this is now a geometric algebra fan blog
in 1987 (not really, it was a long time coming) mathematicians invented new numbers which have the ability to totally simplify all of physics.
All of it.
The Standard Model of Particle Physics, including color charge, falls out of the Cl(1,9) algebra. 1 temporal dimension, 3 spacial dimensions, then 6 dimensions for color charge (3x pairs of complex dimensions). the 64 (2^6) different possible orientations of a multivector in this space correspond exactly to the different particles predicted by the standard model, each multivector representing a single generation of the 16 Weyl fermions: 4 leptons (the left/right-handed electrons and neutrinos), 12 quarks (left/right-handed up & down quarks, 3 colors each). Each Weyl fermion requires 2 complex numbers to describe its quantum state: 16 weyl fermions * 2 complex spin states = 32 complex components, totalling 64 real degrees of freedom.
When you multiply two multivectors together, you take the geometric product: this gets split into the parallel bit (inner product, aka the standard dot product) and the orthogonal bit (outer or wedge product, AKA the cross product in 3 and 7 dimensions).
$$\mathbf{a}\mathbf{b} = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \wedge \mathbf{b}$$
When applying the vector derivative operator, you just left-multiply it with whatever you're deriving, and the result falls out the exact same way:
$$\nabla \mathbf{A} = \nabla \cdot \mathbf{A} + \nabla \wedge \mathbf{A}$$
The inner product of two vectors (grade 1) is a scalar (grade 0) which measures the change along the direction of the vectors - aka, the divergence! The wedge product of two g1 vectors is a bivector (grade 2), which measures the change perpendicular to the direction of the vectors - aka, the curl!
YOU GET DIVERGENCE AND CURL FOR FREE
David Hestenes is my hero. No I don't care that Tumblr doesn't have a way to embed LaTeX, shut up.
