Physics Notes: Clifford Algebras

A Clifford-algebra combines and generalizes the scalar product and the vector product. When you multiply two elements in a Clifford-algebra, the result can be decomposed into a symmetric scalar product and an antisymmetric vector product as follows:

ab = a·b + a^b
a·b = (ab + ba)/2
a^b = (ab – ba)/2

In one dimension, the Clifford-algebra reduces trivially to the algebra of real numbers, and the vector product is identically zero.

In two dimensions, things start to get interesting. One way to generate a Clifford-algebra is through a slightly unconventional application of complex numbers: instead of ordinary multiplication, we may use multiplication on the left by the complex conjugate of the first multiplicand. The result can be decomposed as follows:

a = a₁ + a₂i
b = b₁ + b₂i
ab = (a₁ – a₂i)(b₁ + b₂i) = (a₁b₁ + a₂b₂) + (a₁b₂ – a₂b₁)i

The scalar and vector products, respectively, are (a₁b₁ + a₂b₂) and (a₁b₂ – a₂b₁)i.

There is, however, a more conventional way to represent the two-dimensional Clifford-algebra, a way that makes more structure evident. Instead of starting with complex numbers, we start with two unit vectors, e₁ and e₂ (a = a₁e₁ + a₂e₂, b = b₁e₁ + b₂e₂). The following multiplication rules apply:

e₁e₁ = e₂e₂ = 1
e₁e₂ = –e₂e₁ = e₁^e₂

From this we can compute the Clifford product:

ab = (a₁a₁ + a₂a₂) + (a₁b₂ – a₂b₁)(e₁e₂)

Now is a good time to notice an important coincidence:

(e₁e₂)(e₁e₂) = –(e₁e₂)(e₂e₁) = –e₁(e₂e₂)e₁ = –e₁e₁ = –1

Here we made use without proof or closer scrutiny of the fact that Clifford multiplication is associative. The result is that (e₁e₂) is can be identified with the imaginary unit i, in which case the product of two 2-dimensional "pure" Clifford numbers (no scalar component) is an ordinary complex number.

Generally, an n-dimensional Clifford-algebra can be represented using n unit vectors e₁...e_n, the scalar unit 1, and the pseudoscalar e₁e₂...e_n. In some cases, the pseudoscalar will be equal to 1; when it isn't, the algebra is called a universal Clifford-algebra.

In three dimensions, some real fun begins. To produce a three-dimensional Clifford-algebra, we need three basis units, e₁, e₂ and e₃. The multiplication rules are derived from anticommutativity:

e₁e₁ = e₂e₂ = e₃e₃ = 1
e₁e₂ = –e₂e₁, and similarly e₂e₃ = –e₃e₂ and e₃e₁ = –e₁e₃

From this, the multiplication rule follows:

ab = (a₁a₁ + a₂a₂ + a₃a₃) + (a₁b₂ – a₂b₁)e₁e₂ + (a₂b₃ – a₃b₂)e₂e₃ + (a₃b₁ – a₁b₃)e₃e₁

What we recovered here is the dot product and cross product of ordinary three-dimensional geometry. But notice something: in the cross product part, instead of the base vectors e₁, e₂ and e₃, we have the product terms e₁e₂, e₂e₃ and e₃e₁. The meaning of this is very important. Forget what you were taught in high-school: the cross product is not a vector. It is a bivector. It so happens that in three dimensions (and only in three dimensions) a bivector is also three dimensional, so we have a convenient way to identify bivectors with ordinary vectors, but unfortunately when we do so, we hide some real geometric meaning.

Just as a vector represents the direction and the length of a line segment, a bivector represents the orientation and area of a disk. And unless the 3-dimensional cross product concept, the concept of bivectors naturally generalizes to higher dimensions.

Anyway, back to Clifford algebras. They're generally represented by two numbers: The Clifford algebra Cl(p, q) has p + q base vectors, the square of which will be +1 in p cases, and –1 in the remaining q cases. Clifford algebras can always be represented by matrices, but in some cases, simpler representations are possible: for instance, Cl(0, 1) can be represented by the complex numbers, and Cl(0, 2) by quaternions.

So what makes a Clifford-algebra so special? Take a look at what multiplication by the pseudoscalar does to a unit vector in the two-dimensional case:

(e1e2)e1 = –(e2e1)e1 = –e2(e1e1) = –e2		e1(e1e2) = (e1e1)e2 = e2
(e1e2)e2 = e1(e2e2) = e1		e2(e1e2) = –e2(e2e1) = –(e2e2)e1 = –e1

This, curiously, has a geometric interpretation. If we take e₁ and e₂ to be the unit vectors in the plane pointing in the x and y direction, multiplying on the left rotated e₂ from the y-direction to the x-direction, and e₁ from the x-direction to the –y-direction, i.e., performed a clockwise rotation by 90º. Similarly, multiplying on the right is equivalent to a counterclockwise rotation by 90º.

A rotation of a vector a by an arbitrary angle φ can be expressed as

a' = RaR^*

where R = cos φ/2 + e₁e₂sin φ/2, and R^* = cos φ/2 – e₁e₂sin φ/2. This is very similar to the way rotations in 3D-space were associated with elements of the group SU(2), with one crucial difference: this technique can be generalized to arbitrary dimensions, and doesn't depend on the unique correspondence between SU(2) and SO(3).

Wow.