DFT-27: SQUID Geometry as Global Phase Constraint and Flux Quantization
Posted: Thu Dec 04, 2025 9:40 pm
In DFT-23 through DFT-26 we reached a key structural viewpoint:
a superconducting state is not described by a spatial wave amplitude, but by a coherent T-frame phase winding over a compact manifold.
The S-frame electromagnetic observables are projections of this deep, global phase structure.
A single superconducting segment already exhibits this coherence; but when superconductors form a closed loop, something stronger becomes true:
the phase cannot be defined independently at each local point.
It is globally constrained.
The loop enforces a condition that any T-frame description of the superconducting state must be single-valued.
That is, although the phase may vary around the ring, after traversing the full circuit it must return to the same point in the T-frame.
The way standard superconductivity texts phrase this — “the phase difference must be a multiple of 2𝜋” — sounds like a rule imposed by hand.
In DFT it is automatic: it expresses the fact that the T-frame phase lives in a compact space and its S-frame projection must be globally consistent.
Let the T-frame phase along the SQUID loop be described by a smooth function
)
where 𝜆 is a loop parameter.
After moving around the loop of length 𝐿,
 = \Theta(\lambda) + 2\pi n.
)
This does not require that the phase remain constant.
It only requires that the net phase change per loop equals an integer multiple of 2𝜋:
The integer 𝑛 labels the global T-frame winding class.
This is not an approximation; it is required by the topology of the scalar-motion phase manifold.
Local Phase vs Global Closure
Notice what has happened:
What looks like an “instantaneous nonlocal constraint” in conventional spatial terms is, in DFT, simply the fact that the scalar-motion description supports global single-valuedness.
Connection to Magnetic Flux: Not a Local Interaction
We have not yet mentioned magnetic flux, because magnetic flux is the S-frame shadow of something deeper.
External magnetic flux influences the phase not by exerting force,
but by changing the accessible projection class.
The superconducting loop can only occupy winding classes consistent with the imposed curvature.
DFT formalizes this through the relation
but the meaning here is different from the standard interpretation.
The constant Φ0 is not “measured” or “derived,” but represents the winding–flux correspondence analogous to how Josephson relations connect phase and voltage.
In DFT:
The Mixed Expression: T-frame Winding and S-frame Projection
To make the geometry explicit, the total winding condition in the presence of vector potential 𝐴𝑖 may be written as
=
2\pi n,
\qquad
n \in \mathbb{Z}.
)
The spatially measured flux is just
Thus, one sees directly that
The flux appears as the S-frame projection of that state.
Why a SQUID “Sees Flux Through Shielding”
This is one of the most misunderstood facts in standard superconductivity:
A SQUID still registers flux even when fully shielded spatially.
From the spatial viewpoint, this looks mysterious or “nonlocal.”
In DFT the situation is transparent:
This is not “spooky action.”
It is coherence in the scalar-motion domain, which naturally enforces global phase closure.
The Conceptual Core
The SQUID is therefore not primarily a flux detector.
It is a direct probe of the global T-frame phase manifold,
with the flux appearing only as a projection of the winding constraint.
What the SQUID “detects” is the global winding class.
Flux is just the S-frame bookkeeping shadow of that fact.
Why This Matters for What Comes Next
This single topological fact — that closed loops force distinct winding classes — is exactly what we will use in the next post.
Different winding classes can have different curvature in the T-frame, and thus different energy in the S-frame.
The curvature differences are small, but measurable.
This becomes the conceptual foundation of energy splitting, and eventually the Lamb shift.
The SQUID thus becomes a didactic mirror:
it reveals, macroscopically, the same structural mechanism that atoms exhibit microscopically.
DFT-28 will make this precise.
a superconducting state is not described by a spatial wave amplitude, but by a coherent T-frame phase winding over a compact manifold.
The S-frame electromagnetic observables are projections of this deep, global phase structure.
A single superconducting segment already exhibits this coherence; but when superconductors form a closed loop, something stronger becomes true:
the phase cannot be defined independently at each local point.
It is globally constrained.
The loop enforces a condition that any T-frame description of the superconducting state must be single-valued.
That is, although the phase may vary around the ring, after traversing the full circuit it must return to the same point in the T-frame.
The way standard superconductivity texts phrase this — “the phase difference must be a multiple of 2𝜋” — sounds like a rule imposed by hand.
In DFT it is automatic: it expresses the fact that the T-frame phase lives in a compact space and its S-frame projection must be globally consistent.
Let the T-frame phase along the SQUID loop be described by a smooth function
where 𝜆 is a loop parameter.
After moving around the loop of length 𝐿,
This does not require that the phase remain constant.
It only requires that the net phase change per loop equals an integer multiple of 2𝜋:
The integer 𝑛 labels the global T-frame winding class.
This is not an approximation; it is required by the topology of the scalar-motion phase manifold.
Local Phase vs Global Closure
Notice what has happened:
- Locally, nothing special occurs.
- No local operator enforces the phase condition.
- Nothing “propagates” causally around the loop.
What looks like an “instantaneous nonlocal constraint” in conventional spatial terms is, in DFT, simply the fact that the scalar-motion description supports global single-valuedness.
Connection to Magnetic Flux: Not a Local Interaction
We have not yet mentioned magnetic flux, because magnetic flux is the S-frame shadow of something deeper.
External magnetic flux influences the phase not by exerting force,
but by changing the accessible projection class.
The superconducting loop can only occupy winding classes consistent with the imposed curvature.
DFT formalizes this through the relation
but the meaning here is different from the standard interpretation.
The constant Φ0 is not “measured” or “derived,” but represents the winding–flux correspondence analogous to how Josephson relations connect phase and voltage.
In DFT:
- flux is not inherently quantized;
- what is quantized is winding;
- flux quantization is the S-frame projection of the deeper T-frame topological condition.
The Mixed Expression: T-frame Winding and S-frame Projection
To make the geometry explicit, the total winding condition in the presence of vector potential 𝐴𝑖 may be written as
The spatially measured flux is just
Thus, one sees directly that
- The loop does not measure flux.
- The loop does not probe the field.
- The loop does not mechanically “sense” anything.
The flux appears as the S-frame projection of that state.
Why a SQUID “Sees Flux Through Shielding”
This is one of the most misunderstood facts in standard superconductivity:
A SQUID still registers flux even when fully shielded spatially.
From the spatial viewpoint, this looks mysterious or “nonlocal.”
In DFT the situation is transparent:
- Shielding is an S-frame operation.
- The T-frame coherence is global.
- The winding constraint is topological, not geometric in space.
This is not “spooky action.”
It is coherence in the scalar-motion domain, which naturally enforces global phase closure.
The Conceptual Core
The SQUID is therefore not primarily a flux detector.
It is a direct probe of the global T-frame phase manifold,
with the flux appearing only as a projection of the winding constraint.
What the SQUID “detects” is the global winding class.
Flux is just the S-frame bookkeeping shadow of that fact.
Why This Matters for What Comes Next
This single topological fact — that closed loops force distinct winding classes — is exactly what we will use in the next post.
Different winding classes can have different curvature in the T-frame, and thus different energy in the S-frame.
The curvature differences are small, but measurable.
This becomes the conceptual foundation of energy splitting, and eventually the Lamb shift.
The SQUID thus becomes a didactic mirror:
it reveals, macroscopically, the same structural mechanism that atoms exhibit microscopically.
DFT-28 will make this precise.