Appendix E — Unified Action Principle

E.1 Motivation

Traditional physics relies on multiple, independent fields (metric, gauge potentials, scalar fields) and distinct action principles for gravity, electromagnetism, and quantum dynamics. Phase-Biased Geometry (PBG) replaces this multiplicity with a single scalar coherence field—the coherence phase $\mathrm{\Phi }(t,\mathbf{x})$—governed by one unified action principle. This economy underlies PBG’s predictive power across atomic, gravitational, and cosmological scales.

E.2 Coherence-Anchoring Action

We define the action functional

with Lagrangian density

Here:

• $\alpha$ (J·s²/m³): spatial anchoring stiffness.
• $\beta$ (J/m³): coherence bias (mass) term.
• $\gamma$ (J·s²/m): temporal anchoring inertia.

No additional fields or parameters are assumed.

E.3 Derivation of Lagrangian Terms

Starting from the coherence-cost integral over modal envelopes and imposing:

• global phase-shift invariance ($\mathrm{\Phi }\mapsto \mathrm{\Phi }+\text{const}$),
• spatial homogeneity & isotropy,
• time-translation invariance,
• parity & time-reversal,

one finds that the only relevant or marginal operators (mass dimension $\le 4$) are:

These match the three terms in $\mathcal{L}[\mathrm{\Phi }]$. All higher-derivative or higher-power terms (e.g.\ ${\mathrm{\Phi }}^4$, $({\mathrm{\nabla }}^2\mathrm{\Phi }{)}^2$) are either forbidden by symmetry or irrelevant in the infrared.

E.4 Euler–Lagrange Equation

Variation of $S[\mathrm{\Phi }]$ yields

leading to the field equation

E.5 Physical Corollaries

1. Wave dispersion:
   Plane waves $\mathrm{\Phi }\propto e^{i(\mathbf{k}\cdot \mathbf{x}-\omega t)}$ satisfy

   $${\omega }^2=c^2{k}^2,c^2=\frac{\alpha }{\gamma }.$$

2. Yukawa kernel (static):
   Solve

   $$\alpha {\mathrm{\nabla }}^2\mathrm{\Phi }-\beta \mathrm{\Phi }=-Q{\delta }^3(\mathbf{x}),$$

   yielding

   $$\mathrm{\Phi }(r)=\frac{Q}{4\pi \alpha r}{e}^{-r/\ell },\ell =\sqrt{\frac{\alpha }{\beta }}.$$

3. Cosmological redshift law:
   Coherence turnover rate $k=1/\ell$ gives

   $$D_L(z)=\frac{c}{k}(1+z)\ln (1+z).$$

E.6 Units & Dimensions

E.7 EFT & Symmetry Uniqueness (Sketch)

Up to mass dimension 4, the only allowed relevant or marginal scalar operators are $({\partial }_t\mathrm{\Phi }{)}^2$, $|\mathrm{\nabla }\mathrm{\Phi }{|}^2$, and ${\mathrm{\Phi }}^2$ under the symmetries listed. All other operators are either non-invariant or irrelevant (suppressed by a UV cutoff).

E.8 Cross-References

• See Appendix AJ for Lamb-shift calculations using $\mathrm{\Phi }$.
• Applications in Sections 2–5 deploy these results.

E.9 Closing Summary

From the single action $S[\mathrm{\Phi }]$, PBG derives the full spectrum of modal behavior—relativistic waves, gauge interactions, gravitation, quantum spectra, and cosmological expansion—using only three fundamental constants.

Appendix D | [Index](./Appendix Master) | Appendix F