DFT-20: Fine Structure: A Dual-Frame Interpretation

MWells · Post by **MWells** » Wed Dec 03, 2025 11:08 pm

How the three standard fine-structure corrections (relativistic, spin–orbit, and Darwin) arise naturally from DFT projection geometry — without claiming derivation of numerical coefficients.

In conventional quantum mechanics, fine structure in the hydrogen spectrum is described as a set of corrections to the nonrelativistic Hamiltonian:

$H_{\text{FS}} = H_{\text{rel}} + H_{\text{SO}} + H_{\text{Darwin}}$

where

$H_rel$ is the relativistic kinetic correction,
$H_SO$ is the spin–orbit interaction,
$H_Darwin$ is the Darwin contact term.

These terms are usually obtained by starting from the Dirac equation and expanding to obtain an effective Pauli Hamiltonian.

Dual-Frame Theory (DFT) looks at the same structure from a different angle. In DFT, the three fine-structure contributions are not arbitrary “patches” on a nonrelativistic model. They are the S-frame shadows of three invariant projection effects:

the tradeoff between S-frame translation and T-frame rotation (motion budget),
the alignment mismatch between the S-frame orbital axis and the T-frame spin axis,
unresolved T-frame structure near centers of inward scalar rotation.

The claim here is structural, not numerical:

The familiar fine-structure operators arise in exactly the functional forms they do because S-frame observables must encode these three dual-frame geometric effects in the only ways available to them.

This post does not claim to reproduce the standard numerical coefficients or the exact hydrogen splittings. Instead, it shows how those three correction types are the natural S-frame encodings of dual-frame projection constraints.

1. Standard Fine-Structure Terms (QM Reference)

For orientation, here are the conventional QM expressions in schematic form (in units where the speed of light 𝑐 is explicit but other constants may be suppressed):

1.1 Relativistic kinetic correction

$H_{\text{rel}}^{\text{QM}} = -\frac{p^4}{8 m^3 c^2} \quad\text{(schematic form)}.$

1.2 Spin–orbit coupling

$H_{\text{SO}}^{\text{QM}} \propto \frac{1}{r}\frac{dV}{dr}\,\mathbf{L}\cdot\mathbf{S},$
with the familiar Coulomb potential 𝑉(𝑟)∝1/𝑟.

1.3 Darwin term

$H_{\text{Darwin}}^{\text{QM}} \propto \nabla^2 V(r).$

These will be treated as targets for structural interpretation: we want to see why corrections of exactly these functional forms are forced when we look at hydrogenic motion through the dual-frame projection.

2. DFT Overview: S/T Complementarity and the motion budget

DFT begins with a scalar trajectory in the Natural Reference System:

$\sigma(\lambda) \in \mathbb{R}^3_{\text{NRS}}$

with fixed progression magnitude

$\left\|\frac{d\sigma}{d\lambda}\right\| = \kappa.$

This single process is seen through two complementary projections:

S-frame (spacetime-like):
$F_S : \sigma(\lambda) \mapsto x^\mu(\lambda), \qquad \mu = 0,1,2,3,$

T-frame (phase/rotational):

$F_T : \sigma(\lambda) \mapsto \theta^i(\lambda), \qquad i = 1,2,3,$

with

$\Theta = (\theta^1,\theta^2,\theta^3) \in T^3 = S^1 \times S^1 \times S^1.$

The projections share a fixed motion budget:

$\mathcal{B}_S + \mathcal{B}_T = \mathcal{B}_{\text{total}}.$

Conceptually:

higher S-frame translational “speed” takes more of $\mathcal{B}_S$ leaving less for T-frame rotational structure $\mathcal{B}_T$
S-frame measurements always select some effective T-frame axis or combination, suppressing other internal directions;
where the projection map 𝐹_𝑆 becomes many-to-one, some T-frame microstructure is unresolved and must show up as a coarse-grained correction.

These three features are the dual-frame counterparts of:

relativistic kinetic corrections,
spin–orbit coupling,
Darwin contact corrections.

3. Relativistic Correction: S-Frame Velocity Consumes Budget
3.1 S-frame effective velocity and Lorentz factor

In the S-frame, the spatial velocity emerging from projection is

$v^2 = \frac{dx^i}{d\lambda}\frac{dx_i}{d\lambda}$

Because the S-frame interval must be compatible with the NRS scalar distance (DFT-15), there exists an effective Lorentz-like factor γ_eff relating S-frame time, space, and scalar progression:

$\gamma_{\text{eff}}^{-2} = 1 - \frac{v^2}{c^2},$

with appropriate choice of units and scaling constants absorbed into 𝜅 and 𝑐. This factor is not introduced as a separate postulate; it is the normalization required to keep

$\Delta s^2 \propto d_{\text{NRS}}^2$

consistent across S-frame observers.

3.2 Budget tradeoff and energy shift

As the electron moves into tighter Coulomb orbits, its S-frame velocity 𝑣 increases, so the translational share of the motion budget

$\mathcal{B}_S \propto \gamma_{\text{eff}} v^2$
rises.

Since

$\mathcal{B}_S + \mathcal{B}_T = \mathcal{B}_{\text{total}}$

is fixed, this increased translational cost reduces 𝐵_𝑇, i.e., the capacity for T-frame rotational/vibrational structure. The lost T-frame contribution must reappear in the S-frame as a modification to the kinetic energy–frequency relation.

Expanding the effective energy in powers of 𝑝 or 𝑣 around the nonrelativistic limit naturally yields a correction term proportional to 𝑝⁴. At a purely structural level:

leading kinetic energy is 𝑇∼𝑝²/(2𝑚),
relativistic correction is of order 𝑇²∼𝑝⁴T²∼p⁴,

which matches the functional form of the standard term

$H_{\text{rel}}^{\text{QM}} = -\frac{p^4}{8 m^3 c^2}.$

DFT does not yet derive the specific coefficient −1/(8𝑚³𝑐²); it explains why a 𝑝⁴-type correction must appear once the motion budget is enforced at the dual-frame level.

3.3 DFT interpretation

From the DFT viewpoint:

The relativistic fine-structure correction is the S-frame imprint of the fact that high orbital velocities consume more of the shared motion budget, reducing the T-frame’s rotational contribution and shifting energy levels accordingly.

No additional field or mediator is required; the effect is purely a consequence of how scalar progression is split between translation and rotation.

4. Spin–orbit coupling: S-frame orbital axis vs T-frame spin axis

DFT-19 interpreted spin as a two-dimensional T-frame rotation:

$\Theta_{\text{spin}} = (\theta^1,\theta^2)$

with an associated internal direction 𝑠^𝑖 in T-frame space. Independently, the orbital motion defines a preferred S-frame spatial direction 𝑛^𝑖 (e.g., the normal to the orbital plane or the direction of orbital angular momentum 𝐿).

In dual-frame language, we can write the effective T-frame combinations associated with spin and orbit as:

$\theta^{(\text{spin})} = s^1 \theta^1 + s^2 \theta^2 + s^3 \theta^3,$

$\theta^{(\text{orb})} = n^1 \theta^1 + n^2 \theta^2 + n^3 \theta^3.$

4.1 Axis selection and alignment

A given S-frame measurement axis cannot resolve all three T-frame angles independently. Instead, it selects a particular combination that aligns some S-frame direction with some effective T-frame direction. For orbital motion:

the S-frame orbital geometry selects 𝑛^𝑖,
the T-frame spin structure selects 𝑠^𝑖.

If these directions are misaligned, then maintaining consistency of the motion budget requires a coupling between the spin and orbital structures. The scalar that measures their degree of alignment is

$n^i s_i.$

In S-frame language this appears, after the usual identifications, as an 𝐿⋅𝑆-type coupling:

$H_{\text{SO}}^{\text{DFT}} \sim f(r)\,\mathbf{L}\cdot\mathbf{S},$

where the radial function 𝑓(𝑟) reflects how scalar progression and inward rotation are distributed as a function of distance from the center.

In the Coulombic case, DFT inherits the same radial potential 𝑉(𝑟)∝1/𝑟 from the underlying scalar rotation, so the only rotationally invariant way to build such a coupling in the S-frame is through something of the schematic form

$f(r) \propto \frac{1}{r}\frac{dV}{dr},$

which is precisely the structure of the standard spin–orbit term.

4.2 DFT interpretation

From the dual-frame viewpoint:

Spin–orbit coupling is the energy cost of enforcing projection consistency between the S-frame orbital axis 𝑛^𝑖 and the T-frame spin axis 𝑠^𝑖 under a fixed motion budget. The resulting scalar alignment 𝑛⋅𝑠 appears as the familiar 𝐿⋅𝑆 structure in the S-frame Hamiltonian.

DFT does not, at this stage, derive the precise prefactor or the Thomas-precession factor 1/2; it explains why an 𝐿⋅𝑆-type structure with a Coulomb-like radial factor is inevitable given the dual-frame geometry.

5. Darwin Term: Unresolved T-Frame Structure Near the Center of Inward Scalar Rotation

In the Dirac-to-Pauli reduction, the Darwin term is often motivated as an effective correction arising from short-scale “zitterbewegung” of the electron near a pointlike Coulomb center, leading to a contact-like term proportional to ∇2𝑉(𝑟).

In DFT, an analogous structure appears for a different reason:

the center of an RS/DFT inward scalar rotation is a point of maximal inward displacement,
near this center, the projection 𝐹_𝑆 becomes highly many-to-one: many distinct T-frame histories collapse to the same S-frame location.

Let $x$ be an S-frame point very close to the inward-rotation center. There exists a nontrivial set $\{\Theta_a\}$ of T-frame phase states satisfying:

$F_S(\sigma_a(\lambda)) = x$
for different 𝜎_𝑎 As 𝑟→0:

the number (or measure) of such compatible T-frame configurations grows,
the differential of the projection 𝑑𝐹_𝑆tends toward lower rank,

$\operatorname{rank}(dF_S) \to 0,$

and the map becomes increasingly non-invertible.

In this regime, the S-frame effectively loses position resolution: what looks like a single point in S-frame corresponds to a spread of T-frame microstates. Integrating out this unresolved structure produces a local correction in the effective S-frame dynamics.

5.1 Why the Laplacian appears

In a spherically symmetric situation with potential 𝑉(𝑟), the only rotationally invariant, local, second-order scalar operator available is the Laplacian of the potential:

$\nabla^2 V(r).$

DFT’s claim is that the Darwin-type correction is exactly the S-frame encoding of this local loss of invertibility of 𝐹𝑆 near the inward-rotation center. Once we require:

locality in the S-frame,
rotational symmetry around the center,
a second-order correction in derivatives,

the functional form ∝∇2𝑉(𝑟) is the unique candidate. This matches the standard Darwin term structure.

Note that in conventional hydrogenic QM, this term affects only 𝑠-states, because only they have nonzero wavefunction at the origin. In a full DFT implementation, that selectivity would arise from how the S-frame wavefunctions of different rotational classes sample the region in which 𝐹_𝑆 is most non-invertible. That level of detail is beyond this preliminary post, but the structural correspondence is clear.

5.2 DFT interpretation

From the DFT perspective:

The Darwin term is a correction that encodes the fact that near the center of inward scalar rotation, the S-frame projection collapses many T-frame phase configurations into a single spatial point. Integrating out this unresolved T-frame structure yields a local S-frame correction with the same ∇2𝑉(𝑟) functional form as the QM Darwin term.

This has nothing to do, at root, with a literal pointlike proton; it is a statement about the degeneracy of T-frame structure under S-frame projection at the rotational center.

6. Summary: how the three fine-structure terms arise in DFT
In DFT, the three standard fine-structure corrections correspond one-to-one with three geometric features of dual-frame projection:

Relativistic correction
- As orbital velocity 𝑣 increases, the translational share 𝐵_𝑆 of the motion budget grows.
- The rotational share 𝐵_𝑇 shrinks, shifting the energy–frequency relation.
- Expanding this budget-constrained energetics yields a 𝑝⁴-type correction, structurally matching the standard relativistic fine-structure term.
Spin–orbit coupling
- Orbital motion selects an S-frame direction 𝑛^𝑖; spin is a 2-D T-frame rotation with internal direction 𝑠^𝑖.
- Projection must enforce consistency between these axes under a fixed budget.
- The alignment 𝑛⋅𝑠 appears in S-frame as an 𝐿⋅𝑆-type coupling with a Coulomb-derived radial factor.
Darwin term
- Near the center of inward scalar rotation, many T-frame phase histories collapse to the same S-frame point; 𝐹_𝑆 loses local invertibility.
- Integrating out the unresolved internal structure yields a local, rotationally invariant second-order correction.
- The unique such scalar is ∇2𝑉(𝑟), matching the Darwin term’s functional form.

Throughout this post, the emphasis has been on structural inevitability, not numerical reproduction:

DFT explains why corrections of type 𝑝⁴, 𝐿⋅𝑆, and ∇²𝑉(𝑟) must appear in any dual-frame description of hydrogenic motion.
It does not yet derive the precise coefficients (involving 𝑚, 𝑐, ℏ, and 𝛼) or the exact experimental splittings.

Those quantitative tasks require:

tying the motion-budget scale 𝜅 and projection constants to empirical units,
deriving the Dirac-like dynamics in explicit dual-frame operator form,
and computing level shifts numerically for comparison with data.

Within those limits, DFT-20 closes the conceptual loop that began with the original fine-structure document: the three textbook correction terms are no longer separate, loosely motivated pieces, but the S-frame manifestations of a single, coherent dual-frame geometry of scalar motion.