Interactive Essay

Why a second order equation in Clifford algebra can be a serious alternative to the Dirac equation for the relativistic electron.

Hydrogen-like electron state color = phase

Geometric Arguments Need More Than Papers

Equations are easy to print on paper and in journals. Geometric arguments are harder to judge there, because they depend on motion, slicing, orientation, and how structures transform together. Modern rendering tools such as Three.js make it possible to show those relations directly. What this website offers is therefore not only a chain of symbolic manipulations, but a geometric reading of the argument: interactive figures meant to sharpen intuition, reveal structure, and make the proposal easier to assess.

The Central Claim

This website tests a Clifford-native second-order equation as a serious rival to the Dirac equation for the single relativistic electron.

The goal is to reopen the relativistic-electron discussion from 1925-1928 by putting another equation into the pool: one written directly in native Clifford algebra, a language that never became standard in mainstream physics and therefore remained largely absent from that discussion.

Dirac baseline

Dirac equation

\[ (i\gamma^\mu D_\mu - m)\psi = 0 \]

The standard benchmark is the gamma-matrix first-order equation together with its usual Pauli and Foldy-Wouthuysen reading.

Clifford-native rival

Second-order Clifford equation

\[ (e_i e_j)\cdot D_i D_j\,\phi = m^2\phi \]

This is a second-order equation written directly in Clifford algebra. The rest of the website builds the language needed to read it.

The claim is not that this wins automatically. The task of the website is to test whether this Clifford-native route can compete with, and in some respects improve on, the Dirac route for the single-particle relativistic electron.

Clearer Schrödinger limit. The route down to the single-electron limit becomes easier to track geometrically.
Spin can be visualized. The website treats spin and orientation as geometric structure rather than as hidden matrix data.
Pauli fits into the same operator. The spin coupling is not appended later, but appears inside the same Clifford-native closure.

How the Argument Unfolds

The website is meant to be read top to bottom, but the left navigation also lets you jump into any part of the route once the overall map is clear.

Intro. Build algebra, Clifford algebra, matrix algebra, and Clifford calculus.
Dirac baseline. Set the standard route that the later comparison will use.
Clifford Globe. Visualize spacetime, Maxwell, and Schrödinger structure geometrically.
Dot-Wedge. Build the new equation, the electron current, the Schrödinger limit, and the classical force picture.
Benchmarks. Compare the Dot-Wedge route against Dirac directly.