Editing Openai/691c1dba-9228-800f-8463-13b3a9006306 (section)

===== - ∂μt∂μt\partial_\mu t \partial^\mu t∂μt∂μt — if ttt is literal time coordinate, that object is odd (dimensionally and conceptually). If it’s a clock field call it τ(x)\tau(x)τ(x) or θ(x)\theta(x)θ(x) and use canonical kinetic term ∂μτ∂μτ\partial_\mu \tau \partial^\mu \tau∂μτ∂μτ. =====
* The matrix element ⟨ψi∣E^∣ψj⟩\langle\psi_i|\hat E|\psi_j\rangle⟨ψi∣E^∣ψj⟩ — what index structure? Is this a global factor, or summed over i,j? If it’s part of path integral weighting, embed it into an operator insertion.
* The term ηtransϕψhμνhμν\eta_{\text{trans}} \phi\psi h_{\mu\nu}h^{\mu\nu}ηtransϕψhμνhμν mixes scalar product units with energy density of gravitational waves. If EFT, set scales explicitly: ηtrans/Mpl2\eta_{\text{trans}}/M_{\text{pl}}^2ηtrans/Mpl2 etc.