---
tag: A38B1QFK20
kind: concept
name: Clifford algebra
msc: [15A66, "11E88"]
sources:
  - url: https://en.wikipedia.org/wiki/Clifford_algebra
same_as:
  wikidata: Q674689
  wikipedia: Clifford algebra
---

## Summary

Let $V$ be a vector space with a [[quadratic form|0RBQS1JJCR]] $Q$.
The Clifford algebra of $(V, Q)$ is the associative algebra generated by $V$ in which each vector $v$ squares to the scalar $Q(v)$.
It contains the complex numbers, the quaternions and the [[exterior algebra|0MDMZ965VE]] as special cases.
It is used to build spinors and spin groups, and it gives an algebraic language for rotations and reflections in geometry and physics.

## Explanations

### L1 human

Start with vectors and allow to multiply them, with one rule: a vector times itself is the number $Q(v)$.
On the line $\R$ with $Q(x) = -x^2$, the basis vector $e$ satisfies $e^2 = -1$, so it behaves like the imaginary unit $i$.
In the plane with $Q(x, y) = -(x^2 + y^2)$, the basis vectors $e_1, e_2$ square to $-1$ and anticommute.
Then $e_1$, $e_2$ and $e_1 e_2$ behave like the quaternion units $i$, $j$ and $k$.