Manuscript X-C-ST@1 · X-C-ST.tex · section · b679954e

Scalar and metric-density coefficients

Frozen analytical data and predecessor handoff › Scalar and metric-density coefficients · lines 378–471

X-C-ST@1#eq:density, X-C-ST@1#eq:densityexp, X-C-ST@1#eq:Euler, X-C-ST@1#eq:operators, X-C-ST@1#eq:vertices, X-C-ST@1#eq:Hessian

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Indices are raised with $η$ . Put $H = tr_{η} h$ and define

b_{1} [h] a_{1}^{μν} [h] = \frac{1}{2} H, = \frac{1}{2} H η^{μν} - h^{μν}, b_{2} [h] a_{2}^{μν} [h] = \frac{1}{8} H^{2} - \frac{1}{4} h_{α β} h^{α β}, = h^{μ}_{λ} h^{λ ν} - \frac{1}{2} H h^{μν} + b_{2} [h] η^{μν} .

Set

A_{2} [h, k] = a_{1} [k] + a_{2} [h], B_{2} [h, k] = b_{1} [k] + b_{2} [h] .

Thus

- det q q^{μν} - det q = η^{μν} + κ a_{1}^{μν} [h] + κ^{2} A_{2}^{μν} [h, k], = 1 + κ b_{1} [h] + κ^{2} B_{2} [h, k] .

The switched scalar density is

L_{g} [q, ψ] = \frac{1}{2} [(η^{μν} + g (- det q q^{μν} - η^{μν})) \partial_{μ} ψ \partial_{ν} ψ - m^{2} (1 + g (- det q - 1)) ψ^{2}] .

It is switched before variation. For $P_{m} = □ + m^{2}$ ,

□ = \partial_{t}^{2} - Δ_{x},

its positive Euler operator is

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

The two local differential coefficients are

L_{1, g} [h] u = L_{2, g} [h, k] u = g [\partial_{μ} (a_{1}^{μν} [h] \partial_{ν} u) + m^{2} b_{1} [h] u] + (\partial_{μ} g) a_{1}^{μν} [h] \partial_{ν} u, g [\partial_{μ} (A_{2}^{μν} [h, k] \partial_{ν} u) + m^{2} B_{2} [h, k] u] + (\partial_{μ} g) A_{2}^{μν} [h, k] \partial_{ν} u .

Every displayed derivative of the switch is retained.

The corresponding vertices are

V_{1, g} [h; ψ] V_{2, g} [h, k; ψ] = \frac{1}{2} \int g (a_{1}^{μν} [h] \partial_{μ} ψ \partial_{ν} ψ - m^{2} b_{1} [h] ψ^{2}) d^{4} x, = \frac{1}{2} \int g (A_{2}^{μν} [h, k] \partial_{μ} ψ \partial_{ν} ψ - m^{2} B_{2} [h, k] ψ^{2}) d^{4} x .

Both are quadratic in the scalar and have compact coefficient support. Compact integration by parts gives

S_{0}^{''} = - P_{m}, V_{j, g}^{'} [ψ] = - L_{j, g} ψ, V_{j, g}^{''} = - L_{j, g}, j = 1, 2.

The positive Euler coefficients are not the action Hessians.

Displayed formulas in this passage

X-C-ST@1#eq:density · line 382
$b_{1} [h] a_{1}^{μν} [h] = \frac{1}{2} H, = \frac{1}{2} H η^{μν} - h^{μν}, b_{2} [h] a_{2}^{μν} [h] = \frac{1}{8} H^{2} - \frac{1}{4} h_{α β} h^{α β}, = h^{μ}_{λ} h^{λ ν} - \frac{1}{2} H h^{μν} + b_{2} [h] η^{μν} .$
X-C-ST@1#eq:densityexp · line 400
$- det q q^{μν} - det q = η^{μν} + κ a_{1}^{μν} [h] + κ^{2} A_{2}^{μν} [h, k], = 1 + κ b_{1} [h] + κ^{2} B_{2} [h, k] .$
X-C-ST@1#eq:Euler · line 426
$E_{g} [q] = P_{m} + κ L_{1, g} [h] + κ^{2} L_{2, g} [h, k] .$
X-C-ST@1#eq:operators · line 432
$L_{1, g} [h] u = L_{2, g} [h, k] u = g [\partial_{μ} (a_{1}^{μν} [h] \partial_{ν} u) + m^{2} b_{1} [h] u] + (\partial_{μ} g) a_{1}^{μν} [h] \partial_{ν} u, g [\partial_{μ} (A_{2}^{μν} [h, k] \partial_{ν} u) + m^{2} B_{2} [h, k] u] + (\partial_{μ} g) A_{2}^{μν} [h, k] \partial_{ν} u .$
X-C-ST@1#eq:vertices · line 449
$V_{1, g} [h; ψ] V_{2, g} [h, k; ψ] = \frac{1}{2} \int g (a_{1}^{μν} [h] \partial_{μ} ψ \partial_{ν} ψ - m^{2} b_{1} [h] ψ^{2}) d^{4} x, = \frac{1}{2} \int g (A_{2}^{μν} [h, k] \partial_{μ} ψ \partial_{ν} ψ - m^{2} B_{2} [h, k] ψ^{2}) d^{4} x .$
X-C-ST@1#eq:Hessian · line 463
$S_{0}^{''} = - P_{m}, V_{j, g}^{'} [ψ] = - L_{j, g} ψ, V_{j, g}^{''} = - L_{j, g}, j = 1, 2.$
unlabeled · line 395
$A_{2} [h, k] = a_{1} [k] + a_{2} [h], B_{2} [h, k] = b_{1} [k] + b_{2} [h] .$
unlabeled · line 411
$L_{g} [q, ψ] = \frac{1}{2} [(η^{μν} + g (- det q q^{μν} - η^{μν})) \partial_{μ} ψ \partial_{ν} ψ - m^{2} (1 + g (- det q - 1)) ψ^{2}] .$
unlabeled · line 422
$□ = \partial_{t}^{2} - Δ_{x},$