Geometric algebra

We are a group of physicists who believe that the mathematical language used for physics took a wrong turn at the end of the 19th century. Instead of vector algebra with the dot and cross product, we advocate the use of geometric algebra and the geometric product.

Geometric algebra primarily replaces and improves:

In 1878, William Kingdon Clifford published his work on what would later became known as geometric algebra, building on the work of Hamilton (quaternions), Grassmann (exterior algebra) and Gibbs (vector algebra as we know it today). The algebra envisioned by him unites all of these concepts into a simple and elegantlanguage for geometrical objects. Unfortunately, Clifford already died in 1879 at the age of 33, so his ideas had little chance to catch on. What instead stuck with physicists was the half-baked work of Gibbs. Much later, in the 1960s, Clifford's work was rediscovered by David Hestenes, who reintroduced it to mainstream physics. Unfortunately, most people still do not know about geometric algebra, and we want to change that!

Resources to dive into geometric algebra

Some highlights

Geometric algebra simplifies a lot of tedious calculations and complicated constructs in physics to almost nothing. Here are a few examples:

Bivectors

A vector is a length with an orientation. Geometric algebra generalizes this concept - for instance, a bivector is an area with an orientation. You can imagine it as an oriented plane with a magnitude, similar to how vectors are geometrically depicted as arrows with a magnitude. For instance, x ∧ z is the unit bivector lying in the xz plane.

Rotors

A bivector specifies a plane of rotation. Exponentiating the bivector yields a rotor, the GA analogue of a rotation matrix. In the geometric algebra of spacetime, there even are time-space-like bivectors, which generate Lorentz boosts when exponentiated.

"Imaginary numbers"

We've all been taught imaginary numbers in our first semester, but what are they, actually? Are they 2D vectors? Or rotations? Or spinors? There is no actual geometric picture for them. Luckily, complex numbers can be replaced by the 2D geometric algebra - then, scalars take the place of the real numbers, and the xy bivector, which squares to -1, takes the place of the imaginary unit. In addition, there is a seamless connection between 2D vectors and GA complex numbers via the geometric product.

Maxwell's equation

Yes, that's the singular. Maxwell's equation, not "equations". In the geometric algebra of spacetime, all of Maxwell's equations combined are just ∇F = J, where: The electromagnetic field bivector F is composed of the time-space-like bivectors representing the electric field, and space-space-like bivectors representing the magnetic field. (Note that this distinction is observer-dependent, as it should be! This way, we also get a nice geometric picture of how electric fields transform into Lorentz boosts, and vice versa.)

Space-time observer split

Newtonian physics treats time and space as fully separate, while Einstein's theory of relativity has a unified "spacetime" construct. Normally, we have to tediously translate between these two formalisms, but geometric algebra provides a unique tool to seamlessly link nonrelativistic with relativistic physics - the space-time observer split. Essentially, when no Lorentz boosts are involved in your calculations and your rest frame is consistent, you can reinterpret the 4D time-space-like bivectors as 3D vectors, and the 4D space-space-like bivectors as 3D bivectors. Using this, the 4D electromagnetic field bivector field F decomposes into the electric vector field and the magnetic bivector field.