Appendix A Modal Evolution from Anchoring Cost

(Built on Foundations v1.0 · 2025-06-10)

This appendix derives the second-order wave equation

γ \partial_{t}^{2} ψ - α \nabla^{2} ψ + β ψ = 0

for a mode envelope $ψ (x, t)$ directly from the PBG anchoring-cost action, and then shows how the familiar Schrödinger equation appears as a slow-envelope limit. All constants $α, β, γ$ are the locked values in Foundational Definitions.

A.1 Anchoring-Cost Action (import)

From Foundational Definitions, drop the matter source term ( $ρ_{m} = 0$ for a free mode):

S [ψ] = \int d^{4} x [γ | \partial_{t} ψ |^{2} - α | \nabla ψ |^{2} - β | ψ |^{2}] .

Recall units ([[Foundations/Full-Derivations#§0]]):
$[ψ] = m^{- 3 / 2}$ ,
$[α] = {J m}^{- 1}$ ,
$[β] = {J m}^{- 3}$ ,
$[γ] = {J s}^{2} m^{- 3}$ .

A.2 Euler–Lagrange Variation

Demand $δ S = 0$ under $ψ \to ψ + δ ψ$ with
$δ ψ$ vanishing at the boundary. Integration by parts gives

γ \partial_{t}^{2} ψ - α \nabla^{2} ψ + β ψ = 0.

This is the modal evolution law quoted in Foundations §4.

A.3 Slow-Envelope / Schrödinger Limit

Write the two-time-scale ansatz

ψ (x, t) = ϕ (x, t) e^{- i ω t}, | \partial_{t} ϕ | ≪ ω | ϕ | .

Keeping terms up to first order in $\partial_{t} ϕ$ and dropping
$\partial_{t}^{2} ϕ$ :

- γ ω^{2} ϕ - 2 i γ ω \partial_{t} ϕ = - α \nabla^{2} ϕ + β ϕ .

Solve for $i \partial_{t} ϕ$ :

i \partial_{t} ϕ = - \frac{α}{2 γ ω} \nabla^{2} ϕ + \frac{β - γ ω^{2}}{2 γ ω} ϕ .

Effective mass and potential

Match to the Schrödinger form

i ℏ \partial_{t} ϕ = - \frac{ℏ^{2}}{2 m} \nabla^{2} ϕ + V_{bias} ϕ,

to read off

m = \frac{ℏ^{2} γ ω}{α}, V_{bias} = \frac{β - γ ω^{2}}{2 γ ω} ℏ .

Physical meaning – the inertial mass of the mode grows with carrier
frequency $ω$ and temporal stiffness $γ$ .

A.4 External Potential Consistency

Adding an ambient phase field $Φ_{ext} (x)$ from other modes
(see [[Foundations/Full-Derivations#§3.-Least-Cost-Trajectory]])
amounts to the substitution $β \to β + η ρ_{m}^{ext}$ ,
which reproduces the classical potential

V_{ext} (x) = m c^{2} Φ_{ext} (x),

consistent with the universal force law
$m \ddot{r} = - m c^{2} \nabla Φ_{ext}$ .

A.5 Validity Domain

envelope variation ratio $ϵ = | \partial_{t} ϕ | / (ω | ϕ |) ≪ 1$
characteristic envelope scale $L$ satisfies $k L ≫ 1$ , where $k = \sqrt{β / α}$

Under these conditions the second-order PBG equation reduces to ordinary quantum mechanics; outside them one must revert to the full wave equation.

Last edited 2025-06-10 · based on Foundations v1.0
[Index](./Appendix Master) | Appendix B - Anchoring Cost