Casimir Pressure

The Casimir effect is one of the most striking confirmations of quantum field theory. It describes the attractive force between two uncharged, parallel plates placed in a vacuum—often interpreted as a result of vacuum energy or virtual particle fluctuations.

In modal dynamics, there is no vacuum energy.
There are no virtual particles.
Yet the Casimir pressure arises naturally—from modal anchoring suppression between boundaries.

Coherence and Suppression

In the coherence-based framework, the vacuum is not empty. It is a modal medium—capable of supporting structured phase configurations.

Between two plates, the region becomes constrained. Certain modal structures that could normally anchor cannot form due to geometric limitation.

This leads to:

Reduced anchoring capacity between the plates
Higher modal density outside the plates
A net bias pressure from anchoring asymmetry

Anchoring Cost Differential

Let the anchoring cost density be defined as:

C (x) = γ | \partial_{t} ψ |^{2} + α | \nabla ψ |^{2} + β | ψ |^{2}

In the region between the plates, not all standing wave modes $ψ_{n} (x)$ are permitted. Modes that would normally contribute to anchoring are excluded by boundary conditions.

Outside the plates, no such restriction exists.

This creates a coherence cost differential:

Δ C = C_{outside} - C_{between}

This cost gradient produces a net structural bias—a compression force pushing the plates together.

Pressure Without Vacuum

This anchoring-based pressure matches the Casimir force:

F = - \frac{π^{2} ℏ c}{240 d^{4}}

But here, it is not derived from vacuum energy subtraction. It emerges from a real structural mechanism:

Anchoring modes are excluded between plates
The modal medium pushes inward to rebalance coherence tension

There is no infinite energy and no need for regularisation.
There is only real structure being suppressed, and the system responding accordingly.

Why This Is Important

It reinterprets a quantum effect without requiring field quantisation
It demonstrates that coherence-based structure can produce measurable forces
It shows how modal theory explains boundary effects as naturally as bulk behaviour

The Casimir effect is not the fingerprint of vacuum fluctuations.
It is the modal medium resisting exclusion.

(See Appendix J — Anchoring Suppression and Vacuum Pressure.)

Casimir pressure is not pulled from emptiness.
It is the bias gradient of coherence trying to restore what structure cannot form.