Manuscript VII-A-ST@1 · VII-A-ST.tex · section · c39a67a3

Unswitched harmonic labels and the relational coefficients

Unswitched harmonic labels and the relational coefficients · lines 678–736

VII-A-ST@1#eq:inverse, VII-A-ST@1#eq:observable, VII-A-ST@1#eq:A12, VII-A-ST@1#eq:fullA2

The index $μ$ labels the four scalar functions. It does not make the labels the components of a globally defined conical vector field.

Their formal inverse is

Y_{γ}^{μ} (y) = y^{μ} - κ w^{μ} (y) + κ^{2} (w^{ν} \partial_{ν} w^{μ} - v^{μ}) (y) (mod κ^{3}) .

Substitution in $X_{γ} (Y_{γ} (y))$ proves the formula and uniqueness order by order.

Define

Φ_{g} [γ; ϕ_{0}] = ϕ_{g} [γ; ϕ_{0}] \circ Y_{γ}, O_{f} [g] (γ, ϕ_{0}) = \int f (y) Φ_{g} [γ; ϕ_{0}] (y) d^{4} y .

The smearing uses the fixed label density, not $ρ_{γ} d^{4} y$ . No global numerical invertibility of $X_{γ}$ is required.

Taylor expansion gives

Φ_{g} A_{1}^{(g)} A_{2}^{(g)} = ϕ_{0} + κ A_{1}^{(g)} + κ^{2} A_{2}^{(g)} (mod κ^{3}), = u_{1, g} - w^{μ} \partial_{μ} ϕ_{0}, = u_{2, g} - w^{μ} \partial_{μ} u_{1, g} + (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} .

Write

C_{r} [g] (h, k; ϕ_{0}, f) = \int f A_{r}^{(g)} d^{4} y, r = 1, 2.

In particular, the complete second coefficient is

A_{2}^{(g)} = E_{m} V_{g} E_{m} V_{g} ϕ_{0} - E_{m} W_{g} ϕ_{0} + X_{w} E_{m} V_{g} 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} .

All five displayed contributions are retained.

Displayed formulas in this passage

VII-A-ST@1#eq:inverse · line 685
$Y_{γ}^{μ} (y) = y^{μ} - κ w^{μ} (y) + κ^{2} (w^{ν} \partial_{ν} w^{μ} - v^{μ}) (y) (mod κ^{3}) .$
VII-A-ST@1#eq:observable · line 697
$Φ_{g} [γ; ϕ_{0}] = ϕ_{g} [γ; ϕ_{0}] \circ Y_{γ}, O_{f} [g] (γ, ϕ_{0}) = \int f (y) Φ_{g} [γ; ϕ_{0}] (y) d^{4} y .$
VII-A-ST@1#eq:A12 · line 709
$Φ_{g} A_{1}^{(g)} A_{2}^{(g)} = ϕ_{0} + κ A_{1}^{(g)} + κ^{2} A_{2}^{(g)} (mod κ^{3}), = u_{1, g} - w^{μ} \partial_{μ} ϕ_{0}, = u_{2, g} - w^{μ} \partial_{μ} u_{1, g} + (w^{ν} \partial_{ν} w^{μ} - v^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .$
VII-A-ST@1#eq:fullA2 · line 726
$A_{2}^{(g)} = E_{m} V_{g} E_{m} V_{g} ϕ_{0} - E_{m} W_{g} ϕ_{0} + X_{w} E_{m} V_{g} ϕ_{0} + (w^{ν} \partial_{ν} w^{μ} - v^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .$
unlabeled · line 721
$C_{r} [g] (h, k; ϕ_{0}, f) = \int f A_{r}^{(g)} d^{4} y, r = 1, 2.$