A Primer on Geometric Algebra
Geometric Algebra is — among many other things — a very simple and intuitive yet powerful way to implement rotations and boosts. The core idea is to multiply vectors together. This "geometric product" is not generally commutative, but it has several key properties:
- it is associative,
- it is distributive,
- it commutes with scalar multiplication, and
- the product of a vector with itself is just the inner (dot) product of that vector with itself.
The last property is what makes the geometric product so interesting, because it means that parallel vectors commute, while orthogonal vectors anticommute. This makes it particularly easy to work with orthogonal basis elements. More generally, because of distributivity, we can separate a vector into parts that are parallel and orthogonal to another vector:
\[𝐚𝐛 = 𝐚\left(𝐛_∥ + 𝐛_⟂\right) = \left(𝐛_∥ - 𝐛_⟂\right)𝐚.\]
This last result begins to show that reflections can be implemented by multiplication, and reflections give rise to rotations and boosts. (See the discussion of the Cartan-Dieudonné theorem for details.)
While the idea of a rotation about an axis doesn't generalize, the idea of "rotation in a plane" works in any number of dimensions. For example, in three dimensions, a rotation "about" $𝐳$ should really be thought of as a rotation in the $𝐱$-$𝐲$ plane. The generator of this rotation is simply $𝐱𝐲$ — the pseudoscalar for that plane — times half the angle of rotation. We simply exponentiate this to obtain the full rotation operator, where the exponential can be defined by the usual power series expansion:
\[R = \exp \left[\frac{θ}{2}𝐱𝐲\right] = \cos \frac{θ}{2} + 𝐱𝐲 \sin \frac{θ}{2}.\]
Note the resemblance to Euler's formula for complex numbers, which is not a coincidence — the derivation relies only on the fact that
\[(𝐱𝐲)² = 𝐱(𝐲𝐱)𝐲 = -𝐱(𝐱𝐲)𝐲 = -(𝐱𝐱)(𝐲𝐲) = -1.\]
This $R$ rotates a vector $𝐯$ as
\[𝐯' = R𝐯R̃,\]
where the tilde denotes the reverse operation — which swaps the order of the vectors in any product. In the case of $R$, this just flips the sign of the $𝐱𝐲$ term. Note that $RR̃=1$. Any vector orthogonal to the $𝐱$-$𝐲$ plane anticommutes with both $𝐱$ and $𝐲$, and therefore commutes with $𝐱𝐲$, and is therefore unaffected by the rotation. On the other hand, it is easy to verify that the components of $𝐯$ in the $𝐱$-$𝐲$ plane are rotated by the angle $θ$.
Importantly, this result holds in any number of dimensions, but it does depend on the signature of $𝐱$ and $𝐲$. If we choose, for example, the $𝐭$ and $𝐳$ basis vectors of Minkowski, then $(𝐭𝐳)² = 1$ so the trigonometric functions in the expression for $R$ become hyperbolic:
\[R = \exp \left[\frac{φ}{2}𝐭𝐳\right] = \cosh \frac{φ}{2} + 𝐭𝐳 \sinh \frac{φ}{2}.\]
This "rotation" in the $𝐭$-$𝐳$ plane is just a boost in the $𝐳$ direction with rapidity $φ$. The final case to consider is when the square of the generator is zero. This happens when we multiply a null vector $𝐧$ by another vector $𝐯$ that is orthogonal to it, so that
\[(𝐧𝐯)² = 𝐧(𝐯𝐧)𝐯 = -𝐧(𝐧𝐯)𝐯 = -(𝐧𝐧)(𝐯𝐯) = 0.\]
In this case, the exponential truncates after the linear term:
\[R = \exp \left[\frac{1}{2}𝐧𝐯\right] = 1 + \frac{1}{2}𝐧𝐯.\]
This is called a "null rotation", because it does not affect the null vector $𝐧$, though it does affect other directions. All of these $R$ objects are called rotors, even when they do not represent spatial rotations. Because they act on vectors by conjugation, we can express the composition of transformations simply by multiplying their rotors.
Another very important type of element is the product of all vectors in an orthonormal basis, called the pseudoscalar. In two dimensions, using the usual basis $(𝐱, 𝐲)$, then the product is $𝐈₂ = 𝐱𝐲$. In three dimensions, we include $𝐳$ and get $𝐈₃ = 𝐱𝐲𝐳$. In Minkowski, we include $𝐭$ and get $𝐈₄ = 𝐭𝐱𝐲𝐳$. These all happen to square to $-1$, making them complex structures for their respective spaces. And they are essentially the volume forms of their spaces, and can provide the Hodge dual by simple multiplication.
Geometric Algebra (GA) is a powerful mathematical framework that unifies various algebraic systems, including complex numbers, quaternions, vector calculus, and differential forms. It provides a geometric interpretation of algebraic operations, making it particularly useful in physics, engineering, and computer graphics. Nonetheless, it is simple enough to be accessible to anyone familiar with basic algebra. GA is fundamentally identical to Clifford Algebra; the different names reflect different emphases and traditions in the literature. GA is typically developed over $ℝ$, rather than $ℂ$ — the complex structures we usually encounter in physics appearing naturally within the algebra, rather than being introduced ad hoc. GA also emphasizes the geometric interpretation of the algebraic structures, over what is usually an algebraic emphasis in literature using the name Clifford Algebra.
Before getting into the details of null tetrads and Lorentz transformations, we need to review some of the basics of Geometric Algebra. This is not meant to be a comprehensive introduction to GA, but rather a quick primer on the key concepts and operations that we will need for our purposes. For a more comprehensive introduction, see [7].
The geometric product
Geometric Algebra starts with a real vector space $𝕍$, equipped with an inner product taking a pair of vectors $𝐯, 𝐰 ∈ 𝕍$ to $𝐯⋅𝐰 = 𝐰⋅𝐯 ∈ ℝ$. We then introduce a product called the geometric product, which is associative and distributive, but not necessarily commutative. We express the geometric product between vectors $𝐯$ and $𝐰$ simply as juxtaposition: $𝐯𝐰$. The geometric product is essentially the tensor product, subject to the identification that $𝐯𝐯 = 𝐯⋅𝐯$. We also have compatibility with scalar multiplication, so that we have
\[s(𝐯𝐰) = (s𝐯)𝐰 = 𝐯(s𝐰) = (𝐯𝐰)s \qquad \text{for any }s ∈ ℝ.\]
These rules are enough to extend the algebra to arbitrary dimensions, using inner products of arbitrary (even degenerate) signature.
An example is very helpful in clarifying. Consider the vector space $ℝ²$, with the standard basis vectors $𝐱$ and $𝐲$. Consider the sum $𝐱+𝐲$. The product of this vector with itself is identified with the inner product
\[(𝐱+𝐲)(𝐱+𝐲) = (𝐱+𝐲)⋅(𝐱+𝐲) = 2.\]
On the other hand, we can use the distributive property to derive
\[\begin{aligned} (𝐱+𝐲)(𝐱+𝐲) &= 𝐱𝐱+𝐱𝐲+𝐲𝐱+𝐲𝐲 \\ &= (𝐱𝐱+𝐲𝐲) + (𝐱𝐲+𝐲𝐱) \\ &= (𝐱⋅𝐱+𝐲⋅𝐲) + (𝐱𝐲+𝐲𝐱) \\ &= 2 + (𝐱𝐲+𝐲𝐱). \end{aligned}\]
Comparing the two expressions, we see that the last term, $𝐱𝐲+𝐲𝐱$ must vanish:
\[𝐱𝐲 = -𝐲𝐱.\]
That is, these orthogonal vectors anticommute under the geometric product. This has an important consequence:
\[(𝐱𝐲)(𝐱𝐲) = 𝐱(𝐲𝐱)𝐲 = -𝐱(𝐱𝐲)𝐲 = -(𝐱𝐱)(𝐲𝐲) = -1.\]
That is, $(𝐱𝐲)² = -1$; the product $𝐱𝐲$ is the unit imaginary associated to the $𝐱$-$𝐲$ plane.
Obviously, parallel vectors commute, since they can be expressed as scalar multiples of each other and scalars commute with all vectors. These are the two critical features of the geometric product: parallel vectors commute, while orthogonal vectors anticommute. Combined with associativity and distributivity, these properties allow us to calculate quite general geometric products in arbitrary dimensions.
Specifically, we can decompose the geometric product of two vectors into symmetric and antisymmetric parts:[1]
\[\begin{aligned} 𝐯𝐰 &= \frac{1}{2}(𝐯𝐰 + 𝐰𝐯) + \frac{1}{2}(𝐯𝐰 - 𝐰𝐯) \\ &= (𝐯⋅𝐰) + (𝐯 ∧ 𝐰), \end{aligned}\]
where $𝐯 ∧ 𝐰$ is called the exterior or wedge product, producing a bivector. When $𝐯²≥0$ and $𝐰²≥0$, this result is a general complex number associated with the plane spanned by $𝐯$ and $𝐰$, with $𝐯⋅𝐰$ being the real part and $𝐯 ∧ 𝐰$ being the imaginary part which squares to a negative number. Note that the wedge product corresponds to the usual cross product in three dimensions, but generalizes to arbitrary dimensions and signatures.
The geometric product of two vectors is precisely a complex number, with the real part being the inner product of the vectors, and the imaginary part being a bivector representing the plane spanned by those vectors. This is a generalization of the fact that the product of two orthogonal vectors is a bivector that squares to -1, and thus can be identified with the unit imaginary.
Reflections and rotations
One of the reasons Geometric Algebra is so powerful is that it provides a very natural way to represent reflections — which, in turn, give rise to orthogonal and conformal transformations in all dimensions and (nondegenerate) signatures.
First, note the fact that the geometric product of two vectors tracks both the "dot product" and "cross product" means that we can often find inverses of vectors, which simplifies many calculations. Specifically, if the norm of a vector $𝐧$ is nonzero, then we can define its inverse as
\[𝐧⁻¹ = \frac{𝐧}{𝐧²},\]
where the denominator is just a scalar. Obviously, we then have $𝐧 𝐧⁻¹ = 𝐧 𝐧 / 𝐧² = 1$.
Now, choose any invertible vector $𝐧$. Any other vector $𝐯$ decomposes into a part that commutes with $𝐧$ and a part that anticommutes with $𝐧$ — which we denote as $𝐯_∥$ and $𝐯_⟂$, respectively. Given the properties of the geometric product shown above, we have
\[-𝐧 𝐯 𝐧⁻¹ = -𝐧 𝐯_∥ 𝐧⁻¹ - 𝐧 𝐯_⟂ 𝐧⁻¹ = -𝐧𝐧⁻¹ 𝐯_∥ + 𝐧 𝐧⁻¹ 𝐯_⟂ = -𝐯_∥ + 𝐯_⟂.\]
That is, this negative conjugation by $𝐧$ reflects the vector $𝐯$ along the line defined by $𝐧$; reflections are represented as simple conjugations in the algebra. We can compose reflections, just by applying this transformation repeatedly, which is equivalent to negative conjugation by the product of the vectors defining the reflections. For any such $𝐧$, the negative conjugation results in a reflection, and any reflection can be represented in this way for some choice of $𝐧$. But note that the choice of $𝐧$ is not unique; $-𝐧$ will achieve exactly the same reflection. Thus, the multiplicative group of unit vectors is a double cover of the group of reflections.
This may seem like a trivial curiosity, but it has profound implications because of the Cartan-Dieudonné theorem [8]:
If $T$ is an isometry of a regular quadratic space $(E, Q)$, $T$ is the product of at most $\mathrm{dim}\, E$ simple reflections.
For our purposes, the isometries are just the orthogonal group, and a "regular quadratic space" is just a vector space with a nondegenerate inner product (corresponding to the quadratic form $Q$ above). We are only interested in real vector spaces, so our vector space is essentially just $ℝ^{p,q}$, where $p$ is the number of positive terms in the signature, and $q$ is the number of negative terms. (E.g., we will take Minkowski space as $ℝ^{3,1}$.) Thus, we might rephrase the theorem more simply as
Any orthogonal transformation of $ℝ^{p,q}$ can be expressed as the product of at most $p+q$ simple reflections.
There is a corollary that is also important for our purposes:
Any special orthogonal transformation of $ℝ^{p,q}$ can be expressed as the product of an even number of at most $p+q$ simple reflections.
In fact, the group of special orthogonal transformations is a double cover of the group of even products of unit vectors, which is called the Spin group. This is the key to understanding how spinors arise in physics, and how they are related to
Higher-dimensional products
[#TODO: convert the "details" into a proper section.]
Details on the geometric product
This is part of a broader pattern: the product of two orthogonal vectors — or just the antisymmetric part of the geometric product of any two vectors — is a bivector representing the plane spanned by those vectors, carrying information about the attitude of that plane, orientation of the bivector, and magnitude. That extends to higher dimensions, as well, with the product of three orthogonal vectors being a trivector representing the volume spanned by those vectors, and so on. In particular, the product of $d$ orthogonal vectors in a $d$-dimensional space is a pseudoscalar representing the oriented volume of the entire space, and squaring to either +1 or -1 depending on the signature of the inner product. The pseudoscalar is often denoted as $𝐈$, and it plays a critical role in the algebra, as we will see below.
These products have interesting properties. There is — by definition — just one scalar. And linear dependence shows that the wedge product of more than $d$ vectors must vanish. In between, the space of products of $k$ independent vectors has dimension $\binom{d}{k}$, and the entire algebra has dimension
\[\sum_{k=0}^d \binom{d}{k} = 2^d.\]
Note that there is just one pseudoscalar, which is the product of all the basis vectors. In $d=2$, there is just one bivector, which is the pseudoscalar, which we've seen is the complex unit imaginary. The fact that complex numbers are a linear combination of the scalar and the pseudoscalar for two spatial dimensions, and thus itself has two dimensions, is pure coincidence. Hamilton was misled into believing that there must be a similar structure for $d=3$; here the coincidence is that the number of vectors $\binom{3}{1} = 3$ happens to equal the number of bivectors $\binom{3}{2} = 3$. These seemingly magical coincidences led to confusion that has only recently been resolved by the development of Geometric Algebra.
Structure of the algebra
[Show diagrams of $Cl(2)$, $Cl(3)$, and $Cl(3,1)$ in terms of their bases, noting the numbers as binomial coefficients.]
- 1It is remarkable that this formula actually has a scalar being added to the wedge product of two vectors — which is a rank-2 tensor. In Physics, we are frequently taught that scalars must never be added to vectors — never mind tensors! This is typically good for helping students catch elementary mistakes, but not actually necessary. Mathematicians routinely define the tensor space to allow for adding arbitrary ranks together.