DFT-19: The DFT Interpretation of Spin (Preliminary)

MWells · Post by **MWells** » Wed Dec 03, 2025 10:44 pm

Why 2-D T-frame rotation yields spin-½ behavior, why a 4π cycle is fundamental, and how quaternionic structure arises naturally in the time region.

Spin has always been one of the most conceptually opaque notions in quantum mechanics. In standard physics, spin is:

not actual rotation,
encoded by two-component complex objects (spinors),
changing sign under a 2π rotation,
described by SU(2), a double cover of SO(3),
and represented by Pauli matrices, equivalently, unit quaternions.

In Larson’s Reciprocal System (RS), and especially in Nehru’s later work, spin is instead:

a rotational structure of the time region,
one-dimensional rotation for bosons (integer spin),
two-dimensional rotation for fermions (spin-½),
and a fundamentally geometric.

Dual-Frame Theory (DFT) makes these ideas mathematically coherent by treating spin as a two-dimensional phase-winding structure in the T-frame that, when projected into the S-frame, produces the familiar SU(2)/4π phenomenology.

This post explains, at the framework level:

Why two-dimensional T-frame rotation is the geometric origin of spin-½

why two-dimensional T-frame rotation is the geometric origin of spin-½,
why a 4π cycle is required,
how quaternionic structure naturally appears,
why fermions have internal handedness (helicity-like sectors),
why photons and other bosons behave differently,
why spin appears “pointlike” in measurements,
and why fermionic wavefunctions need 2 and then 4 components.

and why fermionic wavefunctions need 2 and then 4 components.

It does not yet derive the Pauli matrices, the Dirac equation, or the exact value ℏ/2 for spin magnitude. Those require a further layer of work built on top of what follows.

1. Spin as a two-dimensional T-frame rotation

In DFT the T-frame coordinates are three compact angular variables:

$\Theta = (\theta^1,\ \theta^2,\ \theta^3)$

For a one-dimensional rotation, only one angle evolves:

$\dot{\theta}^k \neq 0, \qquad \dot{\theta}^{i \neq k} = 0.$

This yields a single phase circle 𝑆¹. This is the T-frame analog of RS’s one-dimensional rotation associated with integer-spin (boson-like) behavior.

A two-dimensional rotation involves simultaneous evolution in two angles:

$\dot{\theta}^1 \ne 0,\quad \dot{\theta}^2 \ne 0$

The raw configuration space for this internal motion is

$S^1 \times S^1$

topologically a torus. The internal state depends on the ordered pair $(\theta^1,\theta^2)$ , not just on a single phase.

In DFT, spin enters when we look at how this toroidal structure is seen by the S-frame. The projection from T-frame to S-frame does not distinguish all points of the torus separately; instead, it identifies configurations related by a simultaneous half-turn:

$(\theta^1,\theta^2) \sim (\theta^1 + \pi,\ \theta^2 + \pi).$

This identification is part of the projection rule: once an S-frame axis is chosen, configurations differing by this “diagonal” 𝜋-shift cannot be distinguished by the S-frame geometry or motion budget. In effect, the torus is seen through a double-sheeted lens with respect to that axis.

This is the structural reason that the S-frame does not see a simple vector representation of this internal rotation, but something spinor-like: the internal state lives on a double cover, while the S-frame only sees the quotient.

2. Why a 4π cycle is required

Given the identification

$(\theta^1,\theta^2) \sim (\theta^1 + \pi,\ \theta^2 + \pi),$

we can track what a “full rotation” does.

A 2π evolution of the internal motion moves us from (𝜃¹,𝜃²) ) to a point that is equivalent, under the identification, to a different sheet of the same S-frame state. In other words, the underlying T-frame configuration has changed sign in a spinor-like sense, even though the coarse S-frame embedding is the same.
A 4π evolution brings us back to the original T-frame configuration and the same S-frame sheet.

Schematically:
$\text{T-frame}: 4\pi\text{-periodic state} \quad\Rightarrow\quad \text{S-frame}: 2\pi\text{-periodic geometry with a sign ambiguity}.$

This is structurally isomorphic to the familiar SU(2) double cover of SO(3):

SU(2) elements parameterize internal spin states with 4π periodicity,
SO(3) elements describe spatial rotations with 2π periodicity,
the spinor picks up a minus sign under a 2π rotation and returns to itself only after 4π.

DFT does not postulate SU(2); it obtains a double-cover structure from two-dimensional T-frame rotation plus the projection-identification rule. Half-integer spin arises because a 2-D internal rotation cannot be represented in 3-D S-frame geometry by a single-valued vector field; a double cover is forced.

3. Quaternionic structure emerges from 2-D T-frame rotation

A two-dimensional T-frame rotation requires bothL

a magnitude of internal rotation, and
an orientation within the (𝜃¹,𝜃²) plane.

The S-frame cannot represent these two internal angles as a single phase. Instead, it naturally encodes them as a two-component complex object:

$\psi = \begin{pmatrix} \psi_1 \\[4pt] \psi_2 \end{pmatrix},$

the familiar spin-½ spinor.

Internally, the pair (𝜃¹,𝜃²) behaves like:

two orthogonal “imaginary directions” in T-frame,
with a non-commutative rule for composing finite rotations.

The minimal algebra that faithfully captures this structure is the quaternion algebra. A 2-component complex spinor has 4 real degrees of freedom, and there is a well-known one-to-one correspondence between:

unit quaternions,
SU(2) matrices, and
normalized 2-component complex spinors.

The quaternion relations

$i^2 = j^2 = k^2 = -1, \qquad ij = k,\ jk = i,\ ki = j$

provide exactly the algebra needed to represent compositions of these 2-D T-frame rotations. In this sense Nehru’s intuition is realized:

A two-dimensional rotation in the time region is naturally encoded by a quaternionic (SU(2)) structure. The spinor formalism is the S-frame’s way of packaging that structure.

In DFT this is not an arbitrary choice of algebra; it is dictated by:

the dimensionality of the T-frame rotation, and
the need for a double-cover representation under S-frame projection.

4. Internal chirality (helicity-like sectors) from T-frame orientation

Two-dimensional rotation also introduces an internal handedness. In T-frame, consider a spin state

$(\theta^1,\theta^2)$

Its orientation-reversed partner is

$(-\theta^1,-\theta^2)$

These two states:

live on different parts of the torus,
are not equivalent under the (𝜃¹,𝜃²)∼(𝜃¹+𝜋,𝜃²+𝜋) identification,
but project to the same S-frame magnitude of angular momentum.

Thus there are two disjoint chiral sectors:

“right-handed” 2-D T-frame rotation,
“left-handed” 2-D T-frame rotation,

which correspond, at the level of S-frame observables, to the two helicity eigenstates of a spin-½ particle (spin aligned or anti-aligned with a chosen direction of motion).

In this post we are not yet distinguishing in detail between helicity (spin·momentum) and chirality (in the Dirac sense); we are identifying the underlying T-frame origin of a two-sector internal structure that the S-frame sees as two possible “handed” spin states.

5. Why bosons behave differently

For a one-dimensional T-frame rotation, there is only a single phase angle

$\theta$

with the simple periodicity:

$\theta \equiv \theta + 2\pi$

There is:

no torus,
no double-sheeting,
no 4π ambiguity,
and no intrinsic two-sector chirality of the same kind.

This naturally corresponds to:

integer spin,
single-angle rotation,
ordinary vector or tensor representations of spatial rotations rather than spinors,
and symmetric exchange behavior (bosonic statistics at the level of representation theory).

DFT does not, in this post, derive the full spin-statistics theorem. What it does supply is the topological reason why RS one-dimensional rotation aligns with integer-spin, and why an extra internal rotational dimension (2-D T-frame) is required for half-integer spin.

6. Why fermions and photons appear pointlike

Nehru emphasized that rotational motions exist in the time region and therefore occupy rotational space, not ordinary spatial extension. DFT makes this almost tautological:

T-frame rotations live on a compact manifold (like a torus inside 𝑇³).
That manifold has no S-frame spatial extent on its own.
S-frame projection collapses entire loops of T-frame rotation into single points or localized distributions in space.

In other words:

Fermions and photons appear pointlike in ordinary space because their internal structure is rotational in T-frame, not extended in the S-frame.

The “size” of their internal rotation is invisible as spatial radius; it only appears through spin, statistics, and interaction patterns.

7. Why fermion wavefunctions require multiple components

We can now see why multi-component wavefunctions are unavoidable.

A scalar degree of freedom (no T-frame rotation) can be described by a single complex amplitude 𝜓(𝑥).
A one-dimensional T-frame rotation introduces an additional phase structure but still lives on a single 𝑆¹; its S-frame encoding can be treated as a scalar field with an internal phase (spin-0 or effectively spin-1 in RS terms).
A two-dimensional T-frame rotation has two independent internal angles. Compressing that structure into the S-frame requires at least a 2-component complex object:
$\psi(x) = \begin{pmatrix} \psi_1(x) \\ \psi_2(x) \end{pmatrix},$
a Pauli spinor, carrying 4 real degrees of freedom, in one-to-one correspondence with a unit quaternion state.

When we additionally require:

compatibility with Lorentzian S-frame structure (DFT-15), and
separate branches for “particle” and “antiparticle” sectors,

this internal spinor structure must be promoted to a 4-component object, as in the Dirac spinor. From the DFT viewpoint:

Schrödinger/Pauli 2-component spinors and Dirac 4-component spinors are not arbitrary algebraic inventions; they reflect the dimensionality of T-frame rotation plus S-frame relativistic and particle/antiparticle structure.

This post does not yet derive the explicit Dirac equation; it identifies why the number of components is what it is.

8. Spin measurement as projection of T-frame axes onto S-frame axes

A spin measurement along some S-frame axis 𝑛⃗ forces a particular alignment between:

an S-frame spatial direction, and
the internal 2-D T-frame rotational structure.

Operationally, the measurement selects an effective T-frame phase combination

$\theta^{(\text{obs})} = n^1 \theta^1 + n^2 \theta^2 + n^3 \theta^3,$

where the 𝑛^𝑖 encode how the chosen spatial axis couples to the three T-frame directions.

Because of the double-sheet structure discussed earlier, there are generically two inequivalent T-frame branches that:

project to the same S-frame measurement axis,
but differ by an overall sign in their internal phase structure.

These are the two spin-½ eigenstates along axis 𝑛⃗: “spin up” and “spin down”.

In the usual units of quantum mechanics, the corresponding eigenvalues are ±ℏ/2. In DFT, these discrete values arise structurally from:

the double-cover nature of 2-D T-frame rotation, and
the requirement that the motion budget couples T-frame rotation to S-frame angular momentum in a fixed proportion.

A full derivation of the numerical scale ℏ/2 requires tying the motion budget constants (𝜅, the NRS norm, etc.) to empirical units. That calibration step is beyond this preliminary post; here we are isolating the discrete two-branch structure that makes a two-eigenvalue spectrum inevitable.

9. Summary and what remains to be done

DFT reproduces the core phenomenology of spin from geometric first principles:

1. Spin-½ from 2-D T-frame rotation.

Torus topology plus the projection identification (𝜃¹,𝜃²)∼(𝜃¹+𝜋,𝜃²+𝜋) forces a double cover: 4π cycles in T-frame appear as 2π rotations with a sign flip in the S-frame.

2. SU(2) and quaternions from internal dimensionality.

Encoding two internal T-frame angles in a way compatible with spatial rotations leads naturally to SU(2) / quaternion structure and 2-component complex spinors.

3. Internal handedness from T-frame orientation.

(θ¹,θ²) and (−𝜃¹,−𝜃²) form two disjoint chiral sectors that map to “spin up” and “spin down” helicity-like states.

4. Bosons vs fermions from 1-D vs 2-D rotation.

One-dimensional T-frame rotation yields integer spin with no double cover; two-dimensional rotation yields half-integer spin with a 4π cycle.

5. Pointlike appearance from compact rotational space.

T-frame rotations live on compact internal manifolds, not extended spatial volumes. Projection collapses this structure to S-frame points, explaining why spinful particles appear pointlike.

6. Multi-component wavefunctions from T-frame degrees of freedom.

Two T-frame angles require at least a 2-component complex spinor; adding relativistic and particle/antiparticle structure leads naturally to a 4-component Dirac-like object.

What this post does not yet do—but clearly points toward—is:

an explicit derivation of the Pauli matrices and their commutation relations from T-frame geometry,
a derivation of the Dirac equation and γ-matrix algebra from dual-frame structure,
a calculation of the spin magnitude ℏ/2 from motion-budget constants,
and quantitative predictions for Stern–Gerlach deflections, g-factors, and EPR/Bell spin correlations.

Those tasks are the natural next layer on top of the DFT spin framework developed here. DFT-19 should therefore be read as a preliminary geometric explanation of spin’s strange features—not yet as a full replacement for the Pauli–Dirac formalism, but as the scaffolding that such a derivation can attach to.