Manuscript XI-A@1 · XI-A.tex · section · b8211ecc

Metric and inherited density coefficients

X-A premises, conventions, and non-inputs › Metric and inherited density coefficients · lines 353–390

XI-A@1#eq:q, XI-A@1#eq:b, XI-A@1#eq:a1, XI-A@1#eq:a2

The field variable is one classical symmetric tensor $γ$ :

q_{μν} = η_{μν} + κ γ_{μν} .

There are no independent quantum fields $h, k$ . Write $τ = η^{μν} γ_{μν}$ and set

b_{1} [γ] a_{1}^{μν} [γ] a_{2}^{μν} [γ] = \frac{1}{2} τ, = \frac{1}{2} τ η^{μν} - γ^{μν}, = γ^{μ}_{ρ} γ^{ρ ν} - \frac{1}{2} τ γ^{μν} + b_{2} [γ] η^{μν} . b_{2} [γ] = \frac{1}{8} τ^{2} - \frac{1}{4} γ_{μν} γ^{μν},

Thus

- q = 1 + κ b_{1} + κ^{2} b_{2} + O (κ^{3}), A_{q}^{μν} = η^{μν} + κ a_{1}^{μν} + κ^{2} a_{2}^{μν} + O (κ^{3}) .

These symbols are not coefficients of $- q R [q]$ .

The action is defined as a full formal generating expression before its Euler derivative is projected through $κ^{2}$ . In particular, $γ$ may later be evaluated on a formal solution family. This does not introduce another fundamental metric field.

Displayed formulas in this passage

XI-A@1#eq:q · line 356
$q_{μν} = η_{μν} + κ γ_{μν} .$
XI-A@1#eq:b · line 362
$b_{1} [γ] a_{1}^{μν} [γ] a_{2}^{μν} [γ] = \frac{1}{2} τ, = \frac{1}{2} τ η^{μν} - γ^{μν}, = γ^{μ}_{ρ} γ^{ρ ν} - \frac{1}{2} τ γ^{μν} + b_{2} [γ] η^{μν} . b_{2} [γ] = \frac{1}{8} τ^{2} - \frac{1}{4} γ_{μν} γ^{μν},$
XI-A@1#eq:a1 · line 362
$b_{1} [γ] a_{1}^{μν} [γ] a_{2}^{μν} [γ] = \frac{1}{2} τ, = \frac{1}{2} τ η^{μν} - γ^{μν}, = γ^{μ}_{ρ} γ^{ρ ν} - \frac{1}{2} τ γ^{μν} + b_{2} [γ] η^{μν} . b_{2} [γ] = \frac{1}{8} τ^{2} - \frac{1}{4} γ_{μν} γ^{μν},$
XI-A@1#eq:a2 · line 362
$b_{1} [γ] a_{1}^{μν} [γ] a_{2}^{μν} [γ] = \frac{1}{2} τ, = \frac{1}{2} τ η^{μν} - γ^{μν}, = γ^{μ}_{ρ} γ^{ρ ν} - \frac{1}{2} τ γ^{μν} + b_{2} [γ] η^{μν} . b_{2} [γ] = \frac{1}{8} τ^{2} - \frac{1}{4} γ_{μν} γ^{μν},$
unlabeled · line 377
$- q = 1 + κ b_{1} + κ^{2} b_{2} + O (κ^{3}), A_{q}^{μν} = η^{μν} + κ a_{1}^{μν} + κ^{2} a_{2}^{μν} + O (κ^{3}) .$