1 The Idea
▼About 150 years ago, physicists rewrote the equations of electromagnetism. In the process, they threw away a piece — a scalar quantity that their math produced automatically. They called it a "gauge choice" and set it to zero.
That piece was gravity.
This thesis shows that if you keep that discarded scalar, you get a unified description of electromagnetism and gravity. Both forces emerge from a single mathematical object — the electromagnetic four-potential, written as a quaternion — and a single operation: taking its derivative.
The strength of both forces turns out to be determined by one number: 4, the dimension of the quaternion algebra. From this single integer, the theory predicts the fine structure constant (which governs electromagnetism) and the gravitational constant (which governs gravity) with extraordinary precision.
2 One Object, One Operation
▼The object is the electromagnetic four-potential $\chi = (\phi, \mathbf{A})$. It's the same quantity every physicist already uses — the scalar potential $\phi$ (which makes electric fields) and the vector potential $\mathbf{A}$ (which makes magnetic fields) — just packaged as a quaternion instead of a four-vector.
The operation is the quaternion derivative $\nabla$. It's the familiar "take the rate of change" operator, but using the full quaternion multiplication rule instead of ordinary vector calculus.
The key difference: quaternion multiplication naturally produces both a scalar and a vector. Ordinary vector calculus only keeps the vector part. The scalar part is what got thrown away.
The quaternion derivative splits into two pieces:
Applying $\nabla$ a second time gives the dynamics: the full field equation $\nabla\nabla\chi = 0$. When you set $g = 0$, you recover Maxwell's equations exactly. When you keep $g$, you get electromagnetism plus a gradient force — which is exactly the mathematical form of gravity.
3 The Hidden Scalar
▼The quantity $g = \partial\phi/\partial t - \nabla\!\cdot\!\mathbf{A}$ is called the Lorenz gauge condition in standard physics. For 150 years, physicists have set it to zero — treating it as a meaningless mathematical freedom, like choosing which direction is "north."
But the quaternion algebra produces $g$ on completely equal footing with the electric and magnetic fields. It's not optional — it's automatic. Discarding it is a choice, not a necessity.
In the full quaternion field equation $\nabla\nabla\chi = 0$, the scalar $g$ couples to the electromagnetic field $\mathbf{V}$ bidirectionally: electromagnetic configurations generate $g$, and $g$ exerts a gradient force (just like gravity) back on the dynamics. Setting $g = 0$ severs this coupling entirely.
4 Why No Classical Gravity?
▼Here's the puzzle: if $g$ is a real field, why don't we see gravity in the classical equations?
The answer is surprisingly clean. When you work out the quantum mechanics of the quaternion field (technically, the "Dirac bracket" of the constrained system), the scalar $g$ has no propagator. Its response to a source is purely local — a contact interaction, like bumping into something, rather than a long-range force like throwing a ball.
This is actually a feature, not a bug. It explains why gravity wasn't discovered in the quaternion formalism 150 years ago: classical analysis alone can't see it. You need the non-perturbative quantum structure of the theory.
5 The Octonion Emerges
▼There's a beautiful pattern in mathematics called the Cayley–Dickson construction: you can "double" number systems by pairing two numbers into one bigger number. Click the steps below to see how:
The thesis takes the four-potential $\chi$ (a quaternion) and its field $\nabla\chi$ (another quaternion) and pairs them into an octonion:
$\mathcal{O} = [\chi, \; \nabla\chi]$
No new fields are introduced — the octonion emerges from the four-potential interacting with itself through the derivative. It has 7 imaginary directions: 3 for the vector potential $\mathbf{A}$, 1 for the gravitational scalar $g$, and 3 for the electromagnetic field $\mathbf{V}$.
6 Everything from d = 4
▼Starting from just one number — $d = 4$, the dimension of the quaternion algebra — a cascade of geometric quantities follows. Hover over each row to see what it means:
| Quantity | Formula | Value |
|---|---|---|
| Quaternion dimension The starting point: quaternions have 4 components | $d$ | 4 |
| Volume of $S^3$ The "surface area" of the 3-sphere that quaternions live on | $2\pi^2$ | 19.74 |
| Imaginary octonion directions Doubling quaternions gives $2d - 1 = 7$ imaginary directions | $2d - 1$ | 7 |
| $G_2$ generators The symmetry group of the octonions has 14 generators | $2(2d-1)$ | 14 |
| Active generators One generator is frozen by the Cayley–Dickson structure, leaving 13 | $4d - 3$ | 13 |
| Metric components A symmetric rank-2 tensor in 4 dimensions has 10 independent entries | $d(d+1)/2$ | 10 |
| Tunneling exponent Each metric component tunnels independently, each contributing $\alpha^2$ | $d(d+1)$ | 20 |
The gap equation determines the fine structure constant:
$\alpha(1 - \alpha)^2 = e^{-\pi^2/2}$
In words: the coupling constant equals a tunneling probability (the exponential) times a back-reaction factor. Solving this cubic equation gives $1/\alpha = 137.024$ — within 0.009% of the measured value.
The gravitational coupling then follows:
$\dfrac{G \, m_e^2}{k_e \, e^2} = \dfrac{\pi^2}{2} \cdot \dfrac{13}{49} \cdot \alpha^{20}$
Every number on the right comes from $d = 4$. The $\alpha^{20}$ factor is why gravity is so much weaker than electromagnetism: it takes 20 independent tunneling events to build a complete metric tensor.
7 The Predictions
▼The theory makes four testable predictions with no free parameters:
8 The Derivation Chain
▼The complete argument in five steps. Click any step to expand it:
9 The Standard Model Connection
▼The octonion's symmetry group is $G_2$, a 14-dimensional group. It contains a remarkable chain of subgroups:
$G_2 \;\supset\; SU(3) \;\supset\; SU(2) \times U(1)$
This is exactly the Standard Model gauge group:
- SU(3) — the strong force (quantum chromodynamics)
- SU(2) — the weak force
- U(1) — electromagnetism
The gravitational scalar $g$ picks out a preferred direction in the octonion, and the symmetry that stabilizes this direction is $SU(3)$. The Cayley–Dickson structure (the pairing of $\chi$ and $\nabla\chi$) provides the electroweak $SU(2) \times U(1)$ decomposition.
Particles as topological defects:
| Particle | What it is | Charge | Spin |
|---|---|---|---|
| Photon | Ripple in the EM field | 0 | 1 |
| Graviton | Tunneled excitation via $g$ | 0 | 2 |
| Electron | Winding-1 defect in $\mathbf{A}$ | $-e$ | ½ |
| Positron | Winding-(−1) defect in $\mathbf{A}$ | $+e$ | ½ |
| Neutrino | Defect in the $g$-direction | 0 | ½ |
| Quark | Defect in a color direction | fractional | ½ |
| Gluon | Ripple in color directions | 0 | 1 |
| Higgs | Fluctuation of $|\chi|$ | 0 | 0 |
Bosons are waves (ripples). Fermions are topological knots (defects that can't unwind). The topology of $S^3$ gives them spin-½ and conserved charge.
… Appendices
▼The thesis includes five appendices with detailed derivations. Click each to read a summary:
The leading-order gap equation gives $1/\alpha = 137.024$, which is off by 0.009%. This appendix computes the one-loop quantum correction from fluctuations around the $S^3$ instanton. Every ingredient — the fluctuation spectrum, the regularization, the zero-mode Jacobian — is determined by the geometry of $S^3$ and $d = 4$. The corrected value is $1/\alpha = 137.03597$, matching the measured value to 0.3 parts per billion. This is a factor of 30,000 improvement, with no free parameters introduced.
The gap equation contains a factor $(1-\alpha)^2$ that represents self-consistent back-reaction. This appendix derives it from two facts: (1) the effective metric is bilinear in the quaternion field (it has two "legs," each from a spacetime derivative), and (2) each leg independently draws its amplitude from the electromagnetic sector. Since the metric is rank-2, the available amplitude is $(1-\alpha)$ per leg, squared. The exponent 2 is exact — it's fixed by the rank of the tensor, not by a perturbative expansion.
The electron is identified as a degree-1 topological defect (a "knot") of the quaternion field on $S^3$. Its mass is set by a balance between kinetic energy (which favors spreading out) and the octonionic associator (which favors localization). Remarkably, the geometric prefactor cancels exactly due to self-consistency: the soliton lives on the same $S^3$ whose radius it determines. The result: $m_e = m_\text{Planck} \times \alpha^{21/2} \times$ (order-one factor), which predicts the electron mass to 0.10% accuracy.
All dimensionful constants ($G$, $m_e$, $k_e$, $e$, $\hbar$, $c$) cancel from the gravitational coupling formula, leaving the entire theory as just two equations in two unknowns: the gap equation for $\alpha$ and a formula for $m_e^2$ in Planck units. One input ($d = 4$), two outputs. The hierarchy between gravity and electromagnetism is simply $\alpha^{21/2} \approx 4 \times 10^{-23}$ — large, but not mysterious.
Every Standard Model particle maps to a specific configuration of the quaternion field. Small oscillations are bosons (photons, gluons). Topological defects — configurations with nonzero winding number that can't unwind continuously — are fermions (electrons, quarks, neutrinos). The winding direction in the 7-dimensional octonion determines the quantum numbers: electromagnetic, gravitational, or color. The Higgs is a fluctuation in the magnitude of $\chi$, and a black hole is where the tunneling between sectors saturates to $\alpha \to 1$.