Once upon a time, Heron of Alexandria conceived of imaginary numbers. Gerolamo Cardano expanded upon this concept in the 16th century and conceived of complex numbers. Then, in 1843, Sir William Rowan Hamilton expanded upon that concept and created the quaternions, which were essentially complex numbers with three orthogonal imaginary components plus a real component. This was quickly followed in that same year by John T. Graves and Arthur Cayley independently concocting octonions.
The following year, Hermann Grassmann created the exterior algebra, the algebra of anti-commuting entities known as Grassman Numbers. Then along came William Kingdon Clifford, who united Grassman’s algebra with Hamilton’s work on quaternions to create Clifford Algebra.
So, what’s the big deal here? Think spinors. Think fermions.
For quite some time, I’ve been working on a rather lengthy and detailed article on fermions and the Spin-Statistics Theorem. (There is also a lengthy and fairly comprehensive article in the works on neutrino physics, yet another on the Action Principle, and still another on Noether’s Theorem, symmetry, and conservation laws. Working on all of these in parallel is pretty time-consuming.) The aforementioned mathematical constructs are rather critical to understanding these concepts and will have to be fleshed out before I can really present that topic. So watch this space, and consider the little mathematical family tree outlined above as a road-map of what is to come….
For more info, see:
- CLIFHISTORY
- Introduction to Clifford Algebra
- Clifford Algebra — from Wolfram MathWorld
- Clifford Algebras
- Clifford Algebras, Clifford Groups, and a Generalization of the …
- Geometric Algebra (Clifford Algebra)
- The Most Striking Equation in Mathematics « Galileo’s Pendulum
- Why Quaternions Matter « Galileo’s Pendulum
- Everything is Geometrical: Hermann Grassmann’s Algebra « Galileo’s Pendulum
- W. K. Clifford: The Geometry of Physics « Galileo’s Pendulum
- Two Particles Enter, No Particles Leave! « Galileo’s Pendulum
- Spinning Electron Got to Go Round « Galileo’s Pendulum
- Physics Quanta: From Identical Twins to Voltron, Physics Style « Galileo’s Pendulum
- Imaginary Numbers are Real « Galileo’s Pendulum
- The Most Striking Equation in Mathematics « Galileo’s Pendulum
