Uncertainty Principle

Appendix AR — Derivation of the Uncertainty Principle from PBG Anchoring Cost

(Locked derivation • Foundations v1.0 • 2025-06-10)

Result. For any mode in Phase-Biased Geometry
$$

\boxed{;\Delta x,\Delta p ;\ge; \dfrac{\hbar}{2};}

> where $\hbar$ is the anchoring-action quantum derived in >Appendix AS - Planck ℏA. > The bound is **purely structural**—no measurement or observer postulates > are invoked. --- ## 1 Setup and notation * Envelope $\psi(\mathbf x)$ with normalisation $$\int |\psi|^{2} d^{3}x = 1.

Units ([[Foundations/Full-Derivations#§0]]): $[ψ] = m^{- 3 / 2}$ .

C_{stat} = α | \nabla Φ |^{2} + β | ψ |^{2} .

For localisation analysis we set $β$ aside (envelope penalty is fixed)
and focus on the gradient term.

Define expectation values over the envelope:

⟨ f ⟩ = \int | ψ |^{2} f (x) d^{3} x .

Then

(Δ x)^{2} = ⟨ x^{2} ⟩ - ⟨ x ⟩^{2}, (Δ p)^{2} = ⟨ p_{x}^{2} ⟩ - ⟨ p_{x} ⟩^{2} .

Using $p = ℏ \nabla Φ$ ,

α \int | \nabla Φ |^{2} d^{3} x = \frac{α}{ℏ^{2}} \int | p |^{2} d^{3} x .

Expressed as envelope averages:

\begin{matrix} (3.1) & C_{grad} = \frac{α}{ℏ^{2}} ⟨ p_{x}^{2} + p_{y}^{2} + p_{z}^{2} ⟩ . \end{matrix}

(Units: J m⁻¹ · m⁻² · m³ = J.)

Consider the operator pair $x$ and $p_{x} = - i ℏ \partial_{x}$ .
For any normalised $ψ$ :

⟨ [x, p_{x}] ⟩ = ⟨ i ℏ ⟩ = i ℏ .

The Robertson–Schwarz inequality (pure algebra, no measurement lore):

\begin{matrix} (4.1) & Δ x Δ p \geq \frac{1}{2} | ⟨ [x, p_{x}] ⟩ | = \frac{ℏ}{2} . \end{matrix}

Localising $ψ$ — shrinking $Δ x$ forces larger gradients
$\partial_{x} ψ$ , which in turn increase the cost term
$α | \nabla Φ |^{2}$ via (3.1).
Minimising total cost subject to one full phase cycle
( $\int \partial_{t} Φ d t = 2 π$ ) leads to the tight inequality (4.1).
Thus the $ℏ / 2$ bound is the price of maintaining coherence
under simultaneous spatial and momentum confinement.

Take

ψ (x) = \frac{1}{(π σ^{2})^{1 / 4}} e^{- x^{2} / (2 σ^{2})}, Φ = k x .

Then $Δ x = σ / \sqrt{2}$ , $Δ p = ℏ k / \sqrt{2}$ and
$\partial_{x} ψ = - x ψ / σ^{2}$ .

Minimising the total static cost
$C = α ⟨ | \nabla Φ |^{2} ⟩$
with respect to $k$ at fixed $σ$ gives
$k = 1 / σ$ , yielding

Δ x Δ p = \frac{ℏ}{2} .

A Gaussian packet therefore saturates the bound—identical to standard QM.

The inequality $Δ x Δ p \geq ℏ / 2$ is a direct algebraic
consequence of the phase operator pair $(x, - i ℏ \partial_{x})$ acting on a
normalised envelope—no probabilistic or observer postulate used.
In PBG the same bound arises physically because anchoring cost explodes
if one attempts to squeeze both $x$ and the phase gradient (momentum)
simultaneously. The constant $ℏ$ is the action-per-cycle already
derived from the photon packet (Appendix AS - Planck ℏA).

Last edited 2025-06-10 • uses only Foundations equations and $ℏ$ from Appendix ℏA