PBG Variational Derivations

1. Derivation of the Coherence Field $B (\vec{x})$

We begin by defining the coherence anchoring cost functional:

C [B] = \int (α (\nabla B)^{2} + β B^{2}) d^{3} x

where:

$α$ is the phase stiffness coefficient,
$β$ is the saturation penalty coefficient,
$B (\vec{x})$ is the coherence field.

We seek the field configuration that minimises $C [B]$ under variations $δ B$ .

Taking the functional derivative and setting it to zero:

δ C = \int (2 α \nabla B \cdot \nabla δ B + 2 β B δ B) d^{3} x = 0

Integrating by parts the first term:

\int \nabla B \cdot \nabla δ B d^{3} x = - \int (\nabla^{2} B) δ B d^{3} x

(assuming boundary terms vanish).

Thus:

δ C = \int (- 2 α \nabla^{2} B + 2 β B) δ B d^{3} x = 0

Because $δ B$ is arbitrary, the Euler–Lagrange equation follows:

α \nabla^{2} B - β B = 0

This is the Helmholtz equation.

The general solution for a point emitter at the origin is:

B (r) = \frac{A}{r} e^{- k r}

where:

k = \sqrt{\frac{β}{α}}

The bias functional measuring the anchoring cost of a modal configuration $ψ (\vec{x})$ is:

B [ψ] = \int d^{3} x d^{3} x^{'} Γ (\vec{x} - {\vec{x}}^{'}) | ψ (\vec{x}) - ψ ({\vec{x}}^{'}) |^{2}

Expanding:

| ψ (\vec{x}) - ψ ({\vec{x}}^{'}) |^{2} = | ψ (\vec{x}) |^{2} + | ψ ({\vec{x}}^{'}) |^{2} - 2 Re (ψ^{*} (\vec{x}) ψ ({\vec{x}}^{'}))

Thus:

B [ψ] = 2 \int d^{3} x | ψ (\vec{x}) |^{2} (\int Γ (\vec{x} - {\vec{x}}^{'}) d^{3} x^{'}) - 2 \int d^{3} x d^{3} x^{'} Γ (\vec{x} - {\vec{x}}^{'}) Re (ψ^{*} (\vec{x}) ψ ({\vec{x}}^{'}))

Defining:

Γ_{0} (\vec{x}) = \int Γ (\vec{x} - {\vec{x}}^{'}) d^{3} x^{'}

the bias functional simplifies to:

B [ψ] = 2 \int d^{3} x Γ_{0} (\vec{x}) | ψ (\vec{x}) |^{2} - 2 \int d^{3} x d^{3} x^{'} Γ (\vec{x} - {\vec{x}}^{'}) Re (ψ^{*} (\vec{x}) ψ ({\vec{x}}^{'}))

Thus, modal interference lowers anchoring cost when $ψ (\vec{x})$ and $ψ ({\vec{x}}^{'})$ are phase-aligned.

3. Derivation of Motion from Bias Cost Minimisation

The action for a modal packet $ψ (\vec{x}, t)$ is:

S [ψ] = \int d t [i ℏ \int ψ^{*} (\vec{x}, t) \frac{\partial ψ (\vec{x}, t)}{\partial t} d^{3} x - B [ψ]]

Stationarity $δ S = 0$ under variations $δ ψ^{*}$ yields the evolution equation:

i ℏ \frac{\partial ψ}{\partial t} = \frac{δ B [ψ]}{δ ψ^{*}}

Assuming small curvature (long-wavelength limit) and expanding $B [ψ]$ yields:

\frac{δ B [ψ]}{δ ψ^{*}} \approx - α \nabla^{2} ψ + β ψ

Thus the evolution equation becomes:

i ℏ \frac{\partial ψ}{\partial t} = - α \nabla^{2} ψ + β ψ

Identifying modal "mass" $m$ and "bias potential" $V_{bias}$ , we recover the motion equation:

i ℏ \frac{\partial ψ}{\partial t} = - \frac{ℏ^{2}}{2 m} \nabla^{2} ψ + V_{bias} (\vec{x}) ψ

with:

α = \frac{ℏ^{2}}{2 m}, V_{bias} = β

This completes the full, expanded derivations of the coherence field, bias functional, and motion from cost minimisation in the PBG framework.