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

Lemma: Density expansion

Frozen classical prescription › Formal fields and density coefficients · lines 441–468 · proof included or linked

X-A-ST@1#lem:density, X-A-ST@1#eq:densityexp

For $v_{q} = - det q$ and $A_{q}^{μν} = v_{q} q^{μν}$ ,

v_{q} A_{q}^{μν} = 1 + κ b_{1} [h] + κ^{2} (b_{1} [k] + b_{2} [h]), = η^{μν} + κ a_{1}^{μν} [h] + κ^{2} (a_{1}^{μν} [k] + a_{2}^{μν} [h]) .

Proof. The inverse is

q^{μν} = η^{μν} - κ h^{μν} + κ^{2} (h^{μ}_{λ} h^{λ ν} - k^{μν}) .

Expanding $exp {\frac{1}{2} tr lo g (I + κ η^{- 1} h + κ^{2} η^{- 1} k)}$ gives $v_{q}$ . Multiplication gives $A_{q}$ .

Displayed formulas in this passage

X-A-ST@1#eq:densityexp · line 445
$v_{q} A_{q}^{μν} = 1 + κ b_{1} [h] + κ^{2} (b_{1} [k] + b_{2} [h]), = η^{μν} + κ a_{1}^{μν} [h] + κ^{2} (a_{1}^{μν} [k] + a_{2}^{μν} [h]) .$
unlabeled · line 458
$q^{μν} = η^{μν} - κ h^{μν} + κ^{2} (h^{μ}_{λ} h^{λ ν} - k^{μν}) .$