The Quaternion Sphere:
Scattering, Spin, and the Electron from Algebra Alone

Scattering, Spin, and the Electron from Algebra Alone

Martin Scholl — Independent Researcher  ·  It Is All One  ·  July 2026 (working draft)

Independent Researcher May 2026

We begin with the four-dimensional algebra of quaternions and, without invoking any prior physical theory, derive measurable quantities: the angular distribution of charged-particle scattering, the magnetic moment of the electron, and the threshold energy for matter–antimatter pair creation. Every result follows from the geometry of the unit quaternion sphere and the arithmetic of octonion multiplication. We introduce each mathematical object as it is needed and define it in plain language. The final section compares every derived number with laboratory measurements. Keywords: quaternion algebra, unit 3-sphere, scattering cross-section, gyromagnetic ratio, pair production, octonion, division algebra

1The Space We Work In

A quaternion is a number with four parts:

(1)q = w + xi + yj + zk

where w, x, y, and z are ordinary real numbers, and i, j, k are three independent square roots of minus one. They satisfy Hamilton’s rules:

(2)i² = j² = k² = ijk = –1

from which it follows that ij = k, jk = i, ki = j, and—crucially—ji = –k, kj = –i, ik = –j. The order of multiplication matters. This non-commutativity is not a nuisance; it is the engine that drives everything in this paper. The size (or norm) of a quaternion is the four-dimensional analogue of the length of an arrow:

(3)|q| = √(w² + x² + y² + z²)

A unit quaternion has |q| = 1. The set of all unit quaternions forms a surface in four-dimensional space, just as the set of all points at unit distance from the origin in three-dimensional space forms an ordinary sphere. We call this surface S3 — the 3-sphere. It is a three-dimensional surface curved through four dimensions, exactly as the familiar 2-sphere (the surface of a ball) is a two-dimensional surface curved through three dimensions. S3 is the stage on which all the physics in this paper takes place. Any unit quaternion can be written as a rotation:

(4)q = cos(θ/2) + sin(θ/2)(n₁i + n₂j + n₃k)

where θ is an angle and (n1, n2, n3) is a unit direction vector. Notice the half-angle θ/2. A rotation of 180° in ordinary space requires only 90° of quaternion phase. A full 360° turn requires 180° of quaternion phase. This 2-to-1 relationship between physical rotation and quaternion phase—the double cover—is the single most consequential fact in this paper.

2Distance on the Sphere

Given two unit quaternions, qA and qB, there are two natural ways to measure how far apart they are on S3. The geodesic distance is the shortest path along the surface of the sphere, the analogue of a great-circle route on Earth:

(5)d(qₐ, qₑ) = arccos(Re(qₐ* · qₑ)) = θ/2

where qA* denotes the conjugate of qA (flip the sign of the i, j, k parts) and Re extracts the real part. The result is half the physical scattering angle. (One honesty note: this particular half-angle is the geometry of the chord — the same isosceles-triangle arithmetic that gives a momentum transfer 2k·sin(θ/2) in any textbook. The spinor double cover of Section 1 is a second, independent appearance of half-angles in the algebra; the two should not be conflated, and this paper’s scattering result stands on the chord alone.) The chord is the straight-line distance through the interior of the sphere, like a tunnel drilled through the Earth instead of a road over its surface:

(6)chord = |qₑ – qₐ| = 2 sin(θ/2)

This is the Euclidean distance in the embedding four-dimensional space. When the two points are close together (small θ), the chord and the geodesic are nearly the same; when they are far apart (θ approaching 180°), the chord reaches its maximum value of 2 while the geodesic reaches π.

3How Influence Propagates on the Sphere

When a disturbance originates at one point on a surface and spreads outward, the strength of its influence at a distant point is governed by the Green’s function of that surface. The Green’s function answers a simple question: if I place a unit source at point A, how strong is the signal at point B? On a flat three-dimensional space, the response of a unit source weakens as 1/r, where r is the distance from the source; in a flat four-dimensional space, as 1/r². The general rule for the Laplace Green’s function in d flat dimensions is 1/r^(d−2): two powers less than the dimension of the space. S3 is a three-dimensional surface embedded in flat four-dimensional space, and the natural distance between two of its points, seen from the embedding space, is the chord. Evaluating the four-dimensional Green’s function on chords — this is exactly Fock’s 1935 construction for the Coulomb problem [4] — gives a response falling as 1/(chord)2:

(7)G(qₐ, qₑ) ∝ 1 / chord² = 1 / (4 sin²(θ/2))

4Scattering on the Quaternion Sphere

Now we do physics. A particle moves through space. It encounters another particle and deflects by an angle θ. In our framework, this deflection is a rotation in quaternion space: the particle’s state moves from one point on S³ to another. The angular separation on S² maps to a geodesic distance on S³ of θ/2. The transition amplitude—the quantum-mechanical quantity whose square gives the probability of scattering into a given angle—is the Green’s function evaluated between the initial and final quaternion states. This is because the Green’s function is, by definition, the response of the space to a point source; and the Coulomb potential of a charged nucleus, mapped onto S3, is exactly a point source. (The 1/r singularity of the Coulomb force in flat space becomes a smooth delta function on S3; the singularity was never real—it was an artifact of projecting curved quaternion space into flat coordinates.) The transition amplitude is therefore:

(8)f(θ) ∝ 1 / sin²(θ/2)

The differential cross-section—the measurable quantity, with units of area per solid angle—is the square of the amplitude. Writing the proportionality constant explicitly:

(9)dσ/dΩ = (Z₁Z₂e² / 4E)² × 1/sin⁴(θ/2)

where Z1 and Z2 are the electric charges of the two particles (in multiples of the proton charge e), E is the kinetic energy of the projectile, and all other factors come from matching dimensions. The entire angular dependence—the 1/sin4(θ/2)—is the metric of quaternion space. There is nothing else to compute.

5Numbers: Charged-Particle Scattering

Consider a helium-4 nucleus (charge Z1 = 2) with kinetic energy 7.7 MeV approaching a gold nucleus (charge Z2 = 79). The prefactor in Eq. 9 evaluates to:

(10)a/4 = Z₁Z₂e² / (4E) = 7.39 fm

where 1 fm = 10–15 m (roughly the size of a proton). The cross-section at each angle:

Table 1. Predicted cross-sections from Eq. 9 for helium nuclei on gold at 7.7 MeV. One barn = 10–²⁸ m². The cross-section spans four orders of magnitude: nearly all particles pass through at small angles; very few bounce straight back. This is a direct consequence of the chord distance on S³ being small at small deflection angles and reaching its maximum of 2 only at θ = 180°.

6The Magnetic Moment of the Electron

We now derive a second measurable quantity from the quaternion algebra, without reference to any external theory. Recall from Eq. 2 that the quaternion basis elements do not commute: ij = k but ji = –k. Subtracting:

(11)ij – ji = 2k

(12)[i, j] = 2k, [j, k] = 2i, [k, i] = 2j

Now consider a spinning particle in a magnetic field. The particle’s quantum state is described by a unit quaternion. The magnetic field couples to the quaternion phase—it drives the quaternion to rotate around the field axis. Two kinds of angular momentum respond: Orbital angular momentum: a particle physically circling in space. Its wavefunction phase advances at the same rate as its physical position. A full physical orbit (360°) corresponds to a full phase cycle (360°). The coupling between angular momentum and magnetic field gives a precession rate ωL = eB/(2m), known as the Larmor frequency. Spin angular momentum: the intrinsic rotation of the quaternion itself. Because of the double cover (Section 1), 360° of physical rotation requires only 180° of quaternion phase. The magnetic field drives the quaternion phase at the Larmor rate, the same as for orbital motion—the electromagnetic coupling does not know or care whether the angular momentum is orbital or intrinsic. But each degree of quaternion phase now produces two degrees of physical rotation. The spin therefore precesses at twice the Larmor frequency. This factor of 2 is measured as the gyromagnetic ratio g:

(13)g = ω_spin / ω_Larmor = 2

The same result emerges algebraically from the commutator (Eq. 12). The physical observable—the coupling between spin and the electromagnetic field—passes through the quaternion commutator [σa, σb] = 2iσc. The coefficient of this commutator is 2. That is the electron’s g-factor. It is not derived from a wave equation, from matrix algebra, or from quantum field theory. It is read off the multiplication table of quaternions. Numerically:

g = 2.000 000 000 00... (quaternion prediction)

g = 2.002 319 304 36... (laboratory measurement)

7Pair Production: Creating Matter from Geometry

An octonion is a number with eight parts—one real and seven imaginary:

(14)o = a₀ + a₁e₁ + a₂e₂ + a₃e₃ + a₄e₄ + a₅e₅ + a₆e₆ + a₇e₇

An octonion contains two quaternions: the first four components form one quaternion (the left half), and the last four form another (the right half). We can write O = H ⊕ H’, where H denotes a quaternion. A caution before the physics, because the series’ architecture has since been settled (Papers 4–6): the octonion’s two quaternion blocks are not particle and antiparticle — they are the spatial and colour universes of a single quark, and an electron’s colour block is zero. The correct algebra of pair creation is conjugation, not splitting: the vacuum can only add a closed loop, and a loop pinched into two open ends yields a state Q and its conjugate −Q — opposite winding on every axis (Paper 6, Section 4). The particle and the antiparticle are a quaternion and its mirror, twins born of one loop. This splitting requires energy. The minimum energy is the rest-mass energy of the two particles created:

(15)E_threshold = 2mc²

The factor of 2 is simply that twins come in twos: each of the pair costs its rest energy mc². In ladder language this threshold is a door — the price of nucleating the first winding loop at the pair rung (Paper 6, Section 4). For an electron–positron pair, m = me = 9.109 × 10–31 kg:

(16)E_threshold = 2 × 0.511 MeV = 1.022 MeV

A photon with energy above 1.022 MeV, passing near a nucleus (which provides the momentum balance via its quaternion field), splits into an electron and a positron. This has been observed in cloud chambers since 1933. The tracks appear as a matched pair—one curving left, the other right in the magnetic field—emerging from a single point. Each track is one of the conjugate twins — a winding and its anti-winding, keeping every counter of the ledger at zero.

8The Massless Particle: Quaternion Holomorphy

A particle with zero mass (like a photon or a neutrino at high energy) has a particularly elegant description. Define the quaternion derivative:

(17)∇_q = ∂_t + i∂_x + j∂_y + k∂_z

This is the four-dimensional analogue of the ordinary derivative d/dx, extended to quaternion space. Each of the four directions in spacetime gets one quaternion slot. The conjugate derivative ∇*q flips the signs of the three imaginary parts. The equation of motion for a massless particle is simply:

(18)∇_q ψ = 0

This says that the wavefunction ψ is quaternion-holomorphic: it is a smooth, analytic function of the quaternion variable, in the same sense that an analytic function of a complex variable satisfies the Cauchy–Riemann equations. Just as complex analyticity implies f(z) = u + iv with ∂u/∂x = ∂v/∂y, quaternion holomorphy constrains the four components of ψ to satisfy four coupled equations—the quaternion Cauchy–Riemann conditions. For a particle with mass, two quaternion-valued components — a chirality doublet, in the form written by Lanczos in 1929 — couple to each other:

(19a)∇_q ψ_R = m ψ_L

(19b)∇*_q ψ_L = m ψ_R

where ψL and ψR are the left-handed and right-handed quaternion components of the doublet. (Both are citizens of the quaternion level — they are not the octonion’s two blocks, which are spatial and colour; Papers 4–6.) Mass is the coupling strength between the two hands. The derivative in Eq. 17 is the frame operator of this series’ field-equation note, where the same structure carries Maxwell and Einstein. When m = 0, each hand is independent and holomorphic. When m ≠ 0, the left hand cannot exist without the right — the doublet is whole. This is why massless particles (photons) have only one handedness at a time, while massive particles (electrons) always carry both.

9Handedness from Algebra

The split of the octonion into left and right quaternion halves defines a chirality: a handedness. The left half ψL is one handedness; the right half ψR is the other. This is not a metaphor. The quaternion multiplication rule ij = k fixes a specific orientation: i, j, k form a right-handed triple. Reversing the order gives ji = –k—a left-handed triple. The algebra itself has a built-in handedness. This has a physical consequence. The weak nuclear force—the force responsible for radioactive decay—acts only on left-handed particles. This asymmetry, known as parity violation, shocked physics when it was measured in 1957. In our framework it is not surprising, and the responsible algebra is the quaternion itself: ij = k fixes an orientation, and the weak interaction — the curvature of the tightly wound quaternionic space (companion paper, Matter Meets Space) — inherits that orientation. Parity violation is a property of the number system, at the quaternion level; the Postulates paper lists it among P1’s theorems.

10Beyond S³: The Octonion Sphere and Strong-Force Scattering

The unit quaternions form S3, a 3-sphere in 4 dimensions. The unit octonions form S7, a 7-sphere in 8 dimensions. The Green’s function on S7 follows the same dimensional rule: it weakens as 1/(chord)6, because S7 is a 7-dimensional surface in 8-dimensional space and the Green’s function drops as one power less than the dimension of the embedding space.

(20)G₇(qₐ, qₑ) ∝ 1 / chord⁶ = 1 / (2 sin(θ/2))⁶

The scattering cross-section on S7 is |G7|2:

(21)dσ/dΩ ∝ 1 / sin¹²(θ/2)

This is dramatically steeper than the electromagnetic 1/sin4(θ/2). At θ = 10°, the S7 cross-section is roughly 108 times larger than the S3 cross-section at the same angle (relative to their values at 180°). Particles governed by the octonion geometry scatter far more strongly at small angles and far more weakly at large angles.

11Comparison with Experiment

We now list every measurable quantity derived in this paper and compare it with laboratory values. Table 2. Summary of predictions. Every numerical result follows from the quaternion and octonion algebra without adjustable parameters.

12What We Did Not Assume

It is worth listing explicitly what was not used:

The scattering formula, the g-factor, the pair-production threshold, and the parity violation all follow from two facts: the algebra of quaternions (Eq. 2) and the geometry of the unit sphere in four dimensions (Eq. 6). Everything else is arithmetic.

13Notes on Related Work

The angular distribution of charged-particle scattering was first measured by H. Geiger and E. Marsden in 1913 [1] and explained by E. Rutherford in 1911 [2] using classical hyperbolic orbits. The quantum-mechanical derivation via the Born approximation gives the same result [3]. The connection between the Coulomb problem and the 3-sphere was established by V. A. Fock in 1935 [4]. The electron gyromagnetic ratio g = 2 was first derived by P. A. M. Dirac in 1928 [5] from his relativistic wave equation. The anomalous magnetic moment was computed by J. Schwinger in 1948 [6]. Pair production in cloud chambers was first observed by P. M. S. Blackett and G. P. S. Occhialini in 1933 [7]. Parity violation in the weak interaction was demonstrated by C. S. Wu et al. in 1957 [8].

References

[1] H. Geiger and E. Marsden, “The laws of deflexion of α particles through large angles,” Philosophical Magazine, vol. 25, no. 148, pp. 604–623, 1913. [2] E. Rutherford, “The scattering of α and β particles by matter and the structure of the atom,” Philosophical Magazine, vol. 21, no. 125, pp. 669–688, 1911. [3] M. Born, “Quantenmechanik der Stoßvorgänge,” Zeitschrift für Physik, vol. 38, no. 11–12, pp. 803–827, 1926. [4] V. A. Fock, “Zur Theorie des Wasserstoffatoms,” Zeitschrift für Physik, vol. 98, no. 3–4, pp. 145–154, 1935. [5] P. A. M. Dirac, “The quantum theory of the electron,” Proceedings of the Royal Society A, vol. 117, no. 778, pp. 610–624, 1928. [6] J. Schwinger, “On quantum-electrodynamics and the magnetic moment of the electron,” Physical Review, vol. 73, no. 4, pp. 416–417, 1948. [7] P. M. S. Blackett and G. P. S. Occhialini, “Some photographs of the tracks of penetrating radiation,” Proceedings of the Royal Society A, vol. 139, no. 839, pp. 699–726, 1933. [8] C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, “Experimental test of parity conservation in beta decay,” Physical Review, vol. 105, no. 4, pp. 1413–1415, 1957.

Symbols & Terms