DFT-22: The Casimir Effect as Projection-Constrained Phase Adjacency

MWells · Post by **MWells** » Thu Dec 04, 2025 7:48 pm

1. Introduction

The Casimir effect is one of the most frequently misunderstood physical phenomena. In conventional quantum field theory it is explained as a pressure arising from vacuum fluctuations of the electromagnetic field. In many textbooks this is stated without qualification, and virtual particles are invoked as explanatory devices. But virtual particles have no ontological clarity. They are not defined objects, nor do they arise from a first-principles model of space.

In DFT the Casimir effect takes on a different meaning. There is no vacuum. There is no continuum of modes. There are no virtual photons. There are phase relations — and projection constraints. What appears in the S-frame as a force between conducting plates is a necessary consequence of removing allowable T-frame adjacency relationships via S-frame geometric restriction.

This is a structural phenomenon, not a dynamical one. Nothing “pushes.” Something is disallowed, and the rebalancing of projection is seen by the S-frame as a pressure.

2. What “empty space” actually means in DFT

In DFT the “free” electromagnetic region of space corresponds to the T-frame having maximal phase-adjacency freedom consistent with the motion budget. That is, between sufficiently separated surfaces, the T-frame rotational degrees of freedom have maximal combinatorial compatibility under the projection into three-dimensional space.

In this regime we may speak of a set of T-frame phase states:

$\{\,\theta^i_a\,\}$

with adjacency relationships permitted:

$\theta^i_a \sim \theta^i_{a+1}$

meaning that two adjacent T-frame states can project into distinguishable S-frame spatial regions.

This adjacency structure is not “random phase”; it is a maximal freedom of projection compatibility.

3. What conducting boundaries do (in DFT terms)

A perfectly conducting surface maps a boundary condition in the S-frame to a restriction in allowable T-frame adjacency.

This requires careful phrasing.

The surface does not “reflect electromagnetic fields” in the conventional sense.
Rather, it constrains which T-frame phase states can produce consistent S-frame projections near the boundary. Mathematically this appears as a reduction in adjacency freedom:

$\theta^i_a \not\sim \theta^i_{a+1} \quad \text{for certain adjacency classes}$

Thus a “conductor” is not an absorber or reflector; it is a projection filter.

Two parallel conducting surfaces define a region in which certain adjacency relations are suppressed.

Outside the plates, adjacency relations are unconstrained.
Between the plates, they are selectively forbidden.

4. Phase adjacency as an “entropy”

We must be careful here: entropy in DFT has no thermodynamic meaning until S/T-frame representation is chosen. But adjacency freedom can be counted, and the S-frame sees a reduction in permissible adjacencies as a loss of available states.

Thus, outside the plates:

$N_{\text{out}} = \text{number of T-frame adjacency-compatible phase states}$

Inside the plates:

$N_{\text{in}} < N_{\text{out}}$

The key is that nothing is “pushed” from outside to inside. Instead, the projection constraint energy differs.

The energy difference is structural:

$\Delta E \propto \big( N_{\text{out}} - N_{\text{in}} \big)$

This is not a force mediated by quanta; it is an energy gradient created by differential adjacency counting.

5. Why adjacency constraints produce a measurable force

So far we have:

Outside: maximal adjacency compatibility.
Inside: reduced adjacency compatibility.

What is the meaning of “force” here?

In DFT a “force” is not something exerted by one body on another, but rather the manifestation in the S-frame of a reallocation of the motion budget necessary to restore projection consistency.

Between the plates, some adjacency configurations are forbidden, so the projection must reconfigure itself in a way that locally reduces the motion budget.

That reduction must be globally compensated by shifting the projection geometry — which appears in the S-frame as a pressure toward minimizing the constrained region.

Thus the attractive “Casimir force” corresponds to the projection geometry attempting to minimize the region of adjacency suppression, which occurs when the plates approach one another.

6. Why this depends on plate separation

When the plates are widely separated, only a small subset of adjacency relations are constrained. When the plates come closer, the fraction of disallowed adjacency configurations increases. In the notation above:

$\frac{N_{\text{in}}}{N_{\text{out}}} \searrow \quad \text{as plate separation decreases}$

The corresponding energy difference becomes more pronounced:

$\Delta E(d) \propto \big( N_{\text{out}} - N_{\text{in}}(d) \big)$

Since this difference depends on the separation 𝑑, the plates experience an effective S-frame restoring action:

$F(d) = -\frac{\partial}{\partial d}\,\Delta E(d)$

This is not a “push”; it is the projection geometry minimizing constraint.

7. Why this is not “vacuum energy”

We can summarize the difference between QM/QFT and DFT in one sentence:

In QFT, the Casimir force is a difference in vacuum-mode energy.
In DFT, it is a difference in phase adjacency compatibility.

In QFT: virtual modes → vacuum pressure difference
In DFT: adjacency restricted → projection energy difference

The former is heuristic.
The latter arises from structural requirements of projection geometry.

Importantly, DFT does not posit an unobservable vacuum populated with fluctuating quanta. There is only scalar motion and projection constraints.

8. Why this explanation is falsifiable

A good theory does not merely reinterpret. It makes testable structural consequences. DFT predicts:

The Casimir force magnitude should depend only on adjacency constraints, not on the details of conductivity.
Surfaces that constrain adjacency differently (but are electromagnetically similar) should produce different Casimir forces.
Changing projection geometry (e.g., by inserting a transparent dielectric) will modify adjacency compatibility and thus modify the force.

These differ from standard QFT predictions, offering experimental leverage.

9. Summary

The Casimir effect arises in DFT because:

The T-frame contains phase adjacency relations that must be consistent under S-frame projection.
Conducting surfaces restrict which adjacencies are projection-compatible.
Between two surfaces, adjacency compatibility is reduced.
The motion budget must reallocate to maintain projection consistency.
The S-frame sees this reallocation as an effective attraction.

This description requires:

no vacuum,
no fluctuating fields,
no virtual photons,
no zero-point pressure.

It follows directly from:

scalar motion in the NRS,
projection geometry,
adjacency relations,
motion-budget closure.