#Warning: This article contains super advanced content, covering essentially all the basic algebraic theory of four-dimensional Euclidean space. This article emphasizes algebra over geometry, so it may be difficult to understand
This article will introduce a new algebraic system that can “unify” high-dimensional geometry: Geometric Algebra. This algebraic framework encompasses scalars, vectors, multidimensional vectors, as well as various inner products, outer products, and mixed products. It even includes spinors (the culprit that makes electrons need to rotate twice to look the same), spatial rotations, complex numbers and quaternions, various derivative operators in vector fields, and geometric algebra can even give a new definition of determinants…
Introduction
When we discussed magnetic fields in four-dimensional space, we mentioned three multiplication operations between two 2-vectors:
| Operation | $e_{ij}*e_{ij}$ | $e_{ij}*e_{jk}$ | $e_{ij}*e_{kl}$ |
|---|---|---|---|
| Inner product$\cdot$ | 1 | 0 | 0 |
| Mixed product$\times$ | 0 | $e_{ik}$ | 0 |
| Outer product$\wedge$ | 0 | 0 | $e_{ijkl}$ |
At that time, we felt that identical letters could be merged and canceled, while different letters were simply written together to form a multidimensional vector, so that each type of n-vector corresponds exactly to one multiplication operation. Let’s define a new multiplication operation to satisfy all the above conditions simultaneously. To distinguish it from inner and outer products, the new multiplication uses no symbol:
- For vector $\boldsymbol v$, we define $\boldsymbol v^2=\boldsymbol v \boldsymbol v=||\boldsymbol v||$. This is the definition of inner product.
- Then we “forcibly” incorporate the definition of outer product: for mutually perpendicular vectors $\boldsymbol u$ and $\boldsymbol v$, we define $\boldsymbol u \boldsymbol v = -\boldsymbol v \boldsymbol u$.
- We further stipulate that this multiplication satisfies associativity and left-right distributivity for any k-vectors $\boldsymbol A$, $\boldsymbol B$: $(\boldsymbol A \boldsymbol B) \boldsymbol C = \boldsymbol A (\boldsymbol B \boldsymbol C) $, $(\boldsymbol A+ \boldsymbol B) \boldsymbol C = \boldsymbol A \boldsymbol C +\boldsymbol B \boldsymbol C $, $\boldsymbol A( \boldsymbol B+ \boldsymbol C) = \boldsymbol A \boldsymbol B +\boldsymbol A \boldsymbol C $.
This new multiplication is the geometric product. For consistency, we define scalars as 0-vectors.
