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

Scalar and coordinate intertwiners

Plateau covariance and the second-order intertwiner › Scalar and coordinate intertwiners · lines 1222–1242

VII-A@1#eq:phi1transform, VII-A@1#eq:phi2transform, VII-A@1#eq:wtransform, VII-A@1#eq:vtransform

Coefficient extraction yields

ϕ_{1, g}^{'} ϕ_{2, g}^{'} w^{' μ} v^{' μ} = ϕ_{1, g} + X_{ξ} ϕ_{0}, = ϕ_{2, g} + X_{ξ} ϕ_{1, g} + X_{ζ} ϕ_{0} + \frac{1}{2} X_{ξ}^{2} ϕ_{0}, = w^{μ} + ξ^{μ}, = v^{μ} + ξ^{ν} \partial_{ν} w^{μ} + ζ^{μ} + \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ} .

Displayed formulas in this passage

VII-A@1#eq:phi1transform · line 1225
$ϕ_{1, g}^{'} ϕ_{2, g}^{'} w^{' μ} v^{' μ} = ϕ_{1, g} + X_{ξ} ϕ_{0}, = ϕ_{2, g} + X_{ξ} ϕ_{1, g} + X_{ζ} ϕ_{0} + \frac{1}{2} X_{ξ}^{2} ϕ_{0}, = w^{μ} + ξ^{μ}, = v^{μ} + ξ^{ν} \partial_{ν} w^{μ} + ζ^{μ} + \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ} .$
VII-A@1#eq:phi2transform · line 1225
$ϕ_{1, g}^{'} ϕ_{2, g}^{'} w^{' μ} v^{' μ} = ϕ_{1, g} + X_{ξ} ϕ_{0}, = ϕ_{2, g} + X_{ξ} ϕ_{1, g} + X_{ζ} ϕ_{0} + \frac{1}{2} X_{ξ}^{2} ϕ_{0}, = w^{μ} + ξ^{μ}, = v^{μ} + ξ^{ν} \partial_{ν} w^{μ} + ζ^{μ} + \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ} .$
VII-A@1#eq:wtransform · line 1225
$ϕ_{1, g}^{'} ϕ_{2, g}^{'} w^{' μ} v^{' μ} = ϕ_{1, g} + X_{ξ} ϕ_{0}, = ϕ_{2, g} + X_{ξ} ϕ_{1, g} + X_{ζ} ϕ_{0} + \frac{1}{2} X_{ξ}^{2} ϕ_{0}, = w^{μ} + ξ^{μ}, = v^{μ} + ξ^{ν} \partial_{ν} w^{μ} + ζ^{μ} + \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ} .$
VII-A@1#eq:vtransform · line 1225
$ϕ_{1, g}^{'} ϕ_{2, g}^{'} w^{' μ} v^{' μ} = ϕ_{1, g} + X_{ξ} ϕ_{0}, = ϕ_{2, g} + X_{ξ} ϕ_{1, g} + X_{ζ} ϕ_{0} + \frac{1}{2} X_{ξ}^{2} ϕ_{0}, = w^{μ} + ξ^{μ}, = v^{μ} + ξ^{ν} \partial_{ν} w^{μ} + ζ^{μ} + \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ} .$