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

Formal fields and density coefficients

Frozen classical prescription › Formal fields and density coefficients · lines 397–440

X-A-ST@1#eq:densitycoeff

Work over

D_{2} = R [κ] / (κ^{3}) .

An equality modulo $κ^{3}$ is coefficient equality, not a numerical error estimate.

The configuration variables are

q_{μν} = η_{μν} + κ h_{μν} + κ^{2} k_{μν}, ϕ = ϕ_{0} + κ ϕ_{1} + κ^{2} ϕ_{2},

where $h, k$ are arbitrary smooth real symmetric tensors and $ϕ_{j}$ arbitrary smooth real scalars. There is no Einstein equation, free-shell quotient, gauge fixing, or decay condition on this off-shell configuration space.

Indices are raised using $η$ , and parentheses include a factor $1/2$ . Write

H = η^{μν} h_{μν}, h : h = h_{μν} h^{μν} .

Set

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 : h, = h^{μ}_{λ} h^{λ ν} - \frac{1}{2} H h^{μν} + b_{2} [h] η^{μν} .

Polarization is normalized by $a_{2} [h, h] = a_{2} [h]$ and $b_{2} [h, h] = b_{2} [h]$ .

Displayed formulas in this passage

X-A-ST@1#eq:densitycoeff · line 426
$b_{1} [h] a_{1}^{μν} [h] = \frac{1}{2} H, = \frac{1}{2} H η^{μν} - h^{μν}, b_{2} [h] a_{2}^{μν} [h] = \frac{1}{8} H^{2} - \frac{1}{4} h : h, = h^{μ}_{λ} h^{λ ν} - \frac{1}{2} H h^{μν} + b_{2} [h] η^{μν} .$
unlabeled · line 400
$D_{2} = R [κ] / (κ^{3}) .$
unlabeled · line 407
$q_{μν} = η_{μν} + κ h_{μν} + κ^{2} k_{μν}, ϕ = ϕ_{0} + κ ϕ_{1} + κ^{2} ϕ_{2},$
unlabeled · line 421
$H = η^{μν} h_{μν}, h : h = h_{μν} h^{μν} .$

Formal fields and density coefficients

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