λEnter Calculus CompanionRegistered manuscripts, kept distinct

Manuscript VI@1 · VI.tex · figure · b5b60afb

Named coefficient equations

Action, signs, background, and coefficient domain › Formal solution jets · lines 484–503

VI@1#fig:VI-coefficient-domain

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metric / coordinatescalarswitching defectconditional quantumsupport / uncompleted
(η,0) is the exact background,G(γ)=κ24T(γ,ϕ)(\eta,0)\text{ is the exact background},\qquad G(\gamma)=\frac{\kappa^2}{4}T(\gamma,\phi)
γ=η+κh+κ2j,ϕ=ϕ0+κϕ1+κ2ϕ2\gamma=\eta+\kappa h+\kappa^2 j,\qquad \phi=\phi_0+\kappa\phi_1+\kappa^2\phi_2
κ0\kappa^0
κ1\kappa^1
κ2\kappa^2
G(1)(h)=0G^{(1)}(h)=0G(1)(j)+G(2)[h,h]=14T(η,ϕ0)G^{(1)}(j)+G^{(2)}[h,h]=\frac14 T(\eta,\phi_0)
Pϕ0=0P\phi_0=0Pϕ1=−Lhϕ0P\phi_1=-L_h\phi_0Pϕ2=−Lhϕ1−(Lj+Mh)ϕ0P\phi_2=-L_h\phi_1-(L_j+\mathsf M_h)\phi_0
Source factor 1/41/4. ϕ0\phi_0 is a coupling-expansion coefficient, not an exact nonzero Minkowski scalar background.

Coefficient equations on the named jet space. The action normalization fixes the second-order metric source to . The free scalar coefficient belongs to the coupling expansion and is not being treated as a nonzero exact scalar background on Minkowski spacetime. The displayed equations define the admissible jet domain.

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