϕ_{0} + κ ρ_{1, g} + κ^{2} ρ_{2, g}, = - E_{m} V_{1, g} [h] ϕ_{0}, = E_{m} V_{1, g} [h] E_{m} V_{1, g} [h] ϕ_{0} - E_{m} V_{2, g} [h, k] ϕ_{0} .

Displayed formulas in this passage

X-A-ST@1#eq:scalaroperators · line 815
$V_{1, g} [h] ψ V_{2, g} [h, k] ψ = g (\partial_{μ} (a_{1}^{μν} [h] \partial_{ν} ψ) + m^{2} b_{1} [h] ψ) + (\partial_{μ} g) a_{1}^{μν} [h] \partial_{ν} ψ, = g (\partial_{μ} (B^{μν} \partial_{ν} ψ) + m^{2} (b_{1} [k] + b_{2} [h]) ψ) + (\partial_{μ} g) B^{μν} \partial_{ν} ψ .$
X-A-ST@1#eq:scalarsolution · line 832
$ρ_{g} [q; ϕ_{0}] ρ_{1, g} ρ_{2, g} = ϕ_{0} + κ ρ_{1, g} + κ^{2} ρ_{2, g}, = - E_{m} V_{1, g} [h] ϕ_{0}, = E_{m} V_{1, g} [h] E_{m} V_{1, g} [h] ϕ_{0} - E_{m} V_{2, g} [h, k] ϕ_{0} .$
unlabeled · line 810
$E_{g} [q] = P_{m} + κ V_{1, g} [h] + κ^{2} V_{2, g} [h, k],$