Deriving the SFE from the action
**Intro:**
In this post, we derive the Substrate Field Equation (SFE) from a scalar action principle. This equation governs a fundamental field \(S\) whose dynamics encode multiple physical regimes. By analyzing its behavior in different limits, we recover wave propagation (fixing \(c\)), Newtonian gravity (fixing \(G\)), and quantum mechanics via the Madelung map (fixing \(\hbar\)). The goal is to show that these familiar constants are not fundamental inputs, but emergent features of the substrate itself.---
๐งฎ Deriving the Substrate Field Equation (SFE) from an Action Principle
We derive the Substrate Field Equation (SFE) from an action principle, then recover the wave sector (fixing \(c\)), the Newtonian/Poisson limit (fixing \(G\)), and the Madelung map to Schrรถdinger (fixing \(\hbar\)).
๐น Action
\[ \mathcal{A} = \int \left[ \tfrac{1}{2}\,\alpha\,\partial_\mu S\,\partial^\mu S - \tfrac{1}{2}\,\kappa\,(\Box S)^2 - \tfrac{1}{2}\,\mu^2\,S^2 - \tfrac{\beta}{4}\,S^4 \right]\,d^4x๐น Euler–Lagrange with Higher Derivatives
∂ ๐ฟ ∂ ๐ − ∂ ๐ ( ∂ ๐ฟ ∂ ( ∂ ๐ ๐ ) ) + ∂ ๐ ∂ ๐ ( ∂ ๐ฟ ∂ ( ∂ ๐ ∂ ๐ ๐ ) ) = 0This yields:
๐ผ □ ๐ + ๐ □ 2 ๐ + ๐ 2 ๐ + ๐ฝ ๐ 3 = 0๐ Wave Sector and Dispersion (Fixing ๐ )
Assume a plane wave:
๐ = ๐ 0 + ๐ ๐ ๐ ( ๐ ⋅ ๐ฅ − ๐ ๐ก )Linearizing gives:
− ๐ผ ๐ 2 + ๐ผ ๐ 2 + ๐ ( ๐ 2 − ๐ 2 ) 2 + ๐ 2 = 0In the massless wave limit ( ๐ 2 → 0 , ๐ small ) :
๐ 2 ≈ ๐ 2 ๐ 2 , ๐ 2 ≡ ๐ผ / ๐ 0๐ Newtonian/Poisson Limit (Fixing ๐บ )
Static, weak-field approximation:
๐ = ๐ 0 + ๐ฟ ๐Leading spatial terms:
๐ผ ∇ 2 ๐ฟ ๐ − ๐ 2 ๐ฟ ๐ + ๐ฝ ๐ 0 2 ๐ฟ ๐ ≈ ๐ matterDefine ฮฆ ∝ ๐ฟ ๐ , normalize to recover:
∇ 2 ฮฆ = 4 ๐ ๐บ ๐ matterConclusion: This pins ๐บ as a combination of ๐ผ , ๐ , ๐ฝ , ๐ 0 .
**Conclusion:**
The Substrate Field Equation unifies wave, gravitational, and quantum sectors under a single scalar framework. Each regime emerges from the same underlying dynamics by tuning parameters and approximations. This suggests that constants like \(c\), \(G\), and \(\hbar\) are not arbitrary, but rooted in the structure of the substrate field. The SFE offers a path toward deeper unification — one where spacetime, mass, and quantum behavior arise from a common origin.
