Physics Using Geometric Algebra/Basic Geometric Algebra

Clifford described his algebras as geometric because their algebraic operations map naturally into geometric operations. In particular, they provide an elegant representation of rotations and reflections in various spaces. This introduction will explain how geometric algebras work and how they correspond to geometric operations.

The geometric algebras are a family of algebras. From a physics point of view, the most interesting geometric algebras are G₂ (2D), G₃ (Euclidean space), and the Spacetime Algebra (STA). For expository purposes, we will start with G₂ (2D). (We will not discuss the Spacetime Algebra in this section.)

Vector Spaces (2D)

An algebra can be defined as a vector space plus a product operation, where a vector space consists of vectors that can be added and scaled, where scalars are real numbers.

Figure 1: Basis vectors for a 2D space.

e₁ and e₂ are orthonormal basis vectors, which means that  |e₁| = |e₂| = 1 and e₁ and e₂ are orthogonal. (See Figure 1.)

e₁ is the unit vector of the x-axis. e₂ is the unit vector of the y-axis.

Figure 2: Addition of two vectors in 2D

We know what it means to add two vectors, just add components (see Figure 2):

  1e₁ + 3e₂ (u)

+ 2e₁ + 1e₂ (v)

————————

= 3e₁ + 4e₂ (w)

NB: [3,4] is not a vector;  3e₁ + 4e₂ is a vector.

Vector Products (G₂)

What is a vector product in a geometric algebra?

A simple approach to vector multiplication would be to cross-multiply the terms algebraically:

(ae₁ + be₂)(ce₁ + de₂) =

ace₁e₁ + ade₁e₂ + bce₂e₁ + bde₂e₂

But what are e₁e₁, e₁e₂, e₂e₁, and e₂e₂?

Figure 3: Oriented basis planes in a geometric algebra

In partial answer to this, we introduce a new basis element, e₁₂, which corresponds to the x-y plane (see Figure 3). We stipulate that e₁e₂ = e₁₂ and e₂e₁ = -e₁₂.

The product of two basis vectors is a plane.

You can think of e₁e₂ as sweeping e₁ in the direction of e₂.  The area covered is e₁₂.

The basis element has an orientation based on how you rotate when you go from the first vector to the second. If you move in a right-handed way (e.g. from e₁ to e₂ in the first image of Figure 3), then the orientation is positive. If you move in a left-handed way (e.g. from e₂ to e₁ in the second image of Figure 3) then the orientation is negative. ('Right-handed' means the direction that your right hand turns if your thumb is pointing at you and you move your hand in the direction of your fingers.)

Multiplication Table (G₂)

We now have four basis elements in G₂ if you include a scalar element: 1, e₁, e₂, and e₁₂.

The following table defines multiplication between these basis elements, where rows correspond to the left term in a product and columns correspond to the right term:

L\R	1	e1	e2	e12
1	1	e1	e2	e12
e1	e1	1	e12	e2
e2	e2	-e12	1	-e1
e12	e12	-e2	e1	-1

Note that multiplying by e₁₂ rotates basis vectors by 90 degrees:

Multiplying by e₁₂ on the left rotates left-handed (e₁ -> -e₂ -> -e₁ -> e₂).

Multiplying by e₁₂ on the right rotates right-handed (e₁ -> e₂ -> -e₁ -> -e₂).

Note also that e₁₂ squared equals -1, which corresponds to rotation by 180 degrees.

e₁₂ plays the role of i in G₂.

Vector Multiplication (G₂)

We can now say what it means to cross-multiply two vectors in G₂:

(ae₁ + be₂)(ce₁ + de₂)

= ace₁e₁ + ade₁e₂ + bce₂e₁ + bde₂e₂

= ac + ade₁₂ - bce₁₂ + bd

= (ac + bd) + (ad - bc)e₁₂.

This scalar plus a scaled basis plane is isomorphic to a complex number (x +yi), where e₁₂ plays the role of the imaginary i.

We will call any expression of the form a + be₁₂ a spinor (or rotor). (We will see in the next sections why it has this name.)

The product of two vectors is a spinor.

Note that ac + bd is the definition of the dot product (or inner product) of two vectors and ad - bc is the definition of the wedge product (or outer product) of two vectors. Thus, uv = u·v + u∧v.

In general, a multivector in G₂ has four components: a + be₁ + ce₂ + de₁₂.

We will reserve the word 'vector' for a multivector that just consists of basis vectors: be₁ + ce₂.

A spinor is a multivector that just consists of a scalar and a basis plane: a + de₁₂.

Simple Rotation (G₂)

Figure 4: Rotation of a vector by 90 degrees

When we multiply a vector by e₁₂ on the right:

(ae₁ + be₂) e₁₂ = ae₂ - be₁

we see that e₁₂ rotates each basis vector in the vector by 90 degrees:

e₁ -> e₂

e₂ -> -e₁.

This has the effect of rotating the whole vector by 90 degrees (see Figure 4). This is because a vector is just the sum of its basis vectors, so if the basis vectors are rotated, the vector is rotated.

Multiplying any vector by e₁₂ rotates the vector by 90 degrees.

General Rotation (G₂)

Figure 5: Rotation of a vector using a spinor

When we multiply a spinor times a vector we get:

(a + be₁₂) (ce₁ + de₂)

= a(ce₁ + de₂) + be₁₂(ce₁ + de₂)

= a(ce₁ + de₂) + b(de₁- ce₂).

The result has a component in the original direction of the vector (the spinor's scalar times the vector (a in Figure 5)), plus a component at right angles to the vector (the spinor's e₁₂ times the vector, which rotates the vector by 90 degrees (b in Figure 5)). This combination allows for the arbitrary rotation of a vector.

Multiplying a spinor times a vector rotates the vector.

Meaning of the Spinor (G₂)

Let u = ae₁ + be₂

and v = ce₁ + de₂.

Then

uv = (ae₁ + be₂)(ce₁ + de₂)

= (ac + bd) + (ad - bc)e₁₂

= R.

Then uR = (ae₁ + be₂) [[(ac + bd) + (ad - bc)e₁₂]]

= ae₁(ac + bd) + ae₂(ad - bc) + be₂(ac + bd) - be₁(ad - bc)

= [a²c + abd - abd + b²c]e₁ + [a²d - abc + abc + b²d]e₂

= [a²c + b²c]e₁ + [a²d + b²d]e₂

= [a² + b²]ce₁ + [a² + b²]de₂

= [a² + b²](ce₁ + de₂)

= [a² + b²]v.

So, if uv = R, then uR = |u|²v.

R = uv represents the rotation between u to v.

Another way to show this is to note that since uu = |u|² for any vector (the wedge product is zero, so only the dot product remains), then uR = u(uv) = (uu)v = |u|²v. So having R on the right rotates a vector from u to v. Similarly, since Rv = (uv)v = u(vv) = |v|²u, having R on the left rotates a vector from v to u.

Reflection (G₂)

Figure 6: The vector v is rotated to u', producing a reflection of u.

Let R = uv. We know that uR = u(uv) = (uu)v = |u|²v. Now consider vR. We know that R rotates a vector by the angle between u and v. When we rotate v away from u by the angle between u and v, then we get u', which is the reflection of u across v. (See Figure 6.)

Euclidean Space (G₃)

In Euclidean space, there are three basis elements corresponding to the three planes: e₁₂ (the x-y plane), e₂₃ (the y-z plane), and e₃₁ (the x-z plane).  There is also a basis element corresponding to the space: e₁₂₃.

Just as an arbitrary vector can be represented as the sum of scaled basis vectors, an arbitrary plane can be represented as the sum of scaled basis planes. Thus, u∧v is a sum of scaled basis vectors that represents the plane that contains both u and v in it. (u∧v is uv without the scalar element, which can be computed using (uv - vu)/2.)

The G₃ multivector has eight elements in it:

Multivector = a + be₁ + ce₂ + de₃ + fe₁₂ + ge₂₃ + he₃₁ + ke₁₂₃

The following subsets of the multivector are useful:

Vector = be₁ + ce₂ + de₃

Plane = fe₁₂ + ge₂₃ + he₃₁

Spinor = a + fe₁₂ + ge₂₃ + he₃₁

The spinor is isomorphic to Pauli spinors.

Multiplication Table (G₃)

Here is the multiplication table for G3:

	1	e1	e2	e3	e12	e23	e31	e123
1	1	e1	e2	e3	e12	e23	e31	e123
e1	e1	1	e12	-e31	e2	e123	-e3	e23
e2	e2	-e12	1	e23	-e1	e3	e123	e31
e3	e3	e31	-e23	1	e123	-e2	e1	e12
e12	e12	-e2	e1	e123	-1	-e31	e23	-e3
e23	e23	e123	-e3	e2	e31	-1	-e12	-e1
e31	e31	e3	e123	-e1	-e23	e12	-1	-e2
e123	e123	e23	e31	e12	-e3	-e1	-e2	-1

Note that the subset of this table that only includes the rows and columns for 1, e₁, e₂, and e₁₂ is identical to the multiplication table for G₂. Similarly for the subset that only includes 1, e₁, e₃, and e₃₁ and the subset that only includes 1, e₂, e₃, and e₂₃. So the geometry of 2D planes is embedded in the geometry of 3D spaces. The only elements of the table that are new are the ones that involve e₁₂₃ in some way or that involve multiplying two basis planes together.

Multiplying a basis plane times the remaining vector produces e₁₂₃:

e₁₂e₃ = e₁₂₃

e₁e₂₃ = e₁₂₃

e₂e₃₁ = e₁₂₃

e₃₁e₂ = e₁₂₃

Think of the plane sweeping out the space in the direction of the vector.

Multiplying anything by e₁₂₃ produces its dual (NB: vectors and planes are duals of each other, and scalars and spaces are duals of each other):

e₁₂e₁₂₃ = e₃

e₁e₁₂₃ = e₂₃

1e₁₂₃ = e₁₂₃

e₁₂₃e₁₂₃ = -1

This lets us define the cross product in terms of the wedge product:

u⨯v = e₁₂₃u∧v.

The cross product and the wedge product are duals of each other.

Multiplying one plane by another produces the third plane:

e₁₂e₂₃ = -e₃₁

e₁₂e₃₁ = e₂₃

e₂₃e₃₁ = -e₁₂

e₃₁e₂₃ = e₁₂

The three basis planes correspond to i, j, k in quaternions, and the G₃ spinor is isomorphic to quaternions. You can see this by comparing the subset of the table that only includes the rows and columns for 1, e₁₂, e₂₃, and e₃₁ with the following table, where e₁₂ corresponds to i, e₃₁ corresponds to j, and e₂₃ corresponds to k.

Quaternion multiplication

×	1	i	j	k
1	1	i	j	k
i	i	−1	k	−j
j	j	−k	−1	i
k	k	j	−i	−1

The whole G₃ multiplication table is based on just two rules:

e_ie_i = 1

e_ie_j = -e_je_i (i ≠ j)

Here are some examples of how this works (remember that e_ij = e_ie_j and that e_ijk = e_ie_je_k):

e₁₂e₁₂ = e₁e₂e₁e₂ = -e₁e₂e₂e₁ = -e₁e₁ = -1

e₁₂₃e₃ = e₁e₂e₃e₃ = e₁e₂ = e₁₂

e₂e₃₁ = e₂e₃e₁ = -e₂e₁e₃ = e₁e₂e₃ = e₁₂₃

Rotation (G₃)

Rotation in G₃ is a little more complicated than rotation in G₂. Rotation in G₃ works fine if the vector that is being rotated is in the plane of the rotation. But rotation in G₃ does not produce a proper vector if the vector being rotated is out of the plane of the rotation. To see this, consider rotating e₁ + e₂ + e₃ by e₁₂ on the right. Based on the multiplication table for G₃, we get:

e₂ - e₁ + e₁₂₃

The first two elements correspond to rotating e₁ + e₂ by e₁₂. The last element is what you get when you multiply e₃ by e₁₂: the volume e₁₂₃.

The solution to this problem is to rotate the result on the other side by the conjugate of the rotation (the conjugate of a rotation negates all of the non-scalar elements of the rotation):

-e₁₂(e₂ - e₁ + e₁₂₃) = -e₁ - e₂ + e₃

This rotation rotated the first two elements again in the same direction, resulting in a total rotation of 180 degrees. It also restored e₃.

In general, you rotate a vector in G₃ using a sandwich rotation:

v’ = RvR*

where R* is the conjugate of R.

Effectively, you are rotating the vector twice, with each spinor providing half of the spin.

This can be generalized to any multivector V:

V’ = RVR*

In particular, a spinor can be rotated. The result is the sum of the two rotations.