Manuscript VII-A@1 · VII-A.tex · section · 11da3fe6

The full second-order coefficient

Plateau covariance and the second-order intertwiner › The full second-order coefficient · lines 1353–1378

VII-A@1#eq:fullA2

Substituting the constructed scalar coefficients into (eq:A2) gives the explicit expression

A_{2}^{(g)} = E_{m} V_{g} [h] E_{m} V_{g} [h] ϕ_{0} - E_{m} W_{g} [h, k] ϕ_{0} + w^{μ} \partial_{μ} (E_{m} V_{g} [h] ϕ_{0}) + (w^{ν} \partial_{ν} w^{μ} - v^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .

There are five displayed contributions: iterated scalar propagation, the second-order scalar vertex, the mixed first-order term, the second-order inverse-coordinate term, and the quadratic Taylor term.

None may be omitted merely because another term is described as a dressing correction.

Displayed formulas in this passage

VII-A@1#eq:fullA2 · line 1357
$A_{2}^{(g)} = E_{m} V_{g} [h] E_{m} V_{g} [h] ϕ_{0} - E_{m} W_{g} [h, k] ϕ_{0} + w^{μ} \partial_{μ} (E_{m} V_{g} [h] ϕ_{0}) + (w^{ν} \partial_{ν} w^{μ} - v^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .$