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

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* Define ϕ,ψ\phi,\psiϕ,ψ: are they complex scalars? spinors? order parameters? Real vs complex matters for conjugation and hermiticity.
'' Units/dimensions: kinetic-like term ∂μt∂μt\partial_\mu t \partial^\mu t∂μt∂μt is unusual — ttt is normally a coordinate, not a field. If ttt is a scalar clock-field you need to state its dimension; otherwise replace with canonical kinetic terms ∂μϕ∂μϕ∗\partial_\mu\phi\partial^\mu\phi^''∂μϕ∂μϕ∗ or ∂μφ∂μφ\partial_\mu\varphi\partial^\mu\varphi∂μφ∂μφ.
* Hermiticity: terms like ϕψ⟨ψi∣E^∣ψj⟩\phi\psi \langle\psi_i|\hat E|\psi_j\rangleϕψ⟨ψi∣E^∣ψj⟩ need conjugate partners to make the action real if ϕ,ψ\phi,\psiϕ,ψ are complex. Introduce + c.c. where needed.
* ϵ\epsilonϵ regulator: fine, but if it’s to prevent division by zero, document its scale or use ∣ϕψ∣2+ϵ2|\phi\psi|^2 + \epsilon^2∣ϕψ∣2+ϵ2.
* χnon-local\chi_{\text{non-local}}χnon-local: nonlocal objects must be written as integrals or kernels: e.g. ∫d4x′ K(x,x′)O(x′)\int \mathrm{d}^4x'\, K(x,x')\mathcal{O}(x')∫d4x′K(x,x′)O(x′).
* FμνFμνF_{\mu\nu}F^{\mu\nu}FμνFμν: good canonical EM term — but if you couple it add the gauge covariant derivatives for charged fields (if ϕ,ψ\phi,\psiϕ,ψ are charged).
* hμνhμνh_{\mu\nu}h^{\mu\nu}hμνhμν: perturbation-squared is fine, but gauge/GR consistency usually couples via −ggμν∂μϕ∂νϕ\sqrt{-g}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi−ggμν∂μϕ∂νϕ or linear coupling hμνTμνh_{\mu\nu}T^{\mu\nu}hμνTμν. Your coupling is phenomenological — acceptable for EFT, but call it so.
* Stability: the negative exponent and sinusoid could produce sign flips and instabilities; analyze potential minima of V(ϕ,ψ)V(\phi,\psi)V(ϕ,ψ).

===== - ∂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.