DFT-26: Josephson Junctions as Phase Interpolation Interfaces

MWells · Post by **MWells** » Thu Dec 04, 2025 9:23 pm

DFT-26 — Josephson Junctions as Phase Interpolation Interfaces

In Dual-Frame Theory, superconductivity arises when an ensemble of trajectories share a common T-frame phase relation, such that the spatial projection is constrained to reflect a single effective coherence manifold. In the bulk Meissner case (DFT-24), the constraint appears as a suppression of curvature in the T-frame, which expresses as magnetic field exclusion in the S-frame. In the mixed-state case (DFT-25), vortices represent allowable topological defects in the T-frame coherence under external forcing.

A Josephson junction is a different kind of interface. Instead of allowing topological winding within a coherent region, it connects two distinct T-frame coherence domains across a region whose local properties suppress coherence, but without terminating it. The S-frame picture is not “tunneling” of a particle; it is continuity of a phase-defined moving system under a forced interpolation.

To analyze this, consider two superconducting domains, A and B, with well-defined macroscopic phase assignments in the T-frame, denoted 𝜃_𝐴 and 𝜃_𝐵. These phases reflect the local T-frame reference choices for a single coherent mode. The S-frame does not directly “see” these phases; it sees the projection consequences of their difference.

The short region between A and B — the weak link — is where the T-frame coherence is attenuated but not destroyed. The central fact, in DFT terms, is:

The junction region cannot force 𝜃_𝐴=𝜃_𝐵, but it can enforce continuity of the projection of the phase difference.

This continuity condition is the Josephson condition.

1. Phase Difference as an Interpolation Constraint

The physical observable is not b]𝜃[/b]_𝐴 or 𝜃_𝐵 individually, but the difference

$\Delta\theta \equiv \theta_B - \theta_A$

This difference is not a “wavefunction phase” in a literal ontological sense; it is the projection of relative T-frame phase positions of the shared scalar-motion background.

The Josephson junction enforces the relation that the S-frame projection of current is determined by that phase difference. In its simplest expression:

$I = I_c \sin(\Delta\theta)$

This is not an empirical miracle; it is the only possible continuous interpolation between two coherent T-frame assignments when the projection must remain consistent.

2. Why the DC Josephson Relation is Not Hand-Waving

There is no “force” driving coherent pairs across the junction. There is instead a continuity requirement:

The S-frame projection cannot represent two incompatible phase origins at the same spatial location.

When the two superconductors are held at identical chemical potential (no S-frame voltage difference), the phase difference is static. Thus the current is constant.

That is the DC Josephson effect.

3. Introducing a Voltage: Phase Drift

A voltage difference does not “push charges”; in DFT terms it shifts the relative phase advance rates.

Let 𝑉 be the S-frame representation of the imposed energy difference. Then the relative T-frame phase evolves as

$\dot{\Delta\theta} = \frac{2e}{\hbar} \, V$

This is the standard Josephson frequency relation, but here it is understood as:

Not a wavefunction evolution,
Not tunneling of charged pairs,
But a geometric rate of mismatch accumulation between T-frame coherence assignments.

Given this phase drift, the S-frame must represent an oscillatory current:

$I(t) = I_c \sin\!\left(\Delta\theta_0 + \frac{2e}{\hbar} V t\right)$

This is the AC Josephson effect.

4. Why the “Weak Link” Does Not Destroy Coherence

In conventional quantum mechanics, the weak link is treated as a tunneling barrier. In DFT, that language is misleading. The correct interpretation is:

Coherence in the weak link is locally weakened but globally constrained.

The junction region attenuates the coherence magnitude (which determines 𝐼_𝑐), but cannot remove the continuity condition.

In the T-frame, the coherence map remains single-valued:

It can change slope,
But cannot be discontinuous.

Thus we can say precisely:

The Josephson junction enforces a phase-interpolation geometry, not particle transmission.

5. Clarifying “Where the Current Flows”

In RS terms, the current is motion of space through matter. In DFT, that maps to:

A coherent T-frame scalar flux expressed in S-frame geometry.

The Josephson current is therefore not a stream of corpuscles crossing a forbidden region, but rather:

A forced projection of phase continuity across a region where coherence amplitude is small.

This also explains why the Josephson current can persist with negligible dissipation:

There is no motion of spatial objects through the weak link,
There is only a continuity constraint imposed on the projection.

6. Why Josephson Oscillations Define Precise Frequencies

The AC frequency

$\omega = \frac{2e}{\hbar} \, V$

is not a particle-spin effect or a quantization rule; it is:

The rate of phase mismatch accumulation
Under a constant S-frame energy offset
Required to maintain projection continuity

The S-frame sees an oscillation because the T-frame coherence must remain consistent everywhere, and the only consistent projection of a steadily drifting phase difference is a periodic current.

7. Why This is Not Mystical

The Josephson effects hinge on the same fundamental principles already established:

T-frame phase encodes relational structure of scalar-motion.
S-frame current is the geometric projection of phase-gradient relations.
Interfaces cannot terminate coherence; they interpolate it.

There is nothing mysterious or ad hoc here; the Josephson relations emerge as minimal geometric consequences of:

The motion-first ontology,
The dual projection structure,
And the scalar nature of coherence.