Quaternionic conjugation yields three-dimensional rotations and identifies q with −q. A 2π spinor sign becomes measurable only relative to a coherent reference, and does not by itself establish particle identity or statistics.
Why composition needs a carrier
A source process needs a rule for composing its operations. SMM/DTT asks whether that rule can organize phase, orientation, and observable distinctions together. One useful mathematical class consists of finite-dimensional real unital associative algebras with a positive multiplicative quadratic norm. Positivity keeps nonzero operations from having zero norm; multiplicativity makes the norm compatible with composition.
Inside this class, two noncommuting circular phase units force a noncommutative carrier. The conclusion is conditional: the physical source must first justify the algebra class and the accessibility of those phases. A multiplication table selected for mathematical convenience would not establish why the universe must realize it.
Why the stated class leads to quaternions
Suppose N(xy)=N(x)N(y), with N(x)>0 for every nonzero x. If xy=0 and x≠0, positivity forces y=0. Left multiplication by x is therefore injective and, in finite dimension, surjective. The same argument applies on the right. Associativity makes the resulting left and right inverses agree.
The algebra is consequently a finite-dimensional associative real division algebra. Its classical classification leaves the real numbers, complex numbers, and quaternions. Requiring two noncommuting units squaring to −1 excludes the first two. This is the selection argument developed in article A093. Relaxing associativity or the positive norm changes the available alternatives, so these hypotheses remain visible parts of the result.
Constructing the third direction
Use quaternionic units e₁,e₂,e₃ with e₁²=e₂²=e₃²=−1 and e₁e₂=e₃=−e₂e₁. The first two units need not initially be orthogonal. Given unit imaginary directions u,v with inner product η and |η|<1, orthogonalize them by:
The imaginary part then has three orthogonal real directions. A detector insensitive to the scalar part can read this sector, but it must have sufficient controls to distinguish all three components. As |η| approaches one, the input phases nearly commute and reconstruction becomes ill-conditioned. Exact rank is therefore different from stable experimental access.
Half angles and the rotation double cover
Let n be a unit imaginary quaternion and q=cos(θ/2)+n sin(θ/2). For an imaginary vector v, define v′=qvq−1. Quaternion multiplication shows that components parallel to n remain fixed while perpendicular components rotate through angle θ. This operation preserves the imaginary-sector norm.
At θ=2π the quaternion is −1; at θ=4π it is +1. Both ±1 act identically by conjugation on ordinary vectors. Thus unit quaternions cover the three-dimensional rotation group twice. The half angle follows from acting on both sides of v; it is not obtained by declaring that a temporal loop has a particular particle label.
A complex doublet carries the sign
Choosing one circular phase slice identifies the quaternionic carrier with a two-component complex vector. In that spinor representation a rotation can be written U(θ)=exp[−iθ(n·σ)/2]. For a full turn U(2π)=−I, so every spinor changes sign. Yet its density matrix is unchanged: (−ψ)(−ψ)†=ψψ†.
This is the distinction between a vector and its ray. An isolated global sign does not change any ordinary single-state probability. It can matter when a rotation is applied to one branch of a coherent experiment while a reference branch is left unchanged. Then the sign is relative between branches, not global over the whole state.
A worked reference-arm example
Prepare a balanced path superposition (|a⟩+|b⟩)/√2 carrying the same spinor in both arms. Apply the full spin rotation only on arm b. The joint state becomes (|a⟩−|b⟩)ψ/√2. Recombining with the original balanced beam splitter changes which output port has constructive interference.
At zero angle the plus port is certain; at 2π it is dark; at 4π it is certain again. Additional uncontrolled path phases or entanglement with the apparatus alter this ideal law. The experiment reads coherence between two histories, precisely the kind of distinction a completion rule must preserve before final registration.
What remains to identify a particle
The construction joins composition, phase, spinor orientation, and a local rank-three readout. It does not establish electron charge, mass, a relativistic field equation, or fermionic exchange statistics. The spin-statistics relation requires further locality, positivity, and field-theoretic structure; a sign under rotation is not a replacement for those assumptions.
For SMM/DTT, the next productive task is to build a persistent excitation whose allowed source operations realize this representation and whose completed records retain the required relative phases. The nested inner levels would then have specific dynamical jobs. Their connection to spin would be demonstrated by the excitation and its measurement protocol, rather than inferred from counting turns alone.
Continue with the articles
- Why quaternions appear in the reconstruction
Introduce the positive norm, associativity, and incompatible-phase hypotheses that select a quaternionic carrier. Present the conclusion as conditional algebraic rigidity.
- The center-blind route to three dimensions
Explain how removing the scalar center from the quaternionic readout leaves three real directions. Clarify the operational meaning of a detector that cannot distinguish central changes.
- Hopf geometry from a spinor doublet
Introduce the relation between a two-component complex description and a sphere of readout directions. Explain the Hopf bundle’s role in the draft’s reconstruction.
- Retained observations versus predictive states
Ask when retained information is sufficient to predict future outputs autonomously. Explain the completion criterion and why keeping observations need not keep every dynamically relevant distinction.