Appendix J — Derivation 10: Casimir Pressure from Anchoring Suppression

(Foundations v1.0 • 2025-06-10)

No virtual photons needed.
The attraction between two neutral plates appears because the
coherence modes that would anchor in free space are suppressed in the
gap, creating an anchoring-cost imbalance that pulls the plates together.

1 Anchoring cost in free space

With unlocked constants (Foundations §1)

C_{\infty} = \sum_{n} \int [α | \nabla ψ_{n} |^{2} + β | ψ_{n} |^{2}] d^{3} x .

For an unconstrained volume the mode set ${ψ_{n}}$ is continuous.

2 Boundary suppression

Two parallel, perfectly reflecting plates at distance $d$ impose
Dirichlet-type conditions. Allowed modes become

ψ_{n}^{in} (x, y, z) \propto \sin (\frac{n π z}{d}), n = 1, 2, \dots

Many low- $k$ modes present outside are absent inside → higher
anchoring cost per unit volume between plates.

3 Cost imbalance → pressure

Define surface-energy density

Δ C (d) = \frac{1}{A} (C_{in} - C_{out}) < 0.

Net force per area

P (d) = - \frac{\partial}{\partial d} Δ C (d) .

Carrying out the standard mode sum (now over anchoring energies rather
than zero-point energies) one obtains the textbook result

P (d) = - \frac{π^{2} ℏ c}{240 d^{4}} .

No divergent vacuum terms; the factor $ℏ c$ enters because the
lowest-action photon mode ( $S_{min} = 2 π ℏ$ , Appendix ℏA) sets the
normalisation of each suppressed channel.

4 Interpretation

QFT picture	PBG picture
Vacuum fluctuations removed	Coherence modes structurally excluded
Renormalised energy	Anchoring-cost differential
Force emerges after subtracting infinities	Force is direct gradient of $Δ C (d)$

5 Generalisations

Curved plates → replace $\sin (n π z / d)$ by local eigenfunctions.
Layered media → predicts van der Waals forces without dipole
fluctuations.
Cosmology → large-scale voids act as anchoring suppression zones,
yielding small negative pressure akin to dark-energy phenomenology.

Take-home

Casimir attraction is a real structural penalty for excluding
coherence, not a mystical vacuum fluctuation; the formula survives
unchanged because the counting of suppressed modes is identical.

End of SM Companion Derivations

Appendix I - Strong & Weak Phenomena | [Index](./Appendix Master) | Appendix K - Speed of Light